SL3-cup-product-cohomology-sage.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 54,
   "id": "215d7d99",
   "metadata": {},
   "outputs": [],
   "source": [
    "import itertools\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "852193df",
   "metadata": {},
   "source": [
    "# Functions"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2106d3d5",
   "metadata": {},
   "source": [
    "## Functions for finding the basis of $\\operatorname{gr}^{j} \\bigwedge^m \\mathfrak{g}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "id": "c41c86d0",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Example of initialization for I in SL_3(Z_p):\n",
    "\n",
    "# e_sl3 = np.array([1,2,3,4,5,6,7,8], dtype=np.int32)\n",
    "# e_sort_3 = np.array([0,3,4,6,7,1,2,5], dtype=np.int32)\n",
    "# # 1-->0, 2-->3, 3-->4, 4-->6, 5-->7, 6-->1, 7-->2, 8-->5\n",
    "\n",
    "\n",
    "# NOTE: Throughout, we use a tuple to represent a wedge product.\n",
    "\n",
    "\n",
    "\n",
    "def non_zero_wedge(tup, g):\n",
    "    \"\"\"\n",
    "    Return boolean describing whether tup is a non-zero wedge.\n",
    "\n",
    "    Input:\n",
    "    tup = a tuple of integers,\n",
    "    g = np.array([g1,g2,...]) with graded parts of the Lie algebra g\n",
    "    where g1,g2,... are numbers 1,2,... corresponding\n",
    "    to a basis e_1,e_2,... of g\n",
    "\n",
    "    Output:\n",
    "    Boolean\n",
    "\n",
    "    Description:\n",
    "    True if tup contains more k's than dim(g_k)\n",
    "    for any k from range(1, len(g)+1),\n",
    "    where g_k = g[k-1],\n",
    "    which implies that the wedge product is zero.\n",
    "    False otherwise.\n",
    "    \"\"\"\n",
    "    return all(tup.count(k) <= len(g[k - 1]) for k in range(1, len(g) + 1))\n",
    "\n",
    "\n",
    "def grj_nwedge_g(j, n, g):\n",
    "    \"\"\"\n",
    "    Return gradings for grade j of n wedges of g.\n",
    "\n",
    "    Input:\n",
    "    j = grading (≥1),\n",
    "    n = number of wedges (≥1),\n",
    "    g = np.array([g1,g2,...]) with graded parts of the Lie algebra g\n",
    "    where g1,g2,... are lists of numbers 1,2,... corresponding\n",
    "    to a basis e_1,e_2,... of g\n",
    "\n",
    "    Output:\n",
    "    Returns an np.array of lists of which n wedges gives grading j,\n",
    "    each inner list correspond to a direct summand.\n",
    "\n",
    "    Description:\n",
    "    Goes through all ordered combinations (with repeats)\n",
    "    of n numbers from 1 to len(g),\n",
    "    and checks whether they give a non-zero wedge,\n",
    "    and whether they have grading j alltogether.\n",
    "    Also, ignores trivial cases.\n",
    "    \"\"\"\n",
    "    if n <= 0 or j < n:\n",
    "        return []\n",
    "\n",
    "    base_gradings = [\n",
    "        i\n",
    "        for i in itertools.combinations_with_replacement(range(1, len(g) + 1), n)\n",
    "        if non_zero_wedge(i, g) and sum(i) == j\n",
    "    ]\n",
    "    return base_gradings\n",
    "\n",
    "\n",
    "def print_grj_nwedge_g(jmax, nmax, g):\n",
    "    \"\"\"\n",
    "    Print all gradings of all wedges of g.\n",
    "\n",
    "    Input:\n",
    "    jmax = max j to try when finding bases, = max grade,\n",
    "    nmax = max n to try when finding bases, = dim(g),\n",
    "    g = np.array([g1,g2,...]) with graded parts of the Lie algebra g\n",
    "    where g1,g2,... are lists of numbers 1,2,... corresponding\n",
    "    to a basis e_1,e_2,... of g\n",
    "\n",
    "    Output:\n",
    "    No output, just prints gradings.\n",
    "\n",
    "    Description:\n",
    "    Goes through j from 0 to jmax and n from 0 to nmax,\n",
    "    and check if grade j of n wedges of g is non-zero,\n",
    "    and if so prints it.\n",
    "    \"\"\"\n",
    "    print(\n",
    "        \"\"\"\n",
    "        ===================================================\n",
    "        gr^j of n wedges of g in terms of g^1, g^2 and g^3:\n",
    "        =================================================== \\n\n",
    "        \"\"\"\n",
    "    )\n",
    "    print(\n",
    "        \"\"\"\n",
    "        Note:\n",
    "        - This prints [(1)] as [(1,)].\n",
    "        - A tuple corresponds to a wedge product,\n",
    "          e.g., (1,1,2) mean g^1 wedge g^1 wedge g^2.\n",
    "        - Tuples in the list correcsponds to summands,\n",
    "          e.g., [(1,3),(2,2)] means\n",
    "          g^1 wedge g^3 direct sum g^2 wedge g^2.\n",
    "        - This doesn't work for 0 wedges of g nor for\n",
    "          dim(g) wedges of g. \\n\n",
    "        \"\"\"\n",
    "    )\n",
    "    for j in range(jmax + 1):\n",
    "        print(\"------------\")\n",
    "        print(f\"Grade {j}: \\n\")\n",
    "        for n in range(nmax):\n",
    "            grj_nwedge_g_tmp = grj_nwedge_g(j, n, g)\n",
    "            if len(grj_nwedge_g_tmp) != 0:\n",
    "                print(grj_nwedge_g_tmp)\n",
    "\n",
    "\n",
    "def flatten_basis(basis):\n",
    "    \"\"\"\n",
    "    Flattens a list of lists.\n",
    "\n",
    "    Input:\n",
    "    A basis in the form of\n",
    "    np.array([[basis-elements-1],[basis-elements-2],...]),\n",
    "    where each inner list corresponds to a direct summand.\n",
    "\n",
    "    Output:\n",
    "    A flat basis in the form\n",
    "    np.array([basis-elements-1,basis-elements-2,...])\n",
    "\n",
    "    Description:\n",
    "    Flattens bases of direct sums to a list of all basis elements\n",
    "    without separate lists for the direct summands.\n",
    "    \"\"\"\n",
    "    return [item for direct_summand in basis for item in direct_summand]\n",
    "\n",
    "\n",
    "def grj_nwedge_g_basis(j, n, g):\n",
    "    \"\"\"\n",
    "    Return basis for grade j of n wedges of g.\n",
    "\n",
    "    Input:\n",
    "    j = grading (≥1),\n",
    "    n = number of wedges (≥1),\n",
    "    g = np.array([g1,g2,...]) with graded parts of the Lie algebra g\n",
    "    where g1,g2,... are lists of numbers 1,2,... corresponding\n",
    "    to a basis e_1,e_2,... of g\n",
    "\n",
    "    Output:\n",
    "    Returns an np.array of lists of basis elements for\n",
    "    n wedges of g with grading j.\n",
    "    Each inner list correspond to a direct summand.\n",
    "\n",
    "    Description:\n",
    "    Goes through all gradings (grade j with n wedges),\n",
    "    counts the number of g_i for each i,\n",
    "    finds basis for wedges of g_i for each i,\n",
    "    puts the above together to a basis of\n",
    "    grade j of n wedges of g.\n",
    "    Finally, prettify the basis.\n",
    "    \"\"\"\n",
    "    base_gradings = grj_nwedge_g(j, n, g)\n",
    "    basis = []\n",
    "    for grading in base_gradings:\n",
    "        grade_count = [grading.count(i) for i in range(1, len(g) + 1)]\n",
    "        basis_grading_indices = [\n",
    "            tuple(itertools.combinations(range(len(g[i])), grade_count[i]))\n",
    "            for i in range(len(g))\n",
    "        ]\n",
    "        grading_basis_ugly = itertools.product(*basis_grading_indices)\n",
    "        grading_basis_pretty = []\n",
    "        for base in grading_basis_ugly:\n",
    "            pretty_base_element = []\n",
    "            for k in range(len(g)):\n",
    "                for index in base[k]:\n",
    "                    pretty_base_element.append(g[k][index])\n",
    "            grading_basis_pretty.append(tuple(pretty_base_element))\n",
    "        basis.append(grading_basis_pretty)\n",
    "    return np.array(flatten_basis(basis), dtype=np.int32)\n",
    "\n",
    "\n",
    "def store_bases(jmax, nmax, g):\n",
    "    \"\"\"\n",
    "    Store the all bases in an np.array.\n",
    "\n",
    "    Input:\n",
    "    jmax = max j to try when finding bases, = max grade,\n",
    "    nmax = max n to try when finding bases, = dim(g),\n",
    "    g = np.array([g1,g2,...]) with graded parts of the Lie algebra g\n",
    "    where g1,g2,... are lists of numbers 1,2,... corresponding\n",
    "    to a basis e_1,e_2,... of g\n",
    "\n",
    "    Output: \n",
    "    An np.array with a basis for grade j of n wedges of g\n",
    "    in entry array[j][n].\n",
    "\n",
    "    Description:\n",
    "    Goes through j from 0 to jmax+2 and n from 0 to nmax+2,\n",
    "    and saves the bases of grade j of n wedges of g. We go\n",
    "    +2 because we want bases_array[jmax+1][nmax+1] to work\n",
    "    for later use.\n",
    "    \"\"\"\n",
    "    bases_array = np.empty((jmax+2, nmax+2), dtype=object)\n",
    "    for j in range(jmax+2):\n",
    "        for n in range(nmax+2):\n",
    "            bases_array[j][n] = grj_nwedge_g_basis(j, n, g)\n",
    "    return bases_array\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fd0ac335",
   "metadata": {},
   "source": [
    "## Functions for finding matrices of differential maps in the (co-)chain complex"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5052fe47",
   "metadata": {},
   "source": [
    "Note that when going from the chain complex map $d$ described in a basis to the co-chain complex map in the dual basis, we are doing a matrix transpose (so it becomes $d^{\\top}$), but we don't do this in the following, since we are actually finding the transpose matrix directly."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "id": "eb9af93e",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Make sure to implement commutator(a,b) in each example.\n",
    "# E.g. for I in SL_3(Z_p):\n",
    "# def commutator(a, b):\n",
    "# sign = 1\n",
    "# if a > b:\n",
    "#     a_tmp = a\n",
    "#     a = b\n",
    "#     b = a_tmp\n",
    "#     sign = -1\n",
    "# # to make it easier later, we return lists with the answers, where a x+y = [x,y]\n",
    "# if a==1 and b==6:\n",
    "#     return [(-sign,2)]\n",
    "# elif a==1 and b==7:\n",
    "#     return [(sign,3)]\n",
    "# elif a==1 and b==8:\n",
    "#     return [(-sign,5),(-sign,4)] # for the sake of later code, we order these in reverse order\n",
    "# elif a==2 and b==7:\n",
    "#     return [(-sign,4)]\n",
    "# elif a==3 and b==6:\n",
    "#     return [(-sign,5)]\n",
    "# elif a==6 and b==7:\n",
    "#     return [(-sign,8)]\n",
    "# else:\n",
    "#     return [(0,0)]\n",
    "\n",
    "# Here we use a list to represent a sum, which will work with our implementations.\n",
    "\n",
    "\n",
    "def fast_base_sort(g, eis, e_sort):\n",
    "    \"\"\"\n",
    "    Sort our basis to the order we prefer.\n",
    "\n",
    "    Input:\n",
    "    g = np.array([g1,g2,...]) with graded parts of the Lie algebra g\n",
    "    where g1,g2,... are lists of numbers 1,2,... corresponding\n",
    "    to a basis e_1,e_2,... of g,\n",
    "    eis = a tuple (wedge product) of e_i's (basis elements) out of order\n",
    "    with the first entry being the coefficient,\n",
    "    e_sort = an np.array describing our ordering of the e_i's\n",
    "\n",
    "    Output:\n",
    "    A tuple of the e_i's sorted.\n",
    "\n",
    "    Description:\n",
    "    We use that, when constructing the e_i's in the following, \n",
    "    we can only really mess up the order of one element, \n",
    "    so we just linearly check the order of the elements, \n",
    "    and compare to e_sort, remembering to change signs, \n",
    "    when moving elements past wedges.\n",
    "    \n",
    "    Recall that a tuple corresponds to a wedge product.\n",
    "    \"\"\"\n",
    "    eis = list(eis)\n",
    "    sign = 1\n",
    "    # Don't look at eis[0], since that is just the coef\n",
    "    for j in range(1, len(eis) - 1):\n",
    "        if e_sort[eis[j] - 1] > e_sort[eis[j + 1] - 1]:\n",
    "            eis[j], eis[j + 1] = eis[j + 1], eis[j]\n",
    "            # Equivalent to:\n",
    "            # tmp = eis[j]\n",
    "            # eis[j] = eis[j+1]\n",
    "            # eis[j+1] = tmp\n",
    "            sign *= -1\n",
    "    # Put coefficient on first entry\n",
    "    eis[0] = sign * eis[0]\n",
    "    return tuple(eis)\n",
    "\n",
    "\n",
    "def d(g, eis, e_sort, commutator):\n",
    "    \"\"\"\n",
    "    Calculate the d of eis in the (co-)chain complex.\n",
    "\n",
    "    Input:\n",
    "    g = np.array([g1,g2,...]) with graded parts of the Lie algebra g\n",
    "    where g1,g2,... are lists of numbers 1,2,... corresponding\n",
    "    to a basis e_1,e_2,... of g,\n",
    "    eis = a tuple (wedge product) of e_i's (basis elements)\n",
    "    with the first entry being the coefficient,\n",
    "    e_sort = an np.array describing our ordering of the e_i's\n",
    "    commutator = a function (int,int) -> [(int (coefficient),int (results))],\n",
    "    where different elements of the list corespond to summands.\n",
    "\n",
    "    Output:\n",
    "    A list of tuples of the e_i's sorted.\n",
    "    Elements of the list correspond to summands.\n",
    "    We put the coefficient on the first element of each tuple.\n",
    "\n",
    "    Description:\n",
    "    Calculate d(eis) for the wedge product eis of e_i's,\n",
    "    while making sure to place the coefficient in the first entry,\n",
    "    and keeping the e_i's in our desired order.\n",
    "    \n",
    "    NOTE: Remember to implement commutator(a, b) first.\n",
    "    \"\"\"\n",
    "    d_eis = []\n",
    "    eis = np.array(eis)\n",
    "    d_eis_elem = np.zeros(len(eis) - 1, dtype=np.int32)\n",
    "    for j in range(2, len(eis)):\n",
    "        for i in range(1, j):\n",
    "            com = commutator(eis[i], eis[j])\n",
    "            # Use that we put com[0][0] = 0, if the commutator is 0.\n",
    "            if com[0][0] != 0:\n",
    "                sign = (-1) ** (i + j)\n",
    "                d_eis_elem_tmp = [\n",
    "                    eis[k] for k in range(1, len(eis)) if k != i and k != j\n",
    "                ]\n",
    "                # Check if wedge product should be zero, because of repeat e_i\n",
    "                for k in range(len(com)):\n",
    "                    if not com[k][1] in d_eis_elem_tmp:\n",
    "                        d_eis_elem[0] = sign * com[k][0] * eis[0]\n",
    "                        d_eis_elem[1] = com[k][1]\n",
    "                        d_eis_elem[2:] = d_eis_elem_tmp\n",
    "                        d_eis.append(fast_base_sort(g, d_eis_elem, e_sort))\n",
    "    return d_eis\n",
    "\n",
    "\n",
    "def add_one_coef(arr):\n",
    "    \"\"\"\n",
    "    Add one (first) entry to arr with 1.\n",
    "\n",
    "    Input:\n",
    "    An np.array of size n.\n",
    "\n",
    "    Output:\n",
    "    An np.array of size n+1 with 1 in the first entry.\n",
    "\n",
    "    Description:\n",
    "    Construct a new array of size n+1 with all 1's.\n",
    "    Change the last n entries to equal arr.\n",
    "    \"\"\"\n",
    "    arr_with_one = np.ones(len(arr) + 1, dtype=np.int32)\n",
    "    arr_with_one[1:] = arr\n",
    "    return arr_with_one\n",
    "\n",
    "\n",
    "def d_coefs(g, eis, e_sort, codomain_basis, commutator):\n",
    "    \"\"\"\n",
    "    Calculate the coefficients describing the map d in the chain complex.\n",
    "\n",
    "    Input:\n",
    "    g = np.array([g1,g2,...]) with graded parts of the Lie algebra g\n",
    "    where g1,g2,... are lists of numbers 1,2,... corresponding\n",
    "    to a basis e_1,e_2,... of g,\n",
    "    eis = a tuple (wedge product) of e_i's (basis elements)\n",
    "    with the first entry being the coefficient,\n",
    "    e_sort = an np.array describing our ordering of the e_i's,\n",
    "    codomain_basis = basis of codomain of d.\n",
    "    commutator = a function (int,int) -> [(int (coefficient),int (results))],\n",
    "    where different elements of the list corespond to summands.\n",
    "\n",
    "    Output:\n",
    "    An np.array of coefficients for d(eis) in the basis of\n",
    "    the codomain given by basis in the input.\n",
    "\n",
    "    Description:\n",
    "    Return np.array([0]) if d(eis) = 0 (trivially).\n",
    "    Otherwise, calculate d(eis), and find the coefficients\n",
    "    of d(eis) in codomain_basis.\n",
    "    \"\"\"\n",
    "    # len(eis) <= 2 instead of 1, since the first entry is just the coefficient\n",
    "    if len(codomain_basis) == 0 or len(eis) <= 2:\n",
    "        return np.array([0], dtype=np.int32)\n",
    "    d_eis = d(g, eis, e_sort, commutator)\n",
    "    coefs = np.zeros(len(codomain_basis), dtype=np.int32)\n",
    "    for d_ei in d_eis:\n",
    "        find_in_basis = np.all(codomain_basis == np.array(d_ei[1:]), axis=1)\n",
    "        coef_index = np.where(find_in_basis)\n",
    "        coefs[coef_index] = d_ei[0]\n",
    "    return coefs\n",
    "\n",
    "def d_matrix_grj_nwedge_g(j, n, g, e_sort, bases, commutator):\n",
    "    \"\"\"\n",
    "    Return transpose of matrix d out from grade j of n wedges of g.\n",
    "\n",
    "    Input:\n",
    "    j = grading (≥1),\n",
    "    n = number of wedges (≥1),\n",
    "    g = np.array([g1,g2,...]) with graded parts of the Lie algebra g\n",
    "    where g1,g2,... are lists of numbers 1,2,... corresponding\n",
    "    to a basis e_1,e_2,... of g,\n",
    "    e_sort = an np.array describing our ordering of the e_i's,\n",
    "    bases = np.array with all bases in any grade and any wedges.\n",
    "    commutator = a function (int,int) -> [(int (coefficient),int (results))],\n",
    "    where different elements of the list corespond to summands.\n",
    "\n",
    "    Output:\n",
    "    Returns an np.array for the transpose matrix of the map d from\n",
    "    grade j of n wedges of g to grade j of n-1 wedges of g, or\n",
    "    equivalently,\n",
    "    the matrix describing the map d^T\n",
    "    from grade -j of Hom(n-1 wedges of g, k)\n",
    "    to grade -j of Hom(n wedges of g, k).\n",
    "\n",
    "    Description:\n",
    "    First compute the bases of the codomain and domain of d,\n",
    "    then (unless trivial) find coefficients to represent the\n",
    "    image in the codomain basis. Finally, note that we return\n",
    "    the transpose, since we fill in rows instead of columns,\n",
    "    which is the map we actually care about anyways.\n",
    "    \"\"\"\n",
    "    codomain_basis = bases[j][n - 1]\n",
    "    domain_basis = bases[j][n]\n",
    "    if len(codomain_basis) == 0 or len(domain_basis) == 0:\n",
    "        return np.array([0], dtype=np.int32)\n",
    "    # Note, that our basis doesn't contain coefficients,\n",
    "    # so we add the coefficient 1 to eis\n",
    "    matrix = [\n",
    "        d_coefs(g, add_one_coef(eis), e_sort, codomain_basis, commutator)\n",
    "        for eis in domain_basis\n",
    "        if len(eis) > 1\n",
    "    ]\n",
    "    return np.array(matrix, dtype=np.int32)\n",
    "\n",
    "def store_d_matrices(jmax, nmax, g, e_sort, bases, commutator):\n",
    "    \"\"\"\n",
    "    Store the all the matrices describing d in an np.array.\n",
    "\n",
    "    Input:\n",
    "    jmax = max j to try when finding bases, = max grade,\n",
    "    nmax = max n to try when finding bases, = dim(g),\n",
    "    g = np.array([g1,g2,...]) with graded parts of the Lie algebra g\n",
    "    where g1,g2,... are lists of numbers 1,2,... corresponding\n",
    "    to a basis e_1,e_2,... of g,\n",
    "    e_sort = an np.array describing our ordering of the e_i's,\n",
    "    bases = np.array with all bases in any grade any wedges.\n",
    "    commutator = a function (int,int) -> [(int (coefficient),int (results))],\n",
    "    where different elements of the list corespond to summands.\n",
    "\n",
    "    Output:\n",
    "    An np.array with a matrix to the map\n",
    "    gr^(-j) Hom(n-1 wedges of g, k) --> Hom(n wedges of g, k)\n",
    "    in entry array[j][n].\n",
    "\n",
    "    Description:\n",
    "    Goes through j from 0 to jmax+1 and n from 0 to nmax+1,\n",
    "    and saves the matrix d to grade -j of Hom(n wedges of g, k).\n",
    "    We go to +1 to have the entry d_matrices_array[jmax+1][nmax+1]\n",
    "    make sense.\n",
    "    \"\"\"\n",
    "    d_matrices_array = np.empty((jmax+1, nmax+1), dtype=object)\n",
    "    for j in range(jmax+1):\n",
    "        for n in range(nmax+1):\n",
    "            d_matrices_array[j][n] = np.array([0], dtype=np.int32)\n",
    "    for j in range(1, jmax):\n",
    "        for n in range(1, nmax):\n",
    "            d_matrices_array[j][n] = d_matrix_grj_nwedge_g(\n",
    "                j, n, g, e_sort, bases, commutator\n",
    "            )\n",
    "    return d_matrices_array\n",
    "\n",
    "\n",
    "def print_cochain_matrices(d_matrices):\n",
    "    \"\"\"\n",
    "    Print all non-trivial cochain matrices.\n",
    "\n",
    "    Input:\n",
    "    d_matrices = np.array with matrices,\n",
    "    where the matrices describe maps\n",
    "    gr^(-j) Hom(n-1 wedges of g, k) --> gr^(-j) Hom(n wedges of g, k).\n",
    "\n",
    "    Output:\n",
    "    No output, but prints matrices.\n",
    "\n",
    "    Description:\n",
    "    Print (nicely) the matrices, \n",
    "    plus some useful information.\n",
    "    \"\"\"\n",
    "    jmax1, nmax1 = d_matrices.shape\n",
    "    print(\n",
    "        \"\"\"\n",
    "        ===================================================\n",
    "        Non trivial matrices:\n",
    "        =================================================== \\n\n",
    "        \"\"\"\n",
    "    )\n",
    "    for j in range(jmax1):\n",
    "        for n in range(nmax1):\n",
    "            s = -j\n",
    "            t = n - s\n",
    "            # we have the dual map of (j,n) --> (j,n-1), i.e., (j,n-1) --> (j,n)\n",
    "            if np.count_nonzero(d_matrices[j][n]) > 0:\n",
    "                print(f\"s = {s}, t = {t}, s+t = {s+t} ; (j = {j}, n = {n}):\")\n",
    "                print(\n",
    "                    f\"\"\"\n",
    "                    d : Hom^({s})({s+t-1} wedges of g, k) --> Hom^({s})({s+t} wedges of g, k)\n",
    "                    \"\"\"\n",
    "                )\n",
    "                print(f\"matrix shape: {d_matrices[j][n].shape}\")\n",
    "                print(\"\\n Matrix:\\n\")\n",
    "                # NOTE: display/show works in jupyter notebooks instead of print,\n",
    "                # use print to get matrices that look worse but also take up less\n",
    "                # space\n",
    "                show(matrix(d_matrices[j][n]))\n",
    "                print(\"\\n LaTeX code:\")\n",
    "                print(\"Matrix: \" + latex(matrix(d_matrices[j][n])))\n",
    "                print(\"\\n ------------------------------------------- \\n\\n\")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "38d560fe",
   "metadata": {},
   "source": [
    "## Functions for finding bases of the kernel and image of the differential maps"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "id": "b20eeb5a",
   "metadata": {},
   "outputs": [],
   "source": [
    "def kernel_and_image_bases_j_n(j, n, bases, d_matrices):\n",
    "    \"\"\"\n",
    "    Return a basis of the null and column space of the matrix\n",
    "    corresponding to the map\n",
    "    gr^(-j) Hom(n-1 wedges of g, k) --> gr^(-j) Hom(n wedges of g, k).\n",
    "\n",
    "    Input:\n",
    "    j = grading (≥1),\n",
    "    n = number of wedges (≥1, < dim(g)),\n",
    "    bases = np.array with all bases in any grade any wedges,\n",
    "    d_matrices = np.array with matrices,\n",
    "    where the matrices describe maps\n",
    "    gr^(-j) Hom(n-1 wedges of g, k) --> gr^(-j) Hom(n wedges of g, k).\n",
    "\n",
    "    Output:\n",
    "    Returns two lists of bases.\n",
    "\n",
    "    Description:\n",
    "    If we map to or from a space of dimension 0, we handle that\n",
    "    separately, and if that is not the case, we have a matrix\n",
    "    that we can work with and find the null/column space.\n",
    "    \"\"\"\n",
    "    if n == 0:\n",
    "        return [], []\n",
    "    codomain_basis = bases[j][n]\n",
    "    codomain_dim = len(codomain_basis)\n",
    "    domain_basis = bases[j][n - 1]\n",
    "    domain_dim = len(domain_basis)\n",
    "    if codomain_dim == 0 or domain_dim == 0:\n",
    "        this_d = MatrixSpace(IntegerRing(), codomain_dim, domain_dim)(0)\n",
    "        this_image_basis = this_d.column_space().basis()\n",
    "        this_kernel_basis = this_d.right_kernel().basis()\n",
    "        return this_kernel_basis, this_image_basis\n",
    "    this_d = Matrix(d_matrices[j][n], ring=ZZ)\n",
    "    this_image_basis = this_d.column_space().basis()\n",
    "    this_kernel_basis = this_d.right_kernel().basis()\n",
    "    return this_kernel_basis, this_image_basis\n",
    "\n",
    "\n",
    "def store_ker_im_bases(bases, d_matrices):\n",
    "    \"\"\"\n",
    "    Store all the kernel and image bases for the different d\n",
    "    in an np.array.\n",
    "\n",
    "    Input:\n",
    "    bases = np.array with all bases in any grade any wedges,\n",
    "    d_matrices = np.array with matrices,\n",
    "    where the matrices describe maps\n",
    "    gr^(-j) Hom(n-1 wedges of g, k) --> gr^(-j) Hom(n wedges of g, k).\n",
    "\n",
    "    Output:\n",
    "    An np.array with a list of kernel and image bases for each entry.\n",
    "\n",
    "    Description:\n",
    "    Goes through j from 0 to jmax+1 and n from 0 to nmax+1,\n",
    "    and saves the basis corresponding to the matrix d \n",
    "    to grade -j of Hom(n wedges of g, k).\n",
    "    We go to +1 to have the entry d_matrices_array[jmax+1][nmax+1]\n",
    "    make sense.\n",
    "    \"\"\"\n",
    "    (jmax2, nmax2) = bases.shape\n",
    "    ker_im_bases = np.empty((jmax2, nmax2), dtype=object)\n",
    "    for j in range(jmax2):\n",
    "        for n in range(nmax2):\n",
    "            ker_im_bases[j][n] = kernel_and_image_bases_j_n(j, n, bases, d_matrices)\n",
    "    return ker_im_bases\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "id": "d9c9dcc3",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "def print_ker_im_bases(ker_im_bases):\n",
    "    \"\"\"\n",
    "    Print all non-trivial kernel/image bases.\n",
    "\n",
    "    Input:\n",
    "    ker_im_bases = np.array containing bases of the kernel\n",
    "    and the image of the different d.\n",
    "\n",
    "    Output:\n",
    "    No output, but prints bases.\n",
    "\n",
    "    Description:\n",
    "    Print (nicely) the bases, \n",
    "    plus some useful information.\n",
    "    \"\"\"\n",
    "    jmax2, nmax2 = ker_im_bases.shape\n",
    "    print(\n",
    "        \"\"\"\n",
    "        ===================================================\n",
    "        Non trivial kernel and image bases:\n",
    "        =================================================== \\n\n",
    "        \"\"\"\n",
    "    )\n",
    "    for j in range(jmax2):\n",
    "        for n in range(nmax2):\n",
    "            s = -j\n",
    "            t = n - s\n",
    "            # we have the dual map of (j,n) --> (j,n-1), i.e., (j,n-1) --> (j,n)\n",
    "            if len(ker_im_bases[j][n][0]) > 0 or len(ker_im_bases[j][n][1]) > 0:\n",
    "                print(f\"s = {s}, t = {t}, s+t = {s+t} ; (j = {j}, n = {n}):\")\n",
    "                print(\n",
    "                    f\"\"\"\n",
    "                    d : Hom^({s})({s+t-1} wedges of g, k) --> Hom^({s})({s+t} wedges of g, k)\n",
    "                    \"\"\"\n",
    "                )\n",
    "                if len(ker_im_bases[j][n][0]) > 0:\n",
    "                    print(f\"kernel basis:\\n    {ker_im_bases[j][n][0]}\")\n",
    "                if len(ker_im_bases[j][n][1]) > 0:\n",
    "                    print(f\"image basis:\\n    {ker_im_bases[j][n][1]}\")\n",
    "                print(\"\\n ------------------------------------------- \\n\\n\")\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3420a25c",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "3889ea36",
   "metadata": {},
   "source": [
    "## Functions for finding bases of $H^{s,t}$ for each $s,t$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "id": "a69707c3",
   "metadata": {},
   "outputs": [],
   "source": [
    "def H_s_t(j, n, jmax, nmax, ker_im_bases):\n",
    "    \"\"\"\n",
    "    Return the spaces of the kernel, image, and\n",
    "    kernel/image (i.e. cohomology) corresponding\n",
    "    to the map d to gr^(-j) Hom(n wedges of g, k).\n",
    "\n",
    "    Input:\n",
    "    j = grading (≥1),\n",
    "    n = number of wedges (≥1, < dim(g)),\n",
    "    jmax = max j to try when finding bases, = max grade,\n",
    "    nmax = max n to try when finding bases, = dim(g),\n",
    "    ker_im_bases = np.array containing bases of the kernel\n",
    "    and the image of the different d.\n",
    "\n",
    "    Output:\n",
    "    Returns three spaces: kernel, image, kernel/image.\n",
    "\n",
    "    Description:\n",
    "    In trivial cases (n=j=0 and j=jmax,n=nmax) we hard-code\n",
    "    the corresponding spaces. In other cases, we use the\n",
    "    kernel and image bases to find the kernel and image\n",
    "    spaces by looking at the span of the bases. We then\n",
    "    find kernel/image using standard Sage functions. \n",
    "    \"\"\"    \n",
    "    # s = -j\n",
    "    # t = n+j\n",
    "    triv_mod = FreeModule(ZZ, 1)\n",
    "    zero_mod = FreeModule(ZZ, 0)\n",
    "    if j == 0 and n == 0:\n",
    "        return triv_mod, zero_mod, triv_mod/zero_mod\n",
    "    if j == jmax and n == nmax:\n",
    "        return triv_mod, zero_mod, triv_mod/zero_mod\n",
    "    this_d_ker_basis, this_d_im_basis = ker_im_bases[j][n]\n",
    "    next_d_ker_basis, next_d_im_basis = ker_im_bases[j][n+1]\n",
    "    if len(next_d_ker_basis) > 0:\n",
    "        this_dim = len(next_d_ker_basis[0])\n",
    "    elif len(this_d_im_basis) > 0:\n",
    "        this_dim = len(this_d_im_basis[0])\n",
    "    else:\n",
    "        return zero_mod, zero_mod, zero_mod/zero_mod\n",
    "    M = FreeModule(ZZ, this_dim) # ambient module\n",
    "    ker = M.span(next_d_ker_basis, ZZ)\n",
    "    im = M.span(this_d_im_basis, ZZ)\n",
    "    # the image should always be a submodule of the kernel\n",
    "    if not im.is_submodule(ker):\n",
    "        print(\"ERROR!\")\n",
    "    return ker, im, ker/im\n",
    "\n",
    "def fixed_2_H_s_t(j, n, jmax, nmax, ker_im_bases):\n",
    "    \"\"\"\n",
    "    Return the spaces of the kernel, image, and\n",
    "    kernel/image (i.e. cohomology) corresponding\n",
    "    to the map d to gr^(-j) Hom(n wedges of g, k),\n",
    "    where we have removed any 2-torsion.\n",
    "\n",
    "    Input:\n",
    "    j = grading (≥1),\n",
    "    n = number of wedges (≥1, < dim(g)),\n",
    "    jmax = max j to try when finding bases, = max grade,\n",
    "    nmax = max n to try when finding bases, = dim(g),\n",
    "    ker_im_bases = np.array containing bases of the image\n",
    "    and the kernel of the different d.\n",
    "\n",
    "    Output:\n",
    "    Returns three spaces: kernel, image, kernel/image\n",
    "    as H_s_t, but we have removed 2-torsion.\n",
    "\n",
    "    Description:\n",
    "    In trivial cases (n=j=0 and j=jmax,n=nmax) we hard-code\n",
    "    the corresponding spaces. In other cases, we use the\n",
    "    kernel and image bases to find the kernel and image\n",
    "    spaces by looking at the span of the bases. We then\n",
    "    find kernel/image using standard Sage functions.\n",
    "    We double check to see if there is any 2-torsion.\n",
    "    If we find 2-torsion, then we remove it.\n",
    "    \"\"\"   \n",
    "    # s = -j\n",
    "    # t = n+j\n",
    "    triv_mod = FreeModule(ZZ, 1)\n",
    "    zero_mod = FreeModule(ZZ, 0)\n",
    "    if j == 0 and n == 0:\n",
    "        return triv_mod, zero_mod, triv_mod/zero_mod\n",
    "    if j == jmax and n == nmax:\n",
    "        return triv_mod, zero_mod, triv_mod/zero_mod\n",
    "    this_d_ker_basis, this_d_im_basis = ker_im_bases[j][n]\n",
    "    next_d_ker_basis, next_d_im_basis = ker_im_bases[j][n+1]\n",
    "    if len(next_d_ker_basis) > 0:\n",
    "        this_dim = len(next_d_ker_basis[0])\n",
    "    elif len(this_d_im_basis) > 0:\n",
    "        this_dim = len(this_d_im_basis[0])\n",
    "    else:\n",
    "        return zero_mod, zero_mod, zero_mod/zero_mod\n",
    "    M = FreeModule(ZZ, this_dim) # ambient module\n",
    "    ker = M.span(next_d_ker_basis, ZZ)\n",
    "    im = M.span(this_d_im_basis, ZZ)\n",
    "    # the image should always be a submodule of the kernel\n",
    "    if not im.is_submodule(ker):\n",
    "        print(\"ERROR!\")\n",
    "    quot = ker/im\n",
    "    quot_gens = quot.gens()\n",
    "    quot_invs = quot.invariants()\n",
    "    fix_list = []\n",
    "    for i in range(len(quot_gens)):\n",
    "        if quot_invs[i] == 2:\n",
    "            fix_list.append(quot_gens[i].lift())\n",
    "    new_im = M.span(this_d_im_basis + fix_list, ZZ)\n",
    "    return ker, im, ker/new_im\n",
    "\n",
    "\n",
    "def store_H_s_t(ker_im_bases):\n",
    "    \"\"\"\n",
    "    Store all the H^(s,t) spaces in an np.array.\n",
    "\n",
    "    Input:\n",
    "    ker_im_bases = np.array containing bases of the image\n",
    "    and the kernel of the different d.\n",
    "\n",
    "    Output:\n",
    "    An np.array with a H^(s,t) space for each entry.\n",
    "\n",
    "    Description:\n",
    "    Goes through j from 0 to jmax+1 and n from 0 to nmax+1,\n",
    "    and saves the H^(s,t) space in an np.array.\n",
    "    \"\"\"\n",
    "    jmax2, nmax2 = ker_im_bases.shape\n",
    "    H_s_t_array = np.empty((jmax2-1,nmax2-1), dtype=object)\n",
    "    zero_mod = FreeModule(ZZ, 0)\n",
    "    for j in range(jmax2-1):\n",
    "        for n in range(nmax2-1):\n",
    "            _, _, Hst = H_s_t(j, n, jmax2-2, nmax2-2, ker_im_bases)\n",
    "            H_s_t_array[j][n] = Hst\n",
    "    return H_s_t_array\n",
    "\n",
    "\n",
    "def store_fixed_2_H_s_t(ker_im_bases):\n",
    "    \"\"\"\n",
    "    Store all the H^(s,t) spaces with 2-torsion removed\n",
    "    in an np.array.\n",
    "\n",
    "    Input:\n",
    "    ker_im_bases = np.array containing bases of the image\n",
    "    and the kernel of the different d.\n",
    "\n",
    "    Output:\n",
    "    An np.array with a H^(s,t) space for each entry.\n",
    "\n",
    "    Description:\n",
    "    Goes through j from 0 to jmax+1 and n from 0 to nmax+1,\n",
    "    and saves the H^(s,t) space with 2-torsion removed\n",
    "    in an np.array.\n",
    "    \"\"\"\n",
    "    jmax2, nmax2 = ker_im_bases.shape\n",
    "    H_s_t_array = np.empty((jmax2-1,nmax2-1), dtype=object)\n",
    "    zero_mod = FreeModule(ZZ, 0)\n",
    "    for j in range(jmax2-1):\n",
    "        for n in range(nmax2-1):\n",
    "            _, _, Hst = fixed_2_H_s_t(j, n, jmax2-2, nmax2-2, ker_im_bases)\n",
    "            H_s_t_array[j][n] = Hst\n",
    "    return H_s_t_array\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "id": "a106dbb2",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "def print_H_s_t(H_s_t_array):\n",
    "    \"\"\"\n",
    "    Print all H^(s,t) spaces from the np.array.\n",
    "\n",
    "    Input:\n",
    "    H_s_t_array = np.array containing cohomology\n",
    "    spaces H^(s,t) (either with or without 2-torsion).\n",
    "\n",
    "    Output:\n",
    "    No output, but prints spaces.\n",
    "\n",
    "    Description:\n",
    "    Print (nicely) the spaces, \n",
    "    plus some useful information.\n",
    "    \"\"\"\n",
    "    jmax1, nmax1 = H_s_t_array.shape\n",
    "    print(\n",
    "        \"\"\"\n",
    "        ===================================================\n",
    "        Non trivial cohomology H^(s,t) spaces:\n",
    "        =================================================== \\n\n",
    "        \"\"\"\n",
    "    )\n",
    "    for j in range(jmax1):\n",
    "        for n in range(nmax1):\n",
    "            s = -j\n",
    "            t = n - s\n",
    "            # we have the dual map of (j,n) --> (j,n-1), i.e., (j,n-1) --> (j,n)\n",
    "            Hst = H_s_t_array[j][n]\n",
    "            if Hst.ngens() > 0:\n",
    "                print(f\"(s,t) = ({s},{j}): dim H^({s},{t}) = {Hst.ngens()}\\n\")\n",
    "                print(f\"H^({s},{t}) = {Hst}\\n---\\n\")\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "id": "1710be0e",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "\n",
    "def print_H_s_t_bases(H_s_t_array):\n",
    "    \"\"\"\n",
    "    Print bases for all H^(s,t) spaces from the np.array.\n",
    "\n",
    "    Input:\n",
    "    H_s_t_array = np.array containing cohomology\n",
    "    spaces H^(s,t) (either with or without 2-torsion).\n",
    "\n",
    "    Output:\n",
    "    No output, but prints bases of spaces.\n",
    "\n",
    "    Description:\n",
    "    Print (nicely) the bases of the spaces, both in the space\n",
    "    itself and the lift to a basis of the corresponding ambient\n",
    "    space, plus some useful information.\n",
    "    \"\"\"\n",
    "    jmax1, nmax1 = H_s_t_array.shape\n",
    "    print(\n",
    "        \"\"\"\n",
    "        ===================================================\n",
    "        Bases of non trivial cohomology H^(s,t) spaces:\n",
    "        =================================================== \\n\n",
    "        \"\"\"\n",
    "    )\n",
    "    for j in range(jmax1):\n",
    "        for n in range(nmax1):\n",
    "            s = -j\n",
    "            t = n - s\n",
    "            # we have the dual map of (j,n) --> (j,n-1), i.e., (j,n-1) --> (j,n)\n",
    "            Hst = H_s_t_array[j][n]\n",
    "            if Hst.ngens() > 0:\n",
    "                print(f\"(s,t) = ({s},{j}): dim H^({s},{t}) = {Hst.ngens()}\\n\")\n",
    "                print(f\"basis in H^({s},{t}):\\n {Hst.gens()}\\n\")\n",
    "                print(f\"lift of basis to the ambient space:\\n {list(map(lift, Hst.gens()))}\\n---\\n\")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "20d94994",
   "metadata": {},
   "source": [
    "## Functions for finding the cup product"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "id": "1f4e3962",
   "metadata": {},
   "outputs": [],
   "source": [
    "def non_zero_tuple(tup):\n",
    "    \"\"\"\n",
    "    Return boolean describing whether tup is a non-zero wedge.\n",
    "\n",
    "    Input:\n",
    "    tup = a tuple of integers\n",
    "\n",
    "    Output:\n",
    "    Boolean\n",
    "\n",
    "    Description:\n",
    "    True if tup contains distinct integers.\n",
    "    False otherwise.\n",
    "    \"\"\"\n",
    "    return (len(tup) == len(set(tup)))\n",
    "\n",
    "\n",
    "def slow_base_sort(g, eis, e_sort):\n",
    "    \"\"\"\n",
    "    Sort our basis to the order we prefer.\n",
    "\n",
    "    Input:\n",
    "    g = np.array([g1,g2,...]) with graded parts of the Lie algebra g\n",
    "    where g1,g2,... are lists of numbers 1,2,... corresponding\n",
    "    to a basis e_1,e_2,... of g,\n",
    "    eis = a tuple (wedge product) of e_i's (basis elements) out of order\n",
    "    with the first entry being the coefficient,\n",
    "    e_sort = an np.array describing our ordering of the e_i's\n",
    "\n",
    "    Output:\n",
    "    A tuple of the e_i's sorted.\n",
    "\n",
    "    Description:\n",
    "    Compare the order to e_sort, and change it until it matches,\n",
    "    remembering to change signs, when moving elements past wedges.\n",
    "    Recall that a tuple corresponds to a wedge product.\n",
    "    \"\"\"\n",
    "    eis = list(eis)\n",
    "    sign = 1\n",
    "    for i in range(1, len(eis) - 1):\n",
    "        for j in range(1,len(eis)-1):\n",
    "            if e_sort[eis[j] - 1] > e_sort[eis[j + 1] - 1]:\n",
    "                eis[j], eis[j + 1] = eis[j + 1], eis[j]\n",
    "                # Equivalent to:\n",
    "                # tmp = eis[j]\n",
    "                # eis[j] = eis[j+1]\n",
    "                # eis[j+1] = tmp\n",
    "                sign *= -1\n",
    "    # Put coefficient on first entry\n",
    "    eis[0] = sign * eis[0]\n",
    "    return tuple(eis)\n",
    "\n",
    "\n",
    "def hom_cup_product(b1, b2, g, e_sort):\n",
    "    \"\"\"\n",
    "    Find the cup product of b1 and b2: b1 u b2.\n",
    "    \n",
    "    Input:\n",
    "    b1 = np.array([e1,e2,...]) a basis vector (assume sorted)\n",
    "    b2 = np.array([e1',e2',...]) a basis vector (assume sorted)\n",
    "    g = np.array([g1,g2,...]) with graded parts of the Lie algebra g\n",
    "    where g1,g2,... are numbers 1,2,... corresponding\n",
    "    to a basis e_1,e_2,... of g,\n",
    "    e_sort = an np.array describing our ordering of the e_i's\n",
    "    \n",
    "    Output:\n",
    "    The cup product b1 u b2 in the hom-space (not in cohomology).\n",
    "    \n",
    "    Description:\n",
    "    Find the coefficient, if the tuple corresponds to a zero \n",
    "    wedge (i.e., if it has the same element twice) return (0,),\n",
    "    otherwise resorting the indices and keeping track of the \n",
    "    sign will give the correct cupproduct, since \n",
    "    we are assuming that we are only working with a\n",
    "    basis and its dual basis all the the time.\n",
    "    \"\"\"\n",
    "    b1_tmp = tuple(b1[1:]) # b1 without the coef\n",
    "    b2_tmp = tuple(b2[1:]) # b2 without the coef\n",
    "    coef = b1[0] * b2[0]\n",
    "    #tmp_cup = np.concatenate(b1_tmp, b2_tmp)\n",
    "    tmp_cup = b1_tmp + b2_tmp # concatenate tuples\n",
    "    if not (non_zero_tuple(tmp_cup)):\n",
    "        return (0,) #tuple with coef = 0\n",
    "    cup_with_coef = slow_base_sort(g, (coef,) + tmp_cup, e_sort)\n",
    "    return cup_with_coef\n",
    "\n",
    "\n",
    "def vec_to_basis(j, n, vec, bases):\n",
    "    \"\"\"\n",
    "    Given a vector describing coeffients in a basis,\n",
    "    return an np.array with the corresponding basis\n",
    "    elements and their coefficient.\n",
    "\n",
    "    Input:\n",
    "    j = grading (≥1),\n",
    "    n = number of wedges (≥1, < dim(g)),\n",
    "    vec = a vector containing coefficients for each of the\n",
    "    corresponding basis elements from bases,\n",
    "    bases = np.array with all bases in any grade any wedges.\n",
    "\n",
    "    Output:\n",
    "    Returns an np.array with basis elements with a coefficient.\n",
    "\n",
    "    Description:\n",
    "    We look at the basis for the given j and n. If it is empty,\n",
    "    we return an empty list (when using this function, we should\n",
    "    not get to this point). Otherwise make an np.array to contain\n",
    "    all basis elements with a non-zero coefficient in the vector,\n",
    "    and add the basis elements with a coefficient to this array.\n",
    "    \"\"\"\n",
    "    basis = bases[j][n]\n",
    "    if len(basis) == 0:\n",
    "        return []\n",
    "    num_basis, num_wedges = basis.shape\n",
    "    non_zero_vec = np.count_nonzero(vec)\n",
    "    vec_basis = np.zeros((non_zero_vec, num_wedges+1), dtype=np.int32)\n",
    "    j = 0\n",
    "    for i in range(num_basis):\n",
    "        if vec[i] != 0:\n",
    "            coef = vec[i]\n",
    "            vec_basis[j][0] = coef\n",
    "            vec_basis[j][1:] = basis[i]\n",
    "            j += 1\n",
    "    return vec_basis\n",
    "\n",
    "\n",
    "def Hs1t1_Hs2t2_cups(j1, n1, j2, n2, g, e_sort, H_s_t_array, bases):\n",
    "    \"\"\"\n",
    "    Given H^(s1,t1) and H^(s2,t2), we calculate all cup\n",
    "    products (under certain assumptions).\n",
    "\n",
    "    Input:\n",
    "    j1 = grading (≥1),\n",
    "    n1 = number of wedges (≥1, < dim(g)),\n",
    "    j2 = grading (≥1),\n",
    "    n2 = number of wedges (≥1, < dim(g)),\n",
    "    g = np.array([g1,g2,...]) with graded parts of the Lie algebra g\n",
    "    where g1,g2,... are lists of numbers 1,2,... corresponding\n",
    "    to a basis e_1,e_2,... of g,\n",
    "    e_sort = an np.array describing our ordering of the e_i's,\n",
    "    H_s_t_array = np.array containing cohomology\n",
    "    spaces H^(s,t) (either with or without 2-torsion),\n",
    "    bases = np.array with all bases in any grade any wedges.\n",
    "\n",
    "    Output:\n",
    "    Returns an np.array with cup products.\n",
    "\n",
    "    Description:\n",
    "    We lift the H^(s1,t1) and H^(s2,t2) bases to their ambient \n",
    "    space, then we check for a lot of conditions that we assume\n",
    "    are not satisfied here. When there is a possible non-zero\n",
    "    cup product, we go through all the possible cases and \n",
    "    calculate all non-zero cup products. \n",
    "    \"\"\"\n",
    "    Hs1t1_basis = list(map(lift, H_s_t_array[j1][n1].gens()))\n",
    "    Hs1t1_dim = H_s_t_array[j1][n1].ngens()\n",
    "    Hs2t2_basis = list(map(lift, H_s_t_array[j2][n2].gens()))\n",
    "    Hs2t2_dim = H_s_t_array[j2][n2].ngens()\n",
    "    jmax1, nmax1 = H_s_t_array.shape\n",
    "    if (j1+j2 >= jmax1) or (n1+n2 >= nmax1):\n",
    "        # We assume we don't get to here\n",
    "        print(\"ERROR 1\")\n",
    "        return []\n",
    "    Hst_sum = H_s_t_array[j1+j2][n1+n2]\n",
    "    Hst_sum_dim = Hst_sum.ngens()\n",
    "    base_list = bases[j1+j2][n1+n2].tolist()\n",
    "    Hst_sum_am_dim = len(Hst_sum.gen(0).lift())\n",
    "    cups_array = np.empty((Hs1t1_dim, Hs2t2_dim), dtype=object)\n",
    "    if Hs1t1_dim == 0:\n",
    "        print(\"ERROR 2\")\n",
    "        return []\n",
    "    if Hs2t2_dim == 0:\n",
    "        print(\"ERROR 3\")\n",
    "        return []\n",
    "    if Hst_sum_dim == 0:\n",
    "        # WE assume we don't get to here\n",
    "        print(\"ERROR 4\")\n",
    "        return []\n",
    "    for i1 in range(Hs1t1_dim):\n",
    "        vec1 = Hs1t1_basis[i1]\n",
    "        vec1_in_basis = vec_to_basis(j1, n1, vec1, bases)\n",
    "        for i2 in range(Hs2t2_dim):\n",
    "            vec2 = Hs2t2_basis[i2]\n",
    "            vec2_in_basis = vec_to_basis(j2, n2, vec2, bases)\n",
    "            cup_sum_array = np.zeros(Hst_sum_am_dim, dtype=np.int32)\n",
    "            for b1 in vec1_in_basis:\n",
    "                for b2 in vec2_in_basis:\n",
    "                    hom_b1_b2_cup = hom_cup_product(b1, b2, g, e_sort)\n",
    "                    if hom_b1_b2_cup[0] != 0:\n",
    "                        coef = hom_b1_b2_cup[0]\n",
    "                        hom_b1_b2_cup_list = list(hom_b1_b2_cup[1:])\n",
    "                        tmp_b1_b2_cup = np.zeros(Hst_sum_am_dim, dtype=np.int32)\n",
    "                        ind = base_list.index(hom_b1_b2_cup_list)\n",
    "                        tmp_b1_b2_cup[ind] = coef\n",
    "                        cup_sum_array += tmp_b1_b2_cup\n",
    "            cup_sum = tuple(cup_sum_array.tolist())\n",
    "            cups_array[i1][i2] = Hst_sum(cup_sum)\n",
    "    return cups_array\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "id": "3ea7c8ed",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "\n",
    "def print_cup_products(g, e_sort, H_s_t_array, bases):\n",
    "    \"\"\"\n",
    "    Print all non-zero cup products with some extra\n",
    "    information.\n",
    "\n",
    "    Input:\n",
    "    g = np.array([g1,g2,...]) with graded parts of the Lie algebra g\n",
    "    where g1,g2,... are lists of numbers 1,2,... corresponding\n",
    "    to a basis e_1,e_2,... of g,\n",
    "    e_sort = an np.array describing our ordering of the e_i's,\n",
    "    H_s_t_array = np.array containing cohomology\n",
    "    spaces H^(s,t) (either with or without 2-torsion),\n",
    "    bases = np.array with all bases in any grade any wedges.\n",
    "\n",
    "    Output:\n",
    "    No output, but prints non-zero cup products.\n",
    "\n",
    "    Description:\n",
    "    Print (nicely) the cup products of the different H^(s,t)\n",
    "    spsaces, plus some useful information.\n",
    "    \"\"\"\n",
    "    jmax1, nmax1 = H_s_t_array.shape\n",
    "    print(\n",
    "        \"\"\"\n",
    "        ===================================================\n",
    "        Non-zero cup products from H^(s,t) spaces:\n",
    "        =================================================== \\n\n",
    "        \"\"\"\n",
    "    )\n",
    "    # we are not looking at the cup product for H^0\n",
    "    print(\"NOTE: We are not printing any H^(0) cup products.\\n\")\n",
    "    print(\"We write vi^(s,t) for the i'th basis vector of H^(s,t).\\n\")\n",
    "    for j1 in range(1, jmax1):\n",
    "        for n1 in range(1, nmax1):\n",
    "            Hst1 = H_s_t_array[j1][n1]\n",
    "            Hst1_dim = Hst1.ngens()\n",
    "            if Hst1_dim > 0:\n",
    "                for j2 in range(j1, jmax1-j1):\n",
    "                    for n2 in range(n1, nmax1-n1):\n",
    "                        Hst2 = H_s_t_array[j2][n2]\n",
    "                        Hst2_dim = Hst2.ngens()\n",
    "                        Hst_sum = H_s_t_array[j1+j2][n1+n2]\n",
    "                        Hst_sum_dim = Hst_sum.ngens()\n",
    "                        if (Hst2_dim > 0) and (Hst_sum_dim > 0):\n",
    "                            Hst_sum_cups = Hs1t1_Hs2t2_cups(j1, n1, j2, n2, g, e_sort, H_s_t_array, bases)\n",
    "                            print(\"\\n\")\n",
    "                            for i1 in range(Hst1_dim):\n",
    "                                for i2 in range(Hst2_dim):\n",
    "                                    if Hst_sum_cups[i1][i2] != Hst_sum(0):\n",
    "                                        s1, t1, s2, t2 = -j1, n1+j1, -j2, n2+j2\n",
    "                                        s3, t3 = s1+s2, t1+t2\n",
    "                                        current_cup = np.array(list(Hst_sum_cups[i1][i2]), dtype=np.int32)\n",
    "                                        cup_nonzero_index = np.nonzero(current_cup)\n",
    "                                        current_cup_nonzero = current_cup[cup_nonzero_index]\n",
    "                                        current_string = f\"v{i1+1}^({s1},{t1}) cup v{i2+1}^({s2},{t2}) = \"\n",
    "                                        for ind in range(len(current_cup_nonzero)-1):\n",
    "                                            current_string += f\"{current_cup_nonzero[ind]}*v{cup_nonzero_index[0][ind]+1}^({s3},{t3}) + \"\n",
    "                                        current_string += f\"{current_cup_nonzero[-1]}*v{cup_nonzero_index[0][-1]+1}^({s3},{t3})\"\n",
    "                                        print(current_string + \"\\n\")\n",
    "                                        #print(f\"v{i1+1}^({-j1},{n1+j1}) cup v{i2+1}^({-j2},{n2+j2}) = {Hst_sum_cups[i1][i2]} in H^({-j1-j2},{n1+n2+j1+j2})\\n\")\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "id": "5e9ba3cd",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "\n",
    "def print_cup_products_alt(g, e_sort, H_s_t_array, bases):\n",
    "    \"\"\"\n",
    "    Print all non-zero cup products with some extra\n",
    "    information.\n",
    "\n",
    "    Input:\n",
    "    g = np.array([g1,g2,...]) with graded parts of the Lie algebra g\n",
    "    where g1,g2,... are lists of numbers 1,2,... corresponding\n",
    "    to a basis e_1,e_2,... of g,\n",
    "    e_sort = an np.array describing our ordering of the e_i's,\n",
    "    H_s_t_array = np.array containing cohomology\n",
    "    spaces H^(s,t) (either with or without 2-torsion),\n",
    "    bases = np.array with all bases in any grade any wedges.\n",
    "\n",
    "    Output:\n",
    "    No output, but prints non-zero cup products.\n",
    "\n",
    "    Description:\n",
    "    Print (nicely) the cup products of the different H^(s,t)\n",
    "    spsaces, plus some useful information.\n",
    "    \"\"\"\n",
    "    jmax1, nmax1 = H_s_t_array.shape\n",
    "    print(\n",
    "        \"\"\"\n",
    "        ===================================================\n",
    "        Non-zero cup products from H^(s,t) spaces:\n",
    "        =================================================== \\n\n",
    "        \"\"\"\n",
    "    )\n",
    "    # we are not looking at the cup product for H^0\n",
    "    print(\"NOTE: We are not printing any H^(0) cup products.\\n\")\n",
    "    print(\"We write vi^(s,t) for the i'th basis vector of H^(s,t).\\n\")\n",
    "    for j1 in range(1, jmax1):\n",
    "        for n1 in range(1, nmax1):\n",
    "            Hst1 = H_s_t_array[j1][n1]\n",
    "            Hst1_dim = Hst1.ngens()\n",
    "            if Hst1_dim > 0:\n",
    "                for j2 in range(j1, jmax1-j1):\n",
    "                    for n2 in range(n1, nmax1-n1):\n",
    "                        Hst2 = H_s_t_array[j2][n2]\n",
    "                        Hst2_dim = Hst2.ngens()\n",
    "                        Hst_sum = H_s_t_array[j1+j2][n1+n2]\n",
    "                        Hst_sum_dim = Hst_sum.ngens()\n",
    "                        if (Hst2_dim > 0) and (Hst_sum_dim > 0):\n",
    "                            Hst_sum_cups = Hs1t1_Hs2t2_cups(j1, n1, j2, n2, g, e_sort, H_s_t_array, bases)\n",
    "                            print(\"\\n\")\n",
    "                            for i1 in range(Hst1_dim):\n",
    "                                for i2 in range(Hst2_dim):\n",
    "                                    if Hst_sum_cups[i1][i2] != Hst_sum(0):\n",
    "                                        s1, t1, s2, t2 = -j1, n1+j1, -j2, n2+j2\n",
    "                                        s3, t3 = s1+s2, t1+t2\n",
    "                                        print(f\"v{i1+1}^({s1},{t1}) cup v{i2+1}^({s2},{t2}) = {Hst_sum_cups[i1][i2]} in H^({s3},{t3})\\n\")\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "04255519",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "bfadb398",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "fea51209",
   "metadata": {},
   "source": [
    "# Using the functions for $I$ the pro-$p$ Iwahori in $\\operatorname{SL}_3(\\mathbb Z_p)$ ($p\\geq5$)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1ce76b83",
   "metadata": {},
   "source": [
    "## Initialization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "id": "37b08c59",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([array([1, 6, 7], dtype=int32), array([2, 3, 8], dtype=int32),\n",
       "       array([4, 5], dtype=int32)], dtype=object)"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g_sl3_1 = np.array([1,6,7], dtype=np.int32) #1-graded (1,6,7)\n",
    "g_sl3_2 = np.array([2,3,8], dtype=np.int32) #2-graded (2,3,8)\n",
    "g_sl3_3 = np.array([4,5], dtype=np.int32) #3-graded (4,5)\n",
    "g_sl3 = np.array([g_sl3_1,g_sl3_2,g_sl3_3], dtype=object); \n",
    "g_sl3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "d70996c4",
   "metadata": {},
   "outputs": [],
   "source": [
    "# g = g^1 + g^2 + g^3, we order them g^1, g^2, g^3, \n",
    "# where g^1 = span(e_1, e_6, e_7), g^2 = span(e_2, e_3, e_8), g^3 = span(e_4, e_5)\n",
    "# so the prefered order is: \n",
    "# e_1 (1st), e_6 (2nd), e_7 (3rd), \n",
    "# e_2 (4th), e_3 (5th), e_8 (6th), \n",
    "# e_4 (7th), e_5 (8th)\n",
    "# i.e., in a zero indexed list:\n",
    "# (e_1 =) 1-->0, (e_2 =) 2-->3, 3-->4, 4-->6, 5-->7, 6-->1, 7-->2, 8-->5\n",
    "e_sort_sl3 = np.array([0,3,4,6,7,1,2,5], dtype=np.int32)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "id": "78f17430",
   "metadata": {},
   "outputs": [],
   "source": [
    "def commutator_sl3(a, b):\n",
    "    sign = 1\n",
    "    if a > b:\n",
    "        a,b = b,a\n",
    "        sign = -1\n",
    "    # to make it easier later, we return lists with the answers, where a x+y = [x,y]\n",
    "    if a==1 and b==6:\n",
    "        return [(-sign,2)]\n",
    "    elif a==1 and b==7:\n",
    "        return [(sign,3)]\n",
    "    elif a==1 and b==8:\n",
    "        return [(-sign,5),(-sign,4)] # for the sake of later code, we order these in reverse order\n",
    "    elif a==2 and b==7:\n",
    "        return [(-sign,4)]\n",
    "    elif a==3 and b==6:\n",
    "        return [(-sign,5)]\n",
    "    elif a==6 and b==7:\n",
    "        return [(-sign,8)]\n",
    "    else:\n",
    "        return [(0,0)]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b30e134f",
   "metadata": {},
   "source": [
    "## Store relevant information"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "70b5f5d9",
   "metadata": {},
   "source": [
    "Note that $j_{\\max} = 15$ and $n_{\\max} = 8$ for $I$ in $\\operatorname{SL}_3(\\mathbb Z_p)$ ($p \\geq 5$)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "id": "ae01588e",
   "metadata": {},
   "outputs": [],
   "source": [
    "sl3_jmax, sl3_nmax = 15, 8\n",
    "sl3_bases = store_bases(sl3_jmax, sl3_nmax, g_sl3)\n",
    "sl3_d_matrices = store_d_matrices(sl3_jmax, sl3_nmax, g_sl3, e_sort_sl3, sl3_bases, commutator_sl3)\n",
    "sl3_ker_im_bases = store_ker_im_bases(sl3_bases, sl3_d_matrices)\n",
    "sl3_H_s_t_array = store_H_s_t(sl3_ker_im_bases)\n",
    "sl3_H_s_t_fixed_2 = store_fixed_2_H_s_t(sl3_ker_im_bases)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9137ed99",
   "metadata": {},
   "source": [
    "## Print information"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "id": "22ded726",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "        ===================================================\n",
      "        gr^j of n wedges of g in terms of g^1, g^2 and g^3:\n",
      "        =================================================== \n",
      "\n",
      "        \n",
      "\n",
      "        Note:\n",
      "        - This prints [(1)] as [(1,)].\n",
      "        - A tuple corresponds to a wedge product,\n",
      "          e.g., (1,1,2) mean g^1 wedge g^1 wedge g^2.\n",
      "        - Tuples in the list correcsponds to summands,\n",
      "          e.g., [(1,3),(2,2)] means\n",
      "          g^1 wedge g^3 direct sum g^2 wedge g^2.\n",
      "        - This doesn't work for 0 wedges of g nor for\n",
      "          dim(g) wedges of g. \n",
      "\n",
      "        \n",
      "------------\n",
      "Grade 0: \n",
      "\n",
      "------------\n",
      "Grade 1: \n",
      "\n",
      "[(1,)]\n",
      "------------\n",
      "Grade 2: \n",
      "\n",
      "[(2,)]\n",
      "[(1, 1)]\n",
      "------------\n",
      "Grade 3: \n",
      "\n",
      "[(3,)]\n",
      "[(1, 2)]\n",
      "[(1, 1, 1)]\n",
      "------------\n",
      "Grade 4: \n",
      "\n",
      "[(1, 3), (2, 2)]\n",
      "[(1, 1, 2)]\n",
      "------------\n",
      "Grade 5: \n",
      "\n",
      "[(2, 3)]\n",
      "[(1, 1, 3), (1, 2, 2)]\n",
      "[(1, 1, 1, 2)]\n",
      "------------\n",
      "Grade 6: \n",
      "\n",
      "[(3, 3)]\n",
      "[(1, 2, 3), (2, 2, 2)]\n",
      "[(1, 1, 1, 3), (1, 1, 2, 2)]\n",
      "------------\n",
      "Grade 7: \n",
      "\n",
      "[(1, 3, 3), (2, 2, 3)]\n",
      "[(1, 1, 2, 3), (1, 2, 2, 2)]\n",
      "[(1, 1, 1, 2, 2)]\n",
      "------------\n",
      "Grade 8: \n",
      "\n",
      "[(2, 3, 3)]\n",
      "[(1, 1, 3, 3), (1, 2, 2, 3)]\n",
      "[(1, 1, 1, 2, 3), (1, 1, 2, 2, 2)]\n",
      "------------\n",
      "Grade 9: \n",
      "\n",
      "[(1, 2, 3, 3), (2, 2, 2, 3)]\n",
      "[(1, 1, 1, 3, 3), (1, 1, 2, 2, 3)]\n",
      "[(1, 1, 1, 2, 2, 2)]\n",
      "------------\n",
      "Grade 10: \n",
      "\n",
      "[(2, 2, 3, 3)]\n",
      "[(1, 1, 2, 3, 3), (1, 2, 2, 2, 3)]\n",
      "[(1, 1, 1, 2, 2, 3)]\n",
      "------------\n",
      "Grade 11: \n",
      "\n",
      "[(1, 2, 2, 3, 3)]\n",
      "[(1, 1, 1, 2, 3, 3), (1, 1, 2, 2, 2, 3)]\n",
      "------------\n",
      "Grade 12: \n",
      "\n",
      "[(2, 2, 2, 3, 3)]\n",
      "[(1, 1, 2, 2, 3, 3)]\n",
      "[(1, 1, 1, 2, 2, 2, 3)]\n",
      "------------\n",
      "Grade 13: \n",
      "\n",
      "[(1, 2, 2, 2, 3, 3)]\n",
      "[(1, 1, 1, 2, 2, 3, 3)]\n",
      "------------\n",
      "Grade 14: \n",
      "\n",
      "[(1, 1, 2, 2, 2, 3, 3)]\n",
      "------------\n",
      "Grade 15: \n",
      "\n"
     ]
    }
   ],
   "source": [
    "print_grj_nwedge_g(sl3_jmax, sl3_nmax, g_sl3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "id": "19550a8a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "        ===================================================\n",
      "        Non trivial matrices:\n",
      "        =================================================== \n",
      "\n",
      "        \n",
      "s = -2, t = 4, s+t = 2 ; (j = 2, n = 2):\n",
      "\n",
      "                    d : Hom^(-2)(1 wedges of g, k) --> Hom^(-2)(2 wedges of g, k)\n",
      "                    \n",
      "matrix shape: (3, 3)\n",
      "\n",
      " Matrix:\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{rrr}\n",
       "1 & 0 & 0 \\\\\n",
       "0 & -1 & 0 \\\\\n",
       "0 & 0 & 1\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{rrr}\n",
       "1 & 0 & 0 \\\\\n",
       "0 & -1 & 0 \\\\\n",
       "0 & 0 & 1\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[ 1  0  0]\n",
       "[ 0 -1  0]\n",
       "[ 0  0  1]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " LaTeX code:\n",
      "Matrix: \\left(\\begin{array}{rrr}\n",
      "1 & 0 & 0 \\\\\n",
      "0 & -1 & 0 \\\\\n",
      "0 & 0 & 1\n",
      "\\end{array}\\right)\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -3, t = 5, s+t = 2 ; (j = 3, n = 2):\n",
      "\n",
      "                    d : Hom^(-3)(1 wedges of g, k) --> Hom^(-3)(2 wedges of g, k)\n",
      "                    \n",
      "matrix shape: (9, 2)\n",
      "\n",
      " Matrix:\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{rr}\n",
       "0 & 0 \\\\\n",
       "0 & 0 \\\\\n",
       "1 & 1 \\\\\n",
       "0 & 0 \\\\\n",
       "0 & -1 \\\\\n",
       "0 & 0 \\\\\n",
       "-1 & 0 \\\\\n",
       "0 & 0 \\\\\n",
       "0 & 0\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{rr}\n",
       "0 & 0 \\\\\n",
       "0 & 0 \\\\\n",
       "1 & 1 \\\\\n",
       "0 & 0 \\\\\n",
       "0 & -1 \\\\\n",
       "0 & 0 \\\\\n",
       "-1 & 0 \\\\\n",
       "0 & 0 \\\\\n",
       "0 & 0\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[ 0  0]\n",
       "[ 0  0]\n",
       "[ 1  1]\n",
       "[ 0  0]\n",
       "[ 0 -1]\n",
       "[ 0  0]\n",
       "[-1  0]\n",
       "[ 0  0]\n",
       "[ 0  0]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " LaTeX code:\n",
      "Matrix: \\left(\\begin{array}{rr}\n",
      "0 & 0 \\\\\n",
      "0 & 0 \\\\\n",
      "1 & 1 \\\\\n",
      "0 & 0 \\\\\n",
      "0 & -1 \\\\\n",
      "0 & 0 \\\\\n",
      "-1 & 0 \\\\\n",
      "0 & 0 \\\\\n",
      "0 & 0\n",
      "\\end{array}\\right)\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -3, t = 6, s+t = 3 ; (j = 3, n = 3):\n",
      "\n",
      "                    d : Hom^(-3)(2 wedges of g, k) --> Hom^(-3)(3 wedges of g, k)\n",
      "                    \n",
      "matrix shape: (1, 9)\n",
      "\n",
      " Matrix:\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{rrrrrrrrr}\n",
       "0 & 0 & -1 & 0 & -1 & 0 & -1 & 0 & 0\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{rrrrrrrrr}\n",
       "0 & 0 & -1 & 0 & -1 & 0 & -1 & 0 & 0\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[ 0  0 -1  0 -1  0 -1  0  0]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " LaTeX code:\n",
      "Matrix: \\left(\\begin{array}{rrrrrrrrr}\n",
      "0 & 0 & -1 & 0 & -1 & 0 & -1 & 0 & 0\n",
      "\\end{array}\\right)\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -4, t = 7, s+t = 3 ; (j = 4, n = 3):\n",
      "\n",
      "                    d : Hom^(-4)(2 wedges of g, k) --> Hom^(-4)(3 wedges of g, k)\n",
      "                    \n",
      "matrix shape: (9, 9)\n",
      "\n",
      " Matrix:\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{rrrrrrrrr}\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n",
       "0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 \\\\\n",
       "1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & -1 \\\\\n",
       "0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & -1 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{rrrrrrrrr}\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n",
       "0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 \\\\\n",
       "1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & -1 \\\\\n",
       "0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & -1 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[ 0  0  0  0  0  0  0  0  0]\n",
       "[ 0  1  0  0  0  0  1  0  0]\n",
       "[ 0  0  1  1  0  0  0  1  0]\n",
       "[ 1  0  0  0  0  0  1  0  0]\n",
       "[ 0  0  0  0  0  0  0  0  0]\n",
       "[ 0  0  0  0  1  1  0  0 -1]\n",
       "[ 0  0  1  0  0  0  0 -1  0]\n",
       "[ 0  0  0  0  0 -1  0  0 -1]\n",
       "[ 0  0  0  0  0  0  0  0  0]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " LaTeX code:\n",
      "Matrix: \\left(\\begin{array}{rrrrrrrrr}\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n",
      "0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 \\\\\n",
      "1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & -1 \\\\\n",
      "0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & -1 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\n",
      "\\end{array}\\right)\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -5, t = 8, s+t = 3 ; (j = 5, n = 3):\n",
      "\n",
      "                    d : Hom^(-5)(2 wedges of g, k) --> Hom^(-5)(3 wedges of g, k)\n",
      "                    \n",
      "matrix shape: (15, 6)\n",
      "\n",
      " Matrix:\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{rrrrrr}\n",
       "1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 1 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & -1 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & -1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 1 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 1 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "1 & 1 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 1 & 1 & 0 & 0 \\\\\n",
       "0 & -1 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 1 \\\\\n",
       "0 & 0 & 1 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 1 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{rrrrrr}\n",
       "1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 1 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & -1 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & -1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 1 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 1 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "1 & 1 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 1 & 1 & 0 & 0 \\\\\n",
       "0 & -1 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 1 \\\\\n",
       "0 & 0 & 1 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 1 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[ 1  0  0  0  0  0]\n",
       "[ 0  1  0  0  0  0]\n",
       "[ 0  0 -1  0  0  0]\n",
       "[ 0  0  0 -1  0  0]\n",
       "[ 0  0  0  0  1  0]\n",
       "[ 0  0  0  0  0  1]\n",
       "[ 0  0  0  0  0  0]\n",
       "[ 1  1  0  0  0  0]\n",
       "[ 0  0  1  1  0  0]\n",
       "[ 0 -1  0  0  0  0]\n",
       "[ 0  0  0  0  0  0]\n",
       "[ 0  0  0  0  0  1]\n",
       "[ 0  0  1  0  0  0]\n",
       "[ 0  0  0  0  1  0]\n",
       "[ 0  0  0  0  0  0]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " LaTeX code:\n",
      "Matrix: \\left(\\begin{array}{rrrrrr}\n",
      "1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 1 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & -1 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & -1 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 1 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 1 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "1 & 1 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 1 & 1 & 0 & 0 \\\\\n",
      "0 & -1 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 1 \\\\\n",
      "0 & 0 & 1 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 1 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0\n",
      "\\end{array}\\right)\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -5, t = 9, s+t = 4 ; (j = 5, n = 4):\n",
      "\n",
      "                    d : Hom^(-5)(3 wedges of g, k) --> Hom^(-5)(4 wedges of g, k)\n",
      "                    \n",
      "matrix shape: (3, 15)\n",
      "\n",
      " Matrix:\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{rrrrrrrrrrrrrrr}\n",
       "-1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & -1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & -1 & 0\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{rrrrrrrrrrrrrrr}\n",
       "-1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & -1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & -1 & 0\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[-1  0  0  0  0  0  0  1  0  1  0  0  0  0  0]\n",
       "[ 0  0  0  1  0  0  0  0  1  0  0  0 -1  0  0]\n",
       "[ 0  0  0  0  1  1  0  0  0  0  0 -1  0 -1  0]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " LaTeX code:\n",
      "Matrix: \\left(\\begin{array}{rrrrrrrrrrrrrrr}\n",
      "-1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & -1 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & -1 & 0\n",
      "\\end{array}\\right)\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -6, t = 9, s+t = 3 ; (j = 6, n = 3):\n",
      "\n",
      "                    d : Hom^(-6)(2 wedges of g, k) --> Hom^(-6)(3 wedges of g, k)\n",
      "                    \n",
      "matrix shape: (19, 1)\n",
      "\n",
      " Matrix:\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{r}\n",
       "0 \\\\\n",
       "0 \\\\\n",
       "0 \\\\\n",
       "0 \\\\\n",
       "-1 \\\\\n",
       "1 \\\\\n",
       "0 \\\\\n",
       "0 \\\\\n",
       "1 \\\\\n",
       "0 \\\\\n",
       "0 \\\\\n",
       "0 \\\\\n",
       "0 \\\\\n",
       "-1 \\\\\n",
       "0 \\\\\n",
       "0 \\\\\n",
       "0 \\\\\n",
       "0 \\\\\n",
       "0\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{r}\n",
       "0 \\\\\n",
       "0 \\\\\n",
       "0 \\\\\n",
       "0 \\\\\n",
       "-1 \\\\\n",
       "1 \\\\\n",
       "0 \\\\\n",
       "0 \\\\\n",
       "1 \\\\\n",
       "0 \\\\\n",
       "0 \\\\\n",
       "0 \\\\\n",
       "0 \\\\\n",
       "-1 \\\\\n",
       "0 \\\\\n",
       "0 \\\\\n",
       "0 \\\\\n",
       "0 \\\\\n",
       "0\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[ 0]\n",
       "[ 0]\n",
       "[ 0]\n",
       "[ 0]\n",
       "[-1]\n",
       "[ 1]\n",
       "[ 0]\n",
       "[ 0]\n",
       "[ 1]\n",
       "[ 0]\n",
       "[ 0]\n",
       "[ 0]\n",
       "[ 0]\n",
       "[-1]\n",
       "[ 0]\n",
       "[ 0]\n",
       "[ 0]\n",
       "[ 0]\n",
       "[ 0]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " LaTeX code:\n",
      "Matrix: \\left(\\begin{array}{r}\n",
      "0 \\\\\n",
      "0 \\\\\n",
      "0 \\\\\n",
      "0 \\\\\n",
      "-1 \\\\\n",
      "1 \\\\\n",
      "0 \\\\\n",
      "0 \\\\\n",
      "1 \\\\\n",
      "0 \\\\\n",
      "0 \\\\\n",
      "0 \\\\\n",
      "0 \\\\\n",
      "-1 \\\\\n",
      "0 \\\\\n",
      "0 \\\\\n",
      "0 \\\\\n",
      "0 \\\\\n",
      "0\n",
      "\\end{array}\\right)\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -6, t = 10, s+t = 4 ; (j = 6, n = 4):\n",
      "\n",
      "                    d : Hom^(-6)(3 wedges of g, k) --> Hom^(-6)(4 wedges of g, k)\n",
      "                    \n",
      "matrix shape: (11, 19)\n",
      "\n",
      " Matrix:\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{rrrrrrrrrrrrrrrrrrr}\n",
       "0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\n",
       "0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{rrrrrrrrrrrrrrrrrrr}\n",
       "0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\n",
       "0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[ 0  0  0  0 -1  0  0  0 -1  0  0  0 -1  0  0  0  0  0  0]\n",
       "[ 0  0  0  0  0 -1  0  0  0 -1  0  0  0 -1  0  0  0  0  0]\n",
       "[ 0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0]\n",
       "[ 0  0  0  0  0  0  1  1  0  0  0  0  0  0  0  0  0  0  0]\n",
       "[ 0  0  0  0  0 -1  0  0  1  1  0  0  0  0  0  0  0  0  1]\n",
       "[ 0  0 -1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0]\n",
       "[ 0  0  0  0 -1  0  0  0  0  0  0  0  1  1  0  0  0  0  1]\n",
       "[ 0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  1  0  0  0]\n",
       "[ 0  0  0  0  0  0  0  0 -1  0  0  0  0 -1  0  0  0  0  1]\n",
       "[ 0  0  0  0  0  0  0  0  0  0 -1  0  0  0  0  0  0  0  0]\n",
       "[ 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " LaTeX code:\n",
      "Matrix: \\left(\\begin{array}{rrrrrrrrrrrrrrrrrrr}\n",
      "0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\n",
      "0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\n",
      "\\end{array}\\right)\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -7, t = 11, s+t = 4 ; (j = 7, n = 4):\n",
      "\n",
      "                    d : Hom^(-7)(3 wedges of g, k) --> Hom^(-7)(4 wedges of g, k)\n",
      "                    \n",
      "matrix shape: (21, 9)\n",
      "\n",
      " Matrix:\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{rrrrrrrrr}\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "-1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\n",
       "0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & 0 \\\\\n",
       "0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\\\\n",
       "0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\\\\n",
       "0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{rrrrrrrrr}\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "-1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\n",
       "0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & 0 \\\\\n",
       "0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\\\\n",
       "0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\\\\n",
       "0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "21 x 9 dense matrix over Integer Ring (use the '.str()' method to see the entries)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " LaTeX code:\n",
      "Matrix: \\left(\\begin{array}{rrrrrrrrr}\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "-1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\n",
      "0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & 0 \\\\\n",
      "0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\\\\n",
      "0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\\\\n",
      "0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\n",
      "\\end{array}\\right)\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -7, t = 12, s+t = 5 ; (j = 7, n = 5):\n",
      "\n",
      "                    d : Hom^(-7)(4 wedges of g, k) --> Hom^(-7)(5 wedges of g, k)\n",
      "                    \n",
      "matrix shape: (3, 21)\n",
      "\n",
      " Matrix:\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{rrrrrrrrrrrrrrrrrrrrr}\n",
       "0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & -1\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{rrrrrrrrrrrrrrrrrrrrr}\n",
       "0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & -1\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[ 0  0  1  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0 -1  0  0]\n",
       "[ 0  0  0  0  1  0  0  0  0  0  0  0  1  1  0  0  0  0  0  1  0]\n",
       "[ 0  0  0  0  0  0  0  0  0  0  0 -1  0  0  1  1  0  0  0  0 -1]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " LaTeX code:\n",
      "Matrix: \\left(\\begin{array}{rrrrrrrrrrrrrrrrrrrrr}\n",
      "0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & -1\n",
      "\\end{array}\\right)\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -8, t = 12, s+t = 4 ; (j = 8, n = 4):\n",
      "\n",
      "                    d : Hom^(-8)(3 wedges of g, k) --> Hom^(-8)(4 wedges of g, k)\n",
      "                    \n",
      "matrix shape: (21, 3)\n",
      "\n",
      " Matrix:\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{rrr}\n",
       "1 & 0 & 0 \\\\\n",
       "0 & -1 & 0 \\\\\n",
       "0 & 0 & 1 \\\\\n",
       "0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 \\\\\n",
       "-1 & 0 & 0 \\\\\n",
       "1 & 0 & 0 \\\\\n",
       "0 & -1 & 0 \\\\\n",
       "0 & 1 & 0 \\\\\n",
       "1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 \\\\\n",
       "0 & 0 & -1 \\\\\n",
       "0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 \\\\\n",
       "0 & 1 & 0 \\\\\n",
       "0 & 0 & 0 \\\\\n",
       "0 & 0 & 1 \\\\\n",
       "0 & 0 & 0 \\\\\n",
       "0 & 0 & 0\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{rrr}\n",
       "1 & 0 & 0 \\\\\n",
       "0 & -1 & 0 \\\\\n",
       "0 & 0 & 1 \\\\\n",
       "0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 \\\\\n",
       "-1 & 0 & 0 \\\\\n",
       "1 & 0 & 0 \\\\\n",
       "0 & -1 & 0 \\\\\n",
       "0 & 1 & 0 \\\\\n",
       "1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 \\\\\n",
       "0 & 0 & -1 \\\\\n",
       "0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 \\\\\n",
       "0 & 1 & 0 \\\\\n",
       "0 & 0 & 0 \\\\\n",
       "0 & 0 & 1 \\\\\n",
       "0 & 0 & 0 \\\\\n",
       "0 & 0 & 0\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "21 x 3 dense matrix over Integer Ring (use the '.str()' method to see the entries)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " LaTeX code:\n",
      "Matrix: \\left(\\begin{array}{rrr}\n",
      "1 & 0 & 0 \\\\\n",
      "0 & -1 & 0 \\\\\n",
      "0 & 0 & 1 \\\\\n",
      "0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 \\\\\n",
      "-1 & 0 & 0 \\\\\n",
      "1 & 0 & 0 \\\\\n",
      "0 & -1 & 0 \\\\\n",
      "0 & 1 & 0 \\\\\n",
      "1 & 0 & 0 \\\\\n",
      "0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 \\\\\n",
      "0 & 0 & -1 \\\\\n",
      "0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 \\\\\n",
      "0 & 1 & 0 \\\\\n",
      "0 & 0 & 0 \\\\\n",
      "0 & 0 & 1 \\\\\n",
      "0 & 0 & 0 \\\\\n",
      "0 & 0 & 0\n",
      "\\end{array}\\right)\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -8, t = 13, s+t = 5 ; (j = 8, n = 5):\n",
      "\n",
      "                    d : Hom^(-8)(4 wedges of g, k) --> Hom^(-8)(5 wedges of g, k)\n",
      "                    \n",
      "matrix shape: (9, 21)\n",
      "\n",
      " Matrix:\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{rrrrrrrrrrrrrrrrrrrrr}\n",
       "0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "-1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & -1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{rrrrrrrrrrrrrrrrrrrrr}\n",
       "0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "-1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & -1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[ 0  0  0  0  0  1  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0]\n",
       "[-1  0  0  0  0  0  1  0  0  0  1  0  0  0  0  0  0  0  0  0  0]\n",
       "[ 0 -1  0  0  0  0  0  1  0  0  0  0  0  0  0 -1  0  0  0  0  0]\n",
       "[ 0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0 -1  0  0  0  0]\n",
       "[ 0  0 -1  0  0  0  0  0  0  0  0  0  0 -1  0  0  0 -1  0  0  0]\n",
       "[ 0  0  1  0  0  0  0  0  0  0  0  0  0  0 -1  0  0  0 -1  0  0]\n",
       "[ 0  0  0  0  0  0 -1  0  0  1  1  0  0  0  0  0  0  0  0  0  0]\n",
       "[ 0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  1  1  0  0  0  0]\n",
       "[ 0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0  0  0  1  0  0]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " LaTeX code:\n",
      "Matrix: \\left(\\begin{array}{rrrrrrrrrrrrrrrrrrrrr}\n",
      "0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "-1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & -1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0\n",
      "\\end{array}\\right)\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -9, t = 14, s+t = 5 ; (j = 9, n = 5):\n",
      "\n",
      "                    d : Hom^(-9)(4 wedges of g, k) --> Hom^(-9)(5 wedges of g, k)\n",
      "                    \n",
      "matrix shape: (19, 11)\n",
      "\n",
      " Matrix:\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{rrrrrrrrrrr}\n",
       "0 & 0 & -1 & 0 & -1 & 0 & -1 & 0 & 0 & 0 & 0 \\\\\n",
       "-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 0 \\\\\n",
       "0 & 0 & -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{rrrrrrrrrrr}\n",
       "0 & 0 & -1 & 0 & -1 & 0 & -1 & 0 & 0 & 0 & 0 \\\\\n",
       "-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 0 \\\\\n",
       "0 & 0 & -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[ 0  0 -1  0 -1  0 -1  0  0  0  0]\n",
       "[-1  0  0  0  0  0  0  0  0  0  0]\n",
       "[ 0  0  0  0  0  0  0  0  0  0  0]\n",
       "[ 0  0  0 -1  0  0  0  0  0  0  0]\n",
       "[ 0  0  0  1  0  0  0  0  0  0  0]\n",
       "[ 0  0  1  0 -1  0  0  0  0  1  0]\n",
       "[ 0  0  0  0  1  0  0  0  0  0  1]\n",
       "[ 0  0  0  0  0  0  0  0  0  0  0]\n",
       "[ 0 -1  0  0  0  0  0  0  0  0  0]\n",
       "[ 0  0  0  0  0  0 -1  0  0  1  0]\n",
       "[ 0  0 -1  0  0  0  1  0  0  0  1]\n",
       "[ 0  0  0  0  0  0  0 -1  0  0  0]\n",
       "[ 0  0  0  0  0  0  0  1  0  0  0]\n",
       "[ 0  0  0  0  0  0  1  0  0  1  0]\n",
       "[ 0  0  0  0 -1  0  0  0  0  0  1]\n",
       "[ 0  0  0  0  0  0  0  0  0  0  0]\n",
       "[ 0  0  0  0  0 -1  0  0  0  0  0]\n",
       "[ 0  0  0  0  0  0  0  0 -1  0  0]\n",
       "[ 0  0  0  0  0  0  0  0  0  0  0]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " LaTeX code:\n",
      "Matrix: \\left(\\begin{array}{rrrrrrrrrrr}\n",
      "0 & 0 & -1 & 0 & -1 & 0 & -1 & 0 & 0 & 0 & 0 \\\\\n",
      "-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 0 \\\\\n",
      "0 & 0 & -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\n",
      "\\end{array}\\right)\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -9, t = 15, s+t = 6 ; (j = 9, n = 6):\n",
      "\n",
      "                    d : Hom^(-9)(5 wedges of g, k) --> Hom^(-9)(6 wedges of g, k)\n",
      "                    \n",
      "matrix shape: (1, 19)\n",
      "\n",
      " Matrix:\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{rrrrrrrrrrrrrrrrrrr}\n",
       "0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{rrrrrrrrrrrrrrrrrrr}\n",
       "0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[ 0  0  0  0  0 -1  0  0  0  0 -1  0  0  1  1  0  0  0  0]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " LaTeX code:\n",
      "Matrix: \\left(\\begin{array}{rrrrrrrrrrrrrrrrrrr}\n",
      "0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0\n",
      "\\end{array}\\right)\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -10, t = 15, s+t = 5 ; (j = 10, n = 5):\n",
      "\n",
      "                    d : Hom^(-10)(4 wedges of g, k) --> Hom^(-10)(5 wedges of g, k)\n",
      "                    \n",
      "matrix shape: (15, 3)\n",
      "\n",
      " Matrix:\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{rrr}\n",
       "0 & 0 & 0 \\\\\n",
       "1 & 0 & 0 \\\\\n",
       "0 & 1 & 0 \\\\\n",
       "1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 \\\\\n",
       "0 & 0 & -1 \\\\\n",
       "0 & -1 & 0 \\\\\n",
       "0 & 0 & -1 \\\\\n",
       "0 & 0 & 0 \\\\\n",
       "-1 & 0 & 0 \\\\\n",
       "1 & 0 & 0 \\\\\n",
       "0 & -1 & 0 \\\\\n",
       "0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 \\\\\n",
       "0 & 0 & -1\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{rrr}\n",
       "0 & 0 & 0 \\\\\n",
       "1 & 0 & 0 \\\\\n",
       "0 & 1 & 0 \\\\\n",
       "1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 \\\\\n",
       "0 & 0 & -1 \\\\\n",
       "0 & -1 & 0 \\\\\n",
       "0 & 0 & -1 \\\\\n",
       "0 & 0 & 0 \\\\\n",
       "-1 & 0 & 0 \\\\\n",
       "1 & 0 & 0 \\\\\n",
       "0 & -1 & 0 \\\\\n",
       "0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 \\\\\n",
       "0 & 0 & -1\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[ 0  0  0]\n",
       "[ 1  0  0]\n",
       "[ 0  1  0]\n",
       "[ 1  0  0]\n",
       "[ 0  0  0]\n",
       "[ 0  0 -1]\n",
       "[ 0 -1  0]\n",
       "[ 0  0 -1]\n",
       "[ 0  0  0]\n",
       "[-1  0  0]\n",
       "[ 1  0  0]\n",
       "[ 0 -1  0]\n",
       "[ 0  0  0]\n",
       "[ 0  0  0]\n",
       "[ 0  0 -1]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " LaTeX code:\n",
      "Matrix: \\left(\\begin{array}{rrr}\n",
      "0 & 0 & 0 \\\\\n",
      "1 & 0 & 0 \\\\\n",
      "0 & 1 & 0 \\\\\n",
      "1 & 0 & 0 \\\\\n",
      "0 & 0 & 0 \\\\\n",
      "0 & 0 & -1 \\\\\n",
      "0 & -1 & 0 \\\\\n",
      "0 & 0 & -1 \\\\\n",
      "0 & 0 & 0 \\\\\n",
      "-1 & 0 & 0 \\\\\n",
      "1 & 0 & 0 \\\\\n",
      "0 & -1 & 0 \\\\\n",
      "0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 \\\\\n",
      "0 & 0 & -1\n",
      "\\end{array}\\right)\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -10, t = 16, s+t = 6 ; (j = 10, n = 6):\n",
      "\n",
      "                    d : Hom^(-10)(5 wedges of g, k) --> Hom^(-10)(6 wedges of g, k)\n",
      "                    \n",
      "matrix shape: (6, 15)\n",
      "\n",
      " Matrix:\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{rrrrrrrrrrrrrrr}\n",
       "0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & -1\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{rrrrrrrrrrrrrrr}\n",
       "0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & -1\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[ 0  0  0 -1  0  0  0  0  0 -1  0  0  0  0  0]\n",
       "[ 0  1  0  0  0  0  0  0  0  0 -1  0  0  0  0]\n",
       "[ 0  0  0  0  0  0 -1  0  0  0  0  1  0  0  0]\n",
       "[ 0  0  1  0  0  0  1  0  0  0  0  0  1  0  0]\n",
       "[ 0  0  0  0  0  1  0 -1  0  0  0  0  0 -1  0]\n",
       "[ 0  0  0  0  0  0  0  1  0  0  0  0  0  0 -1]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " LaTeX code:\n",
      "Matrix: \\left(\\begin{array}{rrrrrrrrrrrrrrr}\n",
      "0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & -1\n",
      "\\end{array}\\right)\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -11, t = 17, s+t = 6 ; (j = 11, n = 6):\n",
      "\n",
      "                    d : Hom^(-11)(5 wedges of g, k) --> Hom^(-11)(6 wedges of g, k)\n",
      "                    \n",
      "matrix shape: (9, 9)\n",
      "\n",
      " Matrix:\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{rrrrrrrrr}\n",
       "0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 1 & 0 & 0 & 0 & -1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & -1 & 0 & -1 & 0 \\\\\n",
       "0 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\\\\n",
       "0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{rrrrrrrrr}\n",
       "0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 1 & 0 & 0 & 0 & -1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & -1 & 0 & -1 & 0 \\\\\n",
       "0 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\\\\n",
       "0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\\\\n",
       "0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[ 0  1  0  1  0  0  0  0  0]\n",
       "[ 0  0  1  0  0  0 -1  0  0]\n",
       "[ 0  0  0  0  0 -1  0 -1  0]\n",
       "[ 0  1  0 -1  0  0  0  0  0]\n",
       "[ 0  0  0  1  0  0  0  0  0]\n",
       "[ 0  0  0  0  0  0 -1  0  0]\n",
       "[ 0  0  1  0  0  0  1  0  0]\n",
       "[ 0  0  0  0  0  0  0 -1  0]\n",
       "[ 0  0  0  0  0  1  0  0  0]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " LaTeX code:\n",
      "Matrix: \\left(\\begin{array}{rrrrrrrrr}\n",
      "0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 1 & 0 & 0 & 0 & -1 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & -1 & 0 & -1 & 0 \\\\\n",
      "0 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\\\\n",
      "0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\\\\n",
      "0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\n",
      "\\end{array}\\right)\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -12, t = 18, s+t = 6 ; (j = 12, n = 6):\n",
      "\n",
      "                    d : Hom^(-12)(5 wedges of g, k) --> Hom^(-12)(6 wedges of g, k)\n",
      "                    \n",
      "matrix shape: (9, 1)\n",
      "\n",
      " Matrix:\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{r}\n",
       "0 \\\\\n",
       "0 \\\\\n",
       "1 \\\\\n",
       "0 \\\\\n",
       "1 \\\\\n",
       "0 \\\\\n",
       "1 \\\\\n",
       "0 \\\\\n",
       "0\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{r}\n",
       "0 \\\\\n",
       "0 \\\\\n",
       "1 \\\\\n",
       "0 \\\\\n",
       "1 \\\\\n",
       "0 \\\\\n",
       "1 \\\\\n",
       "0 \\\\\n",
       "0\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[0]\n",
       "[0]\n",
       "[1]\n",
       "[0]\n",
       "[1]\n",
       "[0]\n",
       "[1]\n",
       "[0]\n",
       "[0]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " LaTeX code:\n",
      "Matrix: \\left(\\begin{array}{r}\n",
      "0 \\\\\n",
      "0 \\\\\n",
      "1 \\\\\n",
      "0 \\\\\n",
      "1 \\\\\n",
      "0 \\\\\n",
      "1 \\\\\n",
      "0 \\\\\n",
      "0\n",
      "\\end{array}\\right)\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -12, t = 19, s+t = 7 ; (j = 12, n = 7):\n",
      "\n",
      "                    d : Hom^(-12)(6 wedges of g, k) --> Hom^(-12)(7 wedges of g, k)\n",
      "                    \n",
      "matrix shape: (2, 9)\n",
      "\n",
      " Matrix:\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{rrrrrrrrr}\n",
       "0 & 0 & 0 & 0 & 1 & 0 & -1 & 0 & 0 \\\\\n",
       "0 & 0 & -1 & 0 & 0 & 0 & 1 & 0 & 0\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{rrrrrrrrr}\n",
       "0 & 0 & 0 & 0 & 1 & 0 & -1 & 0 & 0 \\\\\n",
       "0 & 0 & -1 & 0 & 0 & 0 & 1 & 0 & 0\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[ 0  0  0  0  1  0 -1  0  0]\n",
       "[ 0  0 -1  0  0  0  1  0  0]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " LaTeX code:\n",
      "Matrix: \\left(\\begin{array}{rrrrrrrrr}\n",
      "0 & 0 & 0 & 0 & 1 & 0 & -1 & 0 & 0 \\\\\n",
      "0 & 0 & -1 & 0 & 0 & 0 & 1 & 0 & 0\n",
      "\\end{array}\\right)\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -13, t = 20, s+t = 7 ; (j = 13, n = 7):\n",
      "\n",
      "                    d : Hom^(-13)(6 wedges of g, k) --> Hom^(-13)(7 wedges of g, k)\n",
      "                    \n",
      "matrix shape: (3, 3)\n",
      "\n",
      " Matrix:\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{rrr}\n",
       "-1 & 0 & 0 \\\\\n",
       "0 & 1 & 0 \\\\\n",
       "0 & 0 & -1\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{rrr}\n",
       "-1 & 0 & 0 \\\\\n",
       "0 & 1 & 0 \\\\\n",
       "0 & 0 & -1\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[-1  0  0]\n",
       "[ 0  1  0]\n",
       "[ 0  0 -1]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " LaTeX code:\n",
      "Matrix: \\left(\\begin{array}{rrr}\n",
      "-1 & 0 & 0 \\\\\n",
      "0 & 1 & 0 \\\\\n",
      "0 & 0 & -1\n",
      "\\end{array}\\right)\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n"
     ]
    }
   ],
   "source": [
    "print_cochain_matrices(sl3_d_matrices)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "id": "0647e61f",
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "        ===================================================\n",
      "        Non trivial kernel and image bases:\n",
      "        =================================================== \n",
      "\n",
      "        \n",
      "s = -1, t = 3, s+t = 2 ; (j = 1, n = 2):\n",
      "\n",
      "                    d : Hom^(-1)(1 wedges of g, k) --> Hom^(-1)(2 wedges of g, k)\n",
      "                    \n",
      "kernel basis:\n",
      "    [\n",
      "(1, 0, 0),\n",
      "(0, 1, 0),\n",
      "(0, 0, 1)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -2, t = 4, s+t = 2 ; (j = 2, n = 2):\n",
      "\n",
      "                    d : Hom^(-2)(1 wedges of g, k) --> Hom^(-2)(2 wedges of g, k)\n",
      "                    \n",
      "image basis:\n",
      "    [\n",
      "(1, 0, 0),\n",
      "(0, 1, 0),\n",
      "(0, 0, 1)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -2, t = 5, s+t = 3 ; (j = 2, n = 3):\n",
      "\n",
      "                    d : Hom^(-2)(2 wedges of g, k) --> Hom^(-2)(3 wedges of g, k)\n",
      "                    \n",
      "kernel basis:\n",
      "    [\n",
      "(1, 0, 0),\n",
      "(0, 1, 0),\n",
      "(0, 0, 1)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -3, t = 5, s+t = 2 ; (j = 3, n = 2):\n",
      "\n",
      "                    d : Hom^(-3)(1 wedges of g, k) --> Hom^(-3)(2 wedges of g, k)\n",
      "                    \n",
      "image basis:\n",
      "    [\n",
      "(0, 0, 1, 0, 0, 0, -1, 0, 0),\n",
      "(0, 0, 0, 0, 1, 0, -1, 0, 0)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -3, t = 6, s+t = 3 ; (j = 3, n = 3):\n",
      "\n",
      "                    d : Hom^(-3)(2 wedges of g, k) --> Hom^(-3)(3 wedges of g, k)\n",
      "                    \n",
      "kernel basis:\n",
      "    [\n",
      "(1, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 1, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 1, 0, 0, 0, -1, 0, 0),\n",
      "(0, 0, 0, 1, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 1, 0, -1, 0, 0),\n",
      "(0, 0, 0, 0, 0, 1, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 1, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 1)\n",
      "]\n",
      "image basis:\n",
      "    [\n",
      "(1)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -3, t = 7, s+t = 4 ; (j = 3, n = 4):\n",
      "\n",
      "                    d : Hom^(-3)(3 wedges of g, k) --> Hom^(-3)(4 wedges of g, k)\n",
      "                    \n",
      "kernel basis:\n",
      "    [\n",
      "(1)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -4, t = 7, s+t = 3 ; (j = 4, n = 3):\n",
      "\n",
      "                    d : Hom^(-4)(2 wedges of g, k) --> Hom^(-4)(3 wedges of g, k)\n",
      "                    \n",
      "kernel basis:\n",
      "    [\n",
      "(1, 1, 0, 0, 0, 0, -1, 0, 0),\n",
      "(0, 0, 1, -2, 0, 0, 0, 1, 0),\n",
      "(0, 0, 0, 0, 2, -1, 0, 0, 1)\n",
      "]\n",
      "image basis:\n",
      "    [\n",
      "(0, 1, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 1, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 1, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 1, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 1, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 1, 0)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -4, t = 8, s+t = 4 ; (j = 4, n = 4):\n",
      "\n",
      "                    d : Hom^(-4)(3 wedges of g, k) --> Hom^(-4)(4 wedges of g, k)\n",
      "                    \n",
      "kernel basis:\n",
      "    [\n",
      "(1, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 1, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 1, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 1, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 1, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 1, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 1, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 1, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 1)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -5, t = 8, s+t = 3 ; (j = 5, n = 3):\n",
      "\n",
      "                    d : Hom^(-5)(2 wedges of g, k) --> Hom^(-5)(3 wedges of g, k)\n",
      "                    \n",
      "image basis:\n",
      "    [\n",
      "(1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 1, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0),\n",
      "(0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0),\n",
      "(0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0),\n",
      "(0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -5, t = 9, s+t = 4 ; (j = 5, n = 4):\n",
      "\n",
      "                    d : Hom^(-5)(3 wedges of g, k) --> Hom^(-5)(4 wedges of g, k)\n",
      "                    \n",
      "kernel basis:\n",
      "    [\n",
      "(1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0),\n",
      "(0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0),\n",
      "(0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0),\n",
      "(0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0),\n",
      "(0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1)\n",
      "]\n",
      "image basis:\n",
      "    [\n",
      "(1, 0, 0),\n",
      "(0, 1, 0),\n",
      "(0, 0, 1)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -5, t = 10, s+t = 5 ; (j = 5, n = 5):\n",
      "\n",
      "                    d : Hom^(-5)(4 wedges of g, k) --> Hom^(-5)(5 wedges of g, k)\n",
      "                    \n",
      "kernel basis:\n",
      "    [\n",
      "(1, 0, 0),\n",
      "(0, 1, 0),\n",
      "(0, 0, 1)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -6, t = 9, s+t = 3 ; (j = 6, n = 3):\n",
      "\n",
      "                    d : Hom^(-6)(2 wedges of g, k) --> Hom^(-6)(3 wedges of g, k)\n",
      "                    \n",
      "image basis:\n",
      "    [\n",
      "(0, 0, 0, 0, 1, -1, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -6, t = 10, s+t = 4 ; (j = 6, n = 4):\n",
      "\n",
      "                    d : Hom^(-6)(3 wedges of g, k) --> Hom^(-6)(4 wedges of g, k)\n",
      "                    \n",
      "kernel basis:\n",
      "    [\n",
      "(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1, 1, 0, 0, 0, 0, 1),\n",
      "(0, 0, 0, 0, 0, 1, 0, 0, 1, -1, 0, 0, -1, 0, 0, 0, 0, 0, 1),\n",
      "(0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0)\n",
      "]\n",
      "image basis:\n",
      "    [\n",
      "(1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0),\n",
      "(0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0),\n",
      "(0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0),\n",
      "(0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -6, t = 11, s+t = 5 ; (j = 6, n = 5):\n",
      "\n",
      "                    d : Hom^(-6)(4 wedges of g, k) --> Hom^(-6)(5 wedges of g, k)\n",
      "                    \n",
      "kernel basis:\n",
      "    [\n",
      "(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -7, t = 11, s+t = 4 ; (j = 7, n = 4):\n",
      "\n",
      "                    d : Hom^(-7)(3 wedges of g, k) --> Hom^(-7)(4 wedges of g, k)\n",
      "                    \n",
      "image basis:\n",
      "    [\n",
      "(0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0),\n",
      "(0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 1, 0),\n",
      "(0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0),\n",
      "(0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 0, 0, 0, 0, 0, 1, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -1, 0, 0, 0, 0, 1)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -7, t = 12, s+t = 5 ; (j = 7, n = 5):\n",
      "\n",
      "                    d : Hom^(-7)(4 wedges of g, k) --> Hom^(-7)(5 wedges of g, k)\n",
      "                    \n",
      "kernel basis:\n",
      "    [\n",
      "(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0),\n",
      "(0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0),\n",
      "(0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0)\n",
      "]\n",
      "image basis:\n",
      "    [\n",
      "(1, 0, 0),\n",
      "(0, 1, 0),\n",
      "(0, 0, 1)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -7, t = 13, s+t = 6 ; (j = 7, n = 6):\n",
      "\n",
      "                    d : Hom^(-7)(5 wedges of g, k) --> Hom^(-7)(6 wedges of g, k)\n",
      "                    \n",
      "kernel basis:\n",
      "    [\n",
      "(1, 0, 0),\n",
      "(0, 1, 0),\n",
      "(0, 0, 1)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -8, t = 12, s+t = 4 ; (j = 8, n = 4):\n",
      "\n",
      "                    d : Hom^(-8)(3 wedges of g, k) --> Hom^(-8)(4 wedges of g, k)\n",
      "                    \n",
      "image basis:\n",
      "    [\n",
      "(1, 0, 0, 0, 0, -1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 1, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0),\n",
      "(0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -8, t = 13, s+t = 5 ; (j = 8, n = 5):\n",
      "\n",
      "                    d : Hom^(-8)(4 wedges of g, k) --> Hom^(-8)(5 wedges of g, k)\n",
      "                    \n",
      "kernel basis:\n",
      "    [\n",
      "(1, 0, 0, 0, 0, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0),\n",
      "(0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0),\n",
      "(0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 2, -1, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 1, -2, 0, 0, 0, 0, 0, 0, 1, -2, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, -1, -1, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1)\n",
      "]\n",
      "image basis:\n",
      "    [\n",
      "(1, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 1, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 1, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 1, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 1, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 1, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 1, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 1, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 1)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -8, t = 14, s+t = 6 ; (j = 8, n = 6):\n",
      "\n",
      "                    d : Hom^(-8)(5 wedges of g, k) --> Hom^(-8)(6 wedges of g, k)\n",
      "                    \n",
      "kernel basis:\n",
      "    [\n",
      "(1, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 1, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 1, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 1, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 1, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 1, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 1, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 1, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 1)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -9, t = 14, s+t = 5 ; (j = 9, n = 5):\n",
      "\n",
      "                    d : Hom^(-9)(4 wedges of g, k) --> Hom^(-9)(5 wedges of g, k)\n",
      "                    \n",
      "image basis:\n",
      "    [\n",
      "(1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0),\n",
      "(0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, -2, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -9, t = 15, s+t = 6 ; (j = 9, n = 6):\n",
      "\n",
      "                    d : Hom^(-9)(5 wedges of g, k) --> Hom^(-9)(6 wedges of g, k)\n",
      "                    \n",
      "kernel basis:\n",
      "    [\n",
      "(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1)\n",
      "]\n",
      "image basis:\n",
      "    [\n",
      "(1)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -9, t = 16, s+t = 7 ; (j = 9, n = 7):\n",
      "\n",
      "                    d : Hom^(-9)(6 wedges of g, k) --> Hom^(-9)(7 wedges of g, k)\n",
      "                    \n",
      "kernel basis:\n",
      "    [\n",
      "(1)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -10, t = 15, s+t = 5 ; (j = 10, n = 5):\n",
      "\n",
      "                    d : Hom^(-10)(4 wedges of g, k) --> Hom^(-10)(5 wedges of g, k)\n",
      "                    \n",
      "image basis:\n",
      "    [\n",
      "(0, 1, 0, 1, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0),\n",
      "(0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -10, t = 16, s+t = 6 ; (j = 10, n = 6):\n",
      "\n",
      "                    d : Hom^(-10)(5 wedges of g, k) --> Hom^(-10)(6 wedges of g, k)\n",
      "                    \n",
      "kernel basis:\n",
      "    [\n",
      "(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0),\n",
      "(0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0),\n",
      "(0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0),\n",
      "(0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, -1, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 1),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0)\n",
      "]\n",
      "image basis:\n",
      "    [\n",
      "(1, 0, 0, 0, 0, 0),\n",
      "(0, 1, 0, 0, 0, 0),\n",
      "(0, 0, 1, 0, 0, 0),\n",
      "(0, 0, 0, 1, 0, 0),\n",
      "(0, 0, 0, 0, 1, 0),\n",
      "(0, 0, 0, 0, 0, 1)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -10, t = 17, s+t = 7 ; (j = 10, n = 7):\n",
      "\n",
      "                    d : Hom^(-10)(6 wedges of g, k) --> Hom^(-10)(7 wedges of g, k)\n",
      "                    \n",
      "kernel basis:\n",
      "    [\n",
      "(1, 0, 0, 0, 0, 0),\n",
      "(0, 1, 0, 0, 0, 0),\n",
      "(0, 0, 1, 0, 0, 0),\n",
      "(0, 0, 0, 1, 0, 0),\n",
      "(0, 0, 0, 0, 1, 0),\n",
      "(0, 0, 0, 0, 0, 1)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -11, t = 17, s+t = 6 ; (j = 11, n = 6):\n",
      "\n",
      "                    d : Hom^(-11)(5 wedges of g, k) --> Hom^(-11)(6 wedges of g, k)\n",
      "                    \n",
      "kernel basis:\n",
      "    [\n",
      "(1, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 1, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 1)\n",
      "]\n",
      "image basis:\n",
      "    [\n",
      "(1, 0, 0, 1, 0, 0, 0, 0, 0),\n",
      "(0, 1, 0, 0, 0, 0, 1, 0, 0),\n",
      "(0, 0, 1, 0, 0, 0, 0, 0, -1),\n",
      "(0, 0, 0, 2, -1, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 1, -2, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 1, 1)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -11, t = 18, s+t = 7 ; (j = 11, n = 7):\n",
      "\n",
      "                    d : Hom^(-11)(6 wedges of g, k) --> Hom^(-11)(7 wedges of g, k)\n",
      "                    \n",
      "kernel basis:\n",
      "    [\n",
      "(1, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 1, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 1, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 1, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 1, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 1, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 1, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 1, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 1)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -12, t = 18, s+t = 6 ; (j = 12, n = 6):\n",
      "\n",
      "                    d : Hom^(-12)(5 wedges of g, k) --> Hom^(-12)(6 wedges of g, k)\n",
      "                    \n",
      "image basis:\n",
      "    [\n",
      "(0, 0, 1, 0, 1, 0, 1, 0, 0)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -12, t = 19, s+t = 7 ; (j = 12, n = 7):\n",
      "\n",
      "                    d : Hom^(-12)(6 wedges of g, k) --> Hom^(-12)(7 wedges of g, k)\n",
      "                    \n",
      "kernel basis:\n",
      "    [\n",
      "(1, 0, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 1, 0, 0, 0, 0, 0, 0, 0),\n",
      "(0, 0, 1, 0, 1, 0, 1, 0, 0),\n",
      "(0, 0, 0, 1, 0, 0, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 1, 0, 0, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 1, 0),\n",
      "(0, 0, 0, 0, 0, 0, 0, 0, 1)\n",
      "]\n",
      "image basis:\n",
      "    [\n",
      "(1, 0),\n",
      "(0, 1)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -12, t = 20, s+t = 8 ; (j = 12, n = 8):\n",
      "\n",
      "                    d : Hom^(-12)(7 wedges of g, k) --> Hom^(-12)(8 wedges of g, k)\n",
      "                    \n",
      "kernel basis:\n",
      "    [\n",
      "(1, 0),\n",
      "(0, 1)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -13, t = 20, s+t = 7 ; (j = 13, n = 7):\n",
      "\n",
      "                    d : Hom^(-13)(6 wedges of g, k) --> Hom^(-13)(7 wedges of g, k)\n",
      "                    \n",
      "image basis:\n",
      "    [\n",
      "(1, 0, 0),\n",
      "(0, 1, 0),\n",
      "(0, 0, 1)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -13, t = 21, s+t = 8 ; (j = 13, n = 8):\n",
      "\n",
      "                    d : Hom^(-13)(7 wedges of g, k) --> Hom^(-13)(8 wedges of g, k)\n",
      "                    \n",
      "kernel basis:\n",
      "    [\n",
      "(1, 0, 0),\n",
      "(0, 1, 0),\n",
      "(0, 0, 1)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -14, t = 22, s+t = 8 ; (j = 14, n = 8):\n",
      "\n",
      "                    d : Hom^(-14)(7 wedges of g, k) --> Hom^(-14)(8 wedges of g, k)\n",
      "                    \n",
      "kernel basis:\n",
      "    [\n",
      "(1, 0, 0),\n",
      "(0, 1, 0),\n",
      "(0, 0, 1)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n",
      "s = -15, t = 24, s+t = 9 ; (j = 15, n = 9):\n",
      "\n",
      "                    d : Hom^(-15)(8 wedges of g, k) --> Hom^(-15)(9 wedges of g, k)\n",
      "                    \n",
      "kernel basis:\n",
      "    [\n",
      "(1)\n",
      "]\n",
      "\n",
      " ------------------------------------------- \n",
      "\n",
      "\n"
     ]
    }
   ],
   "source": [
    "print_ker_im_bases(sl3_ker_im_bases)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "id": "76702b59",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "        ===================================================\n",
      "        Non trivial cohomology H^(s,t) spaces:\n",
      "        =================================================== \n",
      "\n",
      "        \n",
      "(s,t) = (0,0): dim H^(0,0) = 1\n",
      "\n",
      "H^(0,0) = Finitely generated module V/W over Integer Ring with invariants (0)\n",
      "---\n",
      "\n",
      "(s,t) = (-1,1): dim H^(-1,2) = 3\n",
      "\n",
      "H^(-1,2) = Finitely generated module V/W over Integer Ring with invariants (0, 0, 0)\n",
      "---\n",
      "\n",
      "(s,t) = (-3,3): dim H^(-3,5) = 6\n",
      "\n",
      "H^(-3,5) = Finitely generated module V/W over Integer Ring with invariants (0, 0, 0, 0, 0, 0)\n",
      "---\n",
      "\n",
      "(s,t) = (-4,4): dim H^(-4,6) = 3\n",
      "\n",
      "H^(-4,6) = Finitely generated module V/W over Integer Ring with invariants (0, 0, 0)\n",
      "---\n",
      "\n",
      "(s,t) = (-4,4): dim H^(-4,7) = 3\n",
      "\n",
      "H^(-4,7) = Finitely generated module V/W over Integer Ring with invariants (0, 0, 0)\n",
      "---\n",
      "\n",
      "(s,t) = (-5,5): dim H^(-5,8) = 6\n",
      "\n",
      "H^(-5,8) = Finitely generated module V/W over Integer Ring with invariants (0, 0, 0, 0, 0, 0)\n",
      "---\n",
      "\n",
      "(s,t) = (-6,6): dim H^(-6,9) = 7\n",
      "\n",
      "H^(-6,9) = Finitely generated module V/W over Integer Ring with invariants (0, 0, 0, 0, 0, 0, 0)\n",
      "---\n",
      "\n",
      "(s,t) = (-6,6): dim H^(-6,10) = 2\n",
      "\n",
      "H^(-6,10) = Finitely generated module V/W over Integer Ring with invariants (2, 2)\n",
      "---\n",
      "\n",
      "(s,t) = (-7,7): dim H^(-7,11) = 9\n",
      "\n",
      "H^(-7,11) = Finitely generated module V/W over Integer Ring with invariants (0, 0, 0, 0, 0, 0, 0, 0, 0)\n",
      "---\n",
      "\n",
      "(s,t) = (-8,8): dim H^(-8,12) = 9\n",
      "\n",
      "H^(-8,12) = Finitely generated module V/W over Integer Ring with invariants (0, 0, 0, 0, 0, 0, 0, 0, 0)\n",
      "---\n",
      "\n",
      "(s,t) = (-9,9): dim H^(-9,14) = 9\n",
      "\n",
      "H^(-9,14) = Finitely generated module V/W over Integer Ring with invariants (2, 2, 0, 0, 0, 0, 0, 0, 0)\n",
      "---\n",
      "\n",
      "(s,t) = (-10,10): dim H^(-10,15) = 6\n",
      "\n",
      "H^(-10,15) = Finitely generated module V/W over Integer Ring with invariants (0, 0, 0, 0, 0, 0)\n",
      "---\n",
      "\n",
      "(s,t) = (-11,11): dim H^(-11,16) = 3\n",
      "\n",
      "H^(-11,16) = Finitely generated module V/W over Integer Ring with invariants (0, 0, 0)\n",
      "---\n",
      "\n",
      "(s,t) = (-11,11): dim H^(-11,17) = 3\n",
      "\n",
      "H^(-11,17) = Finitely generated module V/W over Integer Ring with invariants (0, 0, 0)\n",
      "---\n",
      "\n",
      "(s,t) = (-12,12): dim H^(-12,18) = 6\n",
      "\n",
      "H^(-12,18) = Finitely generated module V/W over Integer Ring with invariants (0, 0, 0, 0, 0, 0)\n",
      "---\n",
      "\n",
      "(s,t) = (-14,14): dim H^(-14,21) = 3\n",
      "\n",
      "H^(-14,21) = Finitely generated module V/W over Integer Ring with invariants (0, 0, 0)\n",
      "---\n",
      "\n",
      "(s,t) = (-15,15): dim H^(-15,23) = 1\n",
      "\n",
      "H^(-15,23) = Finitely generated module V/W over Integer Ring with invariants (0)\n",
      "---\n",
      "\n"
     ]
    }
   ],
   "source": [
    "print_H_s_t(sl3_H_s_t_array)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "id": "0d11d81d",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "        ===================================================\n",
      "        Non trivial cohomology H^(s,t) spaces:\n",
      "        =================================================== \n",
      "\n",
      "        \n",
      "(s,t) = (0,0): dim H^(0,0) = 1\n",
      "\n",
      "H^(0,0) = Finitely generated module V/W over Integer Ring with invariants (0)\n",
      "---\n",
      "\n",
      "(s,t) = (-1,1): dim H^(-1,2) = 3\n",
      "\n",
      "H^(-1,2) = Finitely generated module V/W over Integer Ring with invariants (0, 0, 0)\n",
      "---\n",
      "\n",
      "(s,t) = (-3,3): dim H^(-3,5) = 6\n",
      "\n",
      "H^(-3,5) = Finitely generated module V/W over Integer Ring with invariants (0, 0, 0, 0, 0, 0)\n",
      "---\n",
      "\n",
      "(s,t) = (-4,4): dim H^(-4,6) = 3\n",
      "\n",
      "H^(-4,6) = Finitely generated module V/W over Integer Ring with invariants (0, 0, 0)\n",
      "---\n",
      "\n",
      "(s,t) = (-4,4): dim H^(-4,7) = 3\n",
      "\n",
      "H^(-4,7) = Finitely generated module V/W over Integer Ring with invariants (0, 0, 0)\n",
      "---\n",
      "\n",
      "(s,t) = (-5,5): dim H^(-5,8) = 6\n",
      "\n",
      "H^(-5,8) = Finitely generated module V/W over Integer Ring with invariants (0, 0, 0, 0, 0, 0)\n",
      "---\n",
      "\n",
      "(s,t) = (-6,6): dim H^(-6,9) = 7\n",
      "\n",
      "H^(-6,9) = Finitely generated module V/W over Integer Ring with invariants (0, 0, 0, 0, 0, 0, 0)\n",
      "---\n",
      "\n",
      "(s,t) = (-7,7): dim H^(-7,11) = 9\n",
      "\n",
      "H^(-7,11) = Finitely generated module V/W over Integer Ring with invariants (0, 0, 0, 0, 0, 0, 0, 0, 0)\n",
      "---\n",
      "\n",
      "(s,t) = (-8,8): dim H^(-8,12) = 9\n",
      "\n",
      "H^(-8,12) = Finitely generated module V/W over Integer Ring with invariants (0, 0, 0, 0, 0, 0, 0, 0, 0)\n",
      "---\n",
      "\n",
      "(s,t) = (-9,9): dim H^(-9,14) = 7\n",
      "\n",
      "H^(-9,14) = Finitely generated module V/W over Integer Ring with invariants (0, 0, 0, 0, 0, 0, 0)\n",
      "---\n",
      "\n",
      "(s,t) = (-10,10): dim H^(-10,15) = 6\n",
      "\n",
      "H^(-10,15) = Finitely generated module V/W over Integer Ring with invariants (0, 0, 0, 0, 0, 0)\n",
      "---\n",
      "\n",
      "(s,t) = (-11,11): dim H^(-11,16) = 3\n",
      "\n",
      "H^(-11,16) = Finitely generated module V/W over Integer Ring with invariants (0, 0, 0)\n",
      "---\n",
      "\n",
      "(s,t) = (-11,11): dim H^(-11,17) = 3\n",
      "\n",
      "H^(-11,17) = Finitely generated module V/W over Integer Ring with invariants (0, 0, 0)\n",
      "---\n",
      "\n",
      "(s,t) = (-12,12): dim H^(-12,18) = 6\n",
      "\n",
      "H^(-12,18) = Finitely generated module V/W over Integer Ring with invariants (0, 0, 0, 0, 0, 0)\n",
      "---\n",
      "\n",
      "(s,t) = (-14,14): dim H^(-14,21) = 3\n",
      "\n",
      "H^(-14,21) = Finitely generated module V/W over Integer Ring with invariants (0, 0, 0)\n",
      "---\n",
      "\n",
      "(s,t) = (-15,15): dim H^(-15,23) = 1\n",
      "\n",
      "H^(-15,23) = Finitely generated module V/W over Integer Ring with invariants (0)\n",
      "---\n",
      "\n"
     ]
    }
   ],
   "source": [
    "print_H_s_t(sl3_H_s_t_fixed_2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "id": "72d415d3",
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "        ===================================================\n",
      "        Bases of non trivial cohomology H^(s,t) spaces:\n",
      "        =================================================== \n",
      "\n",
      "        \n",
      "(s,t) = (0,0): dim H^(0,0) = 1\n",
      "\n",
      "basis in H^(0,0):\n",
      " ((1),)\n",
      "\n",
      "lift of basis to the ambient space:\n",
      " [(1)]\n",
      "---\n",
      "\n",
      "(s,t) = (-1,1): dim H^(-1,2) = 3\n",
      "\n",
      "basis in H^(-1,2):\n",
      " ((1, 0, 0), (0, 1, 0), (0, 0, 1))\n",
      "\n",
      "lift of basis to the ambient space:\n",
      " [(1, 0, 0), (0, 1, 0), (0, 0, 1)]\n",
      "---\n",
      "\n",
      "(s,t) = (-3,3): dim H^(-3,5) = 6\n",
      "\n",
      "basis in H^(-3,5):\n",
      " ((1, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0), (0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 1))\n",
      "\n",
      "lift of basis to the ambient space:\n",
      " [(1, 0, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 0, 1)]\n",
      "---\n",
      "\n",
      "(s,t) = (-4,4): dim H^(-4,6) = 3\n",
      "\n",
      "basis in H^(-4,6):\n",
      " ((1, 0, 0), (0, 1, 0), (0, 0, 1))\n",
      "\n",
      "lift of basis to the ambient space:\n",
      " [(1, 1, 0, 0, 0, 0, -1, 0, 0), (0, 0, 1, -2, 0, 0, 0, 1, 0), (0, 0, 0, 0, 2, -1, 0, 0, 1)]\n",
      "---\n",
      "\n",
      "(s,t) = (-4,4): dim H^(-4,7) = 3\n",
      "\n",
      "basis in H^(-4,7):\n",
      " ((1, 0, 0), (0, 1, 0), (0, 0, 1))\n",
      "\n",
      "lift of basis to the ambient space:\n",
      " [(1, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 0, 1)]\n",
      "---\n",
      "\n",
      "(s,t) = (-5,5): dim H^(-5,8) = 6\n",
      "\n",
      "basis in H^(-5,8):\n",
      " ((1, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0), (0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 1))\n",
      "\n",
      "lift of basis to the ambient space:\n",
      " [(1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1)]\n",
      "---\n",
      "\n",
      "(s,t) = (-6,6): dim H^(-6,9) = 7\n",
      "\n",
      "basis in H^(-6,9):\n",
      " ((1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1))\n",
      "\n",
      "lift of basis to the ambient space:\n",
      " [(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1, 1, 0, 0, 0, 0, 1), (0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0)]\n",
      "---\n",
      "\n",
      "(s,t) = (-7,7): dim H^(-7,11) = 9\n",
      "\n",
      "basis in H^(-7,11):\n",
      " ((1, 0, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 0, 1))\n",
      "\n",
      "lift of basis to the ambient space:\n",
      " [(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0)]\n",
      "---\n",
      "\n",
      "(s,t) = (-8,8): dim H^(-8,12) = 9\n",
      "\n",
      "basis in H^(-8,12):\n",
      " ((1, 0, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 0, 1))\n",
      "\n",
      "lift of basis to the ambient space:\n",
      " [(1, 0, 0, 0, 0, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1)]\n",
      "---\n",
      "\n",
      "(s,t) = (-9,9): dim H^(-9,14) = 7\n",
      "\n",
      "basis in H^(-9,14):\n",
      " ((1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1))\n",
      "\n",
      "lift of basis to the ambient space:\n",
      " [(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1)]\n",
      "---\n",
      "\n",
      "(s,t) = (-10,10): dim H^(-10,15) = 6\n",
      "\n",
      "basis in H^(-10,15):\n",
      " ((1, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0), (0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 1))\n",
      "\n",
      "lift of basis to the ambient space:\n",
      " [(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0)]\n",
      "---\n",
      "\n",
      "(s,t) = (-11,11): dim H^(-11,16) = 3\n",
      "\n",
      "basis in H^(-11,16):\n",
      " ((1, 0, 0), (0, 1, 0), (0, 0, 1))\n",
      "\n",
      "lift of basis to the ambient space:\n",
      " [(1, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 0, 1)]\n",
      "---\n",
      "\n",
      "(s,t) = (-11,11): dim H^(-11,17) = 3\n",
      "\n",
      "basis in H^(-11,17):\n",
      " ((1, 0, 0), (0, 1, 0), (0, 0, 1))\n",
      "\n",
      "lift of basis to the ambient space:\n",
      " [(1, 0, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0, 0)]\n",
      "---\n",
      "\n",
      "(s,t) = (-12,12): dim H^(-12,18) = 6\n",
      "\n",
      "basis in H^(-12,18):\n",
      " ((1, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0), (0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 1))\n",
      "\n",
      "lift of basis to the ambient space:\n",
      " [(1, 0, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 0, 1)]\n",
      "---\n",
      "\n",
      "(s,t) = (-14,14): dim H^(-14,21) = 3\n",
      "\n",
      "basis in H^(-14,21):\n",
      " ((1, 0, 0), (0, 1, 0), (0, 0, 1))\n",
      "\n",
      "lift of basis to the ambient space:\n",
      " [(1, 0, 0), (0, 1, 0), (0, 0, 1)]\n",
      "---\n",
      "\n",
      "(s,t) = (-15,15): dim H^(-15,23) = 1\n",
      "\n",
      "basis in H^(-15,23):\n",
      " ((1),)\n",
      "\n",
      "lift of basis to the ambient space:\n",
      " [(1)]\n",
      "---\n",
      "\n"
     ]
    }
   ],
   "source": [
    "print_H_s_t_bases(sl3_H_s_t_fixed_2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "28f72633",
   "metadata": {},
   "source": [
    "# Cup products:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "id": "f578d1d2",
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "        ===================================================\n",
      "        Non-zero cup products from H^(s,t) spaces:\n",
      "        =================================================== \n",
      "\n",
      "        \n",
      "NOTE: We are not printing any H^(0) cup products.\n",
      "\n",
      "We write vi^(s,t) for the i'th basis vector of H^(s,t).\n",
      "\n",
      "\n",
      "\n",
      "v1^(-1,2) cup v3^(-3,5) = 1*v1^(-4,7)\n",
      "\n",
      "v1^(-1,2) cup v5^(-3,5) = 1*v2^(-4,7)\n",
      "\n",
      "v2^(-1,2) cup v1^(-3,5) = -1*v1^(-4,7)\n",
      "\n",
      "v2^(-1,2) cup v6^(-3,5) = 1*v3^(-4,7)\n",
      "\n",
      "v3^(-1,2) cup v2^(-3,5) = -1*v2^(-4,7)\n",
      "\n",
      "v3^(-1,2) cup v4^(-3,5) = -1*v3^(-4,7)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-1,2) cup v1^(-4,6) = -1*v4^(-5,8)\n",
      "\n",
      "v1^(-1,2) cup v2^(-4,6) = -2*v1^(-5,8)\n",
      "\n",
      "v1^(-1,2) cup v3^(-4,6) = 2*v2^(-5,8)\n",
      "\n",
      "v2^(-1,2) cup v1^(-4,6) = -2*v1^(-5,8)\n",
      "\n",
      "v2^(-1,2) cup v2^(-4,6) = 1*v5^(-5,8)\n",
      "\n",
      "v2^(-1,2) cup v3^(-4,6) = -2*v3^(-5,8)\n",
      "\n",
      "v3^(-1,2) cup v1^(-4,6) = -2*v2^(-5,8)\n",
      "\n",
      "v3^(-1,2) cup v2^(-4,6) = 2*v3^(-5,8)\n",
      "\n",
      "v3^(-1,2) cup v3^(-4,6) = 1*v6^(-5,8)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-1,2) cup v3^(-6,9) = 3*v3^(-7,11)\n",
      "\n",
      "v1^(-1,2) cup v4^(-6,9) = 1*v1^(-7,11) + -1*v2^(-7,11)\n",
      "\n",
      "v1^(-1,2) cup v5^(-6,9) = -1*v4^(-7,11)\n",
      "\n",
      "v1^(-1,2) cup v6^(-6,9) = 1*v5^(-7,11) + -1*v6^(-7,11)\n",
      "\n",
      "v1^(-1,2) cup v7^(-6,9) = 1*v7^(-7,11)\n",
      "\n",
      "v2^(-1,2) cup v1^(-6,9) = -1*v1^(-7,11)\n",
      "\n",
      "v2^(-1,2) cup v2^(-6,9) = 1*v3^(-7,11)\n",
      "\n",
      "v2^(-1,2) cup v3^(-6,9) = 3*v4^(-7,11)\n",
      "\n",
      "v2^(-1,2) cup v6^(-6,9) = 1*v7^(-7,11)\n",
      "\n",
      "v2^(-1,2) cup v7^(-6,9) = 1*v8^(-7,11)\n",
      "\n",
      "v3^(-1,2) cup v1^(-6,9) = 1*v3^(-7,11)\n",
      "\n",
      "v3^(-1,2) cup v2^(-6,9) = -1*v6^(-7,11)\n",
      "\n",
      "v3^(-1,2) cup v3^(-6,9) = -3*v7^(-7,11)\n",
      "\n",
      "v3^(-1,2) cup v4^(-6,9) = 1*v4^(-7,11)\n",
      "\n",
      "v3^(-1,2) cup v5^(-6,9) = -1*v9^(-7,11)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-1,2) cup v1^(-8,12) = 1*v2^(-9,14)\n",
      "\n",
      "v1^(-1,2) cup v2^(-8,12) = -1*v4^(-9,14)\n",
      "\n",
      "v1^(-1,2) cup v3^(-8,12) = 3*v1^(-9,14)\n",
      "\n",
      "v1^(-1,2) cup v6^(-8,12) = 1*v3^(-9,14)\n",
      "\n",
      "v1^(-1,2) cup v7^(-8,12) = 1*v3^(-9,14)\n",
      "\n",
      "v1^(-1,2) cup v8^(-8,12) = 1*v5^(-9,14)\n",
      "\n",
      "v1^(-1,2) cup v9^(-8,12) = 1*v5^(-9,14)\n",
      "\n",
      "v2^(-1,2) cup v1^(-8,12) = -1*v3^(-9,14)\n",
      "\n",
      "v2^(-1,2) cup v2^(-8,12) = -3*v1^(-9,14)\n",
      "\n",
      "v2^(-1,2) cup v3^(-8,12) = -1*v6^(-9,14)\n",
      "\n",
      "v2^(-1,2) cup v5^(-8,12) = -1*v2^(-9,14)\n",
      "\n",
      "v2^(-1,2) cup v9^(-8,12) = 1*v7^(-9,14)\n",
      "\n",
      "v3^(-1,2) cup v1^(-8,12) = 3*v1^(-9,14)\n",
      "\n",
      "v3^(-1,2) cup v2^(-8,12) = -1*v5^(-9,14)\n",
      "\n",
      "v3^(-1,2) cup v3^(-8,12) = -1*v7^(-9,14)\n",
      "\n",
      "v3^(-1,2) cup v4^(-8,12) = -1*v4^(-9,14)\n",
      "\n",
      "v3^(-1,2) cup v6^(-8,12) = -1*v6^(-9,14)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-1,2) cup v3^(-10,15) = 2*v1^(-11,17)\n",
      "\n",
      "v1^(-1,2) cup v5^(-10,15) = -2*v2^(-11,17)\n",
      "\n",
      "v1^(-1,2) cup v6^(-10,15) = 1*v3^(-11,17)\n",
      "\n",
      "v2^(-1,2) cup v2^(-10,15) = 2*v1^(-11,17)\n",
      "\n",
      "v2^(-1,2) cup v4^(-10,15) = -1*v2^(-11,17)\n",
      "\n",
      "v2^(-1,2) cup v5^(-10,15) = -2*v3^(-11,17)\n",
      "\n",
      "v3^(-1,2) cup v1^(-10,15) = 1*v1^(-11,17)\n",
      "\n",
      "v3^(-1,2) cup v2^(-10,15) = 2*v2^(-11,17)\n",
      "\n",
      "v3^(-1,2) cup v3^(-10,15) = 2*v3^(-11,17)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-1,2) cup v2^(-11,16) = 1*v2^(-12,18)\n",
      "\n",
      "v1^(-1,2) cup v3^(-11,16) = 1*v4^(-12,18)\n",
      "\n",
      "v2^(-1,2) cup v1^(-11,16) = -1*v1^(-12,18)\n",
      "\n",
      "v2^(-1,2) cup v3^(-11,16) = 1*v6^(-12,18)\n",
      "\n",
      "v3^(-1,2) cup v1^(-11,16) = -1*v3^(-12,18)\n",
      "\n",
      "v3^(-1,2) cup v2^(-11,16) = -1*v5^(-12,18)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-1,2) cup v3^(-14,21) = 1*v1^(-15,23)\n",
      "\n",
      "v2^(-1,2) cup v2^(-14,21) = -1*v1^(-15,23)\n",
      "\n",
      "v3^(-1,2) cup v1^(-14,21) = 1*v1^(-15,23)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-3,5) cup v2^(-4,6) = -1*v1^(-7,11) + 2*v2^(-7,11)\n",
      "\n",
      "v1^(-3,5) cup v3^(-4,6) = 3*v3^(-7,11)\n",
      "\n",
      "v2^(-3,5) cup v2^(-4,6) = -3*v3^(-7,11)\n",
      "\n",
      "v2^(-3,5) cup v3^(-4,6) = -2*v5^(-7,11) + 1*v6^(-7,11)\n",
      "\n",
      "v3^(-3,5) cup v1^(-4,6) = 1*v1^(-7,11) + 1*v2^(-7,11)\n",
      "\n",
      "v3^(-3,5) cup v3^(-4,6) = 3*v4^(-7,11)\n",
      "\n",
      "v4^(-3,5) cup v1^(-4,6) = -3*v4^(-7,11)\n",
      "\n",
      "v4^(-3,5) cup v3^(-4,6) = -2*v8^(-7,11) + 1*v9^(-7,11)\n",
      "\n",
      "v5^(-3,5) cup v1^(-4,6) = 1*v5^(-7,11) + 1*v6^(-7,11)\n",
      "\n",
      "v5^(-3,5) cup v2^(-4,6) = 3*v7^(-7,11)\n",
      "\n",
      "v6^(-3,5) cup v1^(-4,6) = 3*v7^(-7,11)\n",
      "\n",
      "v6^(-3,5) cup v2^(-4,6) = 1*v8^(-7,11) + -2*v9^(-7,11)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-3,5) cup v3^(-6,9) = 1*v2^(-9,14)\n",
      "\n",
      "v1^(-3,5) cup v5^(-6,9) = -1*v3^(-9,14)\n",
      "\n",
      "v1^(-3,5) cup v6^(-6,9) = -1*v4^(-9,14)\n",
      "\n",
      "v1^(-3,5) cup v7^(-6,9) = 1*v1^(-9,14)\n",
      "\n",
      "v2^(-3,5) cup v3^(-6,9) = -1*v4^(-9,14)\n",
      "\n",
      "v2^(-3,5) cup v4^(-6,9) = -1*v2^(-9,14)\n",
      "\n",
      "v2^(-3,5) cup v5^(-6,9) = -1*v1^(-9,14)\n",
      "\n",
      "v2^(-3,5) cup v7^(-6,9) = -1*v5^(-9,14)\n",
      "\n",
      "v3^(-3,5) cup v2^(-6,9) = 1*v2^(-9,14)\n",
      "\n",
      "v3^(-3,5) cup v3^(-6,9) = 1*v3^(-9,14)\n",
      "\n",
      "v3^(-3,5) cup v6^(-6,9) = -1*v1^(-9,14)\n",
      "\n",
      "v3^(-3,5) cup v7^(-6,9) = -1*v6^(-9,14)\n",
      "\n",
      "v4^(-3,5) cup v1^(-6,9) = -1*v3^(-9,14)\n",
      "\n",
      "v4^(-3,5) cup v2^(-6,9) = -1*v1^(-9,14)\n",
      "\n",
      "v4^(-3,5) cup v3^(-6,9) = -1*v6^(-9,14)\n",
      "\n",
      "v4^(-3,5) cup v6^(-6,9) = -1*v7^(-9,14)\n",
      "\n",
      "v5^(-3,5) cup v1^(-6,9) = -1*v4^(-9,14)\n",
      "\n",
      "v5^(-3,5) cup v3^(-6,9) = 1*v5^(-9,14)\n",
      "\n",
      "v5^(-3,5) cup v4^(-6,9) = -1*v1^(-9,14)\n",
      "\n",
      "v5^(-3,5) cup v5^(-6,9) = 1*v7^(-9,14)\n",
      "\n",
      "v6^(-3,5) cup v1^(-6,9) = 1*v1^(-9,14)\n",
      "\n",
      "v6^(-3,5) cup v2^(-6,9) = -1*v5^(-9,14)\n",
      "\n",
      "v6^(-3,5) cup v3^(-6,9) = 1*v7^(-9,14)\n",
      "\n",
      "v6^(-3,5) cup v4^(-6,9) = -1*v6^(-9,14)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-3,5) cup v3^(-8,12) = 3*v1^(-11,17)\n",
      "\n",
      "v1^(-3,5) cup v8^(-8,12) = 2*v2^(-11,17)\n",
      "\n",
      "v1^(-3,5) cup v9^(-8,12) = 1*v2^(-11,17)\n",
      "\n",
      "v2^(-3,5) cup v3^(-8,12) = 3*v2^(-11,17)\n",
      "\n",
      "v2^(-3,5) cup v6^(-8,12) = -1*v1^(-11,17)\n",
      "\n",
      "v2^(-3,5) cup v7^(-8,12) = -2*v1^(-11,17)\n",
      "\n",
      "v3^(-3,5) cup v2^(-8,12) = -3*v1^(-11,17)\n",
      "\n",
      "v3^(-3,5) cup v8^(-8,12) = 1*v3^(-11,17)\n",
      "\n",
      "v3^(-3,5) cup v9^(-8,12) = -1*v3^(-11,17)\n",
      "\n",
      "v4^(-3,5) cup v2^(-8,12) = -3*v3^(-11,17)\n",
      "\n",
      "v4^(-3,5) cup v4^(-8,12) = -1*v1^(-11,17)\n",
      "\n",
      "v4^(-3,5) cup v5^(-8,12) = -2*v1^(-11,17)\n",
      "\n",
      "v5^(-3,5) cup v1^(-8,12) = 3*v2^(-11,17)\n",
      "\n",
      "v5^(-3,5) cup v6^(-8,12) = 1*v3^(-11,17)\n",
      "\n",
      "v5^(-3,5) cup v7^(-8,12) = -1*v3^(-11,17)\n",
      "\n",
      "v6^(-3,5) cup v1^(-8,12) = 3*v3^(-11,17)\n",
      "\n",
      "v6^(-3,5) cup v4^(-8,12) = -2*v2^(-11,17)\n",
      "\n",
      "v6^(-3,5) cup v5^(-8,12) = -1*v2^(-11,17)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-3,5) cup v3^(-11,16) = -1*v2^(-14,21)\n",
      "\n",
      "v2^(-3,5) cup v2^(-11,16) = 1*v1^(-14,21)\n",
      "\n",
      "v3^(-3,5) cup v3^(-11,16) = -1*v3^(-14,21)\n",
      "\n",
      "v4^(-3,5) cup v1^(-11,16) = 1*v1^(-14,21)\n",
      "\n",
      "v5^(-3,5) cup v2^(-11,16) = -1*v3^(-14,21)\n",
      "\n",
      "v6^(-3,5) cup v1^(-11,16) = 1*v2^(-14,21)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-3,5) cup v6^(-12,18) = 1*v1^(-15,23)\n",
      "\n",
      "v2^(-3,5) cup v5^(-12,18) = -1*v1^(-15,23)\n",
      "\n",
      "v3^(-3,5) cup v4^(-12,18) = -1*v1^(-15,23)\n",
      "\n",
      "v4^(-3,5) cup v3^(-12,18) = -1*v1^(-15,23)\n",
      "\n",
      "v5^(-3,5) cup v2^(-12,18) = -1*v1^(-15,23)\n",
      "\n",
      "v6^(-3,5) cup v1^(-12,18) = 1*v1^(-15,23)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-4,6) cup v1^(-4,6) = -2*v4^(-8,12) + -2*v5^(-8,12)\n",
      "\n",
      "v1^(-4,6) cup v2^(-4,6) = 2*v1^(-8,12)\n",
      "\n",
      "v1^(-4,6) cup v3^(-4,6) = 2*v2^(-8,12)\n",
      "\n",
      "v2^(-4,6) cup v1^(-4,6) = 2*v1^(-8,12)\n",
      "\n",
      "v2^(-4,6) cup v2^(-4,6) = 2*v6^(-8,12) + -4*v7^(-8,12)\n",
      "\n",
      "v2^(-4,6) cup v3^(-4,6) = -2*v3^(-8,12)\n",
      "\n",
      "v3^(-4,6) cup v1^(-4,6) = 2*v2^(-8,12)\n",
      "\n",
      "v3^(-4,6) cup v2^(-4,6) = -2*v3^(-8,12)\n",
      "\n",
      "v3^(-4,6) cup v3^(-4,6) = 4*v8^(-8,12) + -2*v9^(-8,12)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-4,6) cup v1^(-5,8) = -1*v2^(-9,14)\n",
      "\n",
      "v1^(-4,6) cup v2^(-5,8) = -1*v4^(-9,14)\n",
      "\n",
      "v1^(-4,6) cup v3^(-5,8) = 3*v1^(-9,14)\n",
      "\n",
      "v1^(-4,6) cup v5^(-5,8) = -2*v3^(-9,14)\n",
      "\n",
      "v1^(-4,6) cup v6^(-5,8) = -2*v5^(-9,14)\n",
      "\n",
      "v2^(-4,6) cup v1^(-5,8) = 1*v3^(-9,14)\n",
      "\n",
      "v2^(-4,6) cup v2^(-5,8) = -3*v1^(-9,14)\n",
      "\n",
      "v2^(-4,6) cup v3^(-5,8) = -1*v6^(-9,14)\n",
      "\n",
      "v2^(-4,6) cup v4^(-5,8) = -2*v2^(-9,14)\n",
      "\n",
      "v2^(-4,6) cup v6^(-5,8) = 2*v7^(-9,14)\n",
      "\n",
      "v3^(-4,6) cup v1^(-5,8) = 3*v1^(-9,14)\n",
      "\n",
      "v3^(-4,6) cup v2^(-5,8) = 1*v5^(-9,14)\n",
      "\n",
      "v3^(-4,6) cup v3^(-5,8) = 1*v7^(-9,14)\n",
      "\n",
      "v3^(-4,6) cup v4^(-5,8) = 2*v4^(-9,14)\n",
      "\n",
      "v3^(-4,6) cup v5^(-5,8) = 2*v6^(-9,14)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-4,6) cup v3^(-6,9) = -3*v2^(-10,15)\n",
      "\n",
      "v1^(-4,6) cup v4^(-6,9) = -2*v1^(-10,15)\n",
      "\n",
      "v1^(-4,6) cup v5^(-6,9) = 1*v3^(-10,15)\n",
      "\n",
      "v1^(-4,6) cup v6^(-6,9) = -2*v4^(-10,15)\n",
      "\n",
      "v1^(-4,6) cup v7^(-6,9) = -1*v5^(-10,15)\n",
      "\n",
      "v2^(-4,6) cup v1^(-6,9) = -2*v1^(-10,15)\n",
      "\n",
      "v2^(-4,6) cup v2^(-6,9) = -1*v2^(-10,15)\n",
      "\n",
      "v2^(-4,6) cup v3^(-6,9) = -3*v3^(-10,15)\n",
      "\n",
      "v2^(-4,6) cup v6^(-6,9) = -1*v5^(-10,15)\n",
      "\n",
      "v2^(-4,6) cup v7^(-6,9) = 2*v6^(-10,15)\n",
      "\n",
      "v3^(-4,6) cup v1^(-6,9) = 1*v2^(-10,15)\n",
      "\n",
      "v3^(-4,6) cup v2^(-6,9) = -2*v4^(-10,15)\n",
      "\n",
      "v3^(-4,6) cup v3^(-6,9) = -3*v5^(-10,15)\n",
      "\n",
      "v3^(-4,6) cup v4^(-6,9) = 1*v3^(-10,15)\n",
      "\n",
      "v3^(-4,6) cup v5^(-6,9) = -2*v6^(-10,15)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-4,6) cup v4^(-7,11) = -2*v1^(-11,17)\n",
      "\n",
      "v1^(-4,6) cup v7^(-7,11) = 2*v2^(-11,17)\n",
      "\n",
      "v1^(-4,6) cup v8^(-7,11) = 2*v3^(-11,17)\n",
      "\n",
      "v1^(-4,6) cup v9^(-7,11) = -2*v3^(-11,17)\n",
      "\n",
      "v2^(-4,6) cup v3^(-7,11) = -2*v1^(-11,17)\n",
      "\n",
      "v2^(-4,6) cup v5^(-7,11) = 4*v2^(-11,17)\n",
      "\n",
      "v2^(-4,6) cup v6^(-7,11) = 2*v2^(-11,17)\n",
      "\n",
      "v2^(-4,6) cup v7^(-7,11) = 2*v3^(-11,17)\n",
      "\n",
      "v3^(-4,6) cup v1^(-7,11) = -2*v1^(-11,17)\n",
      "\n",
      "v3^(-4,6) cup v2^(-7,11) = -4*v1^(-11,17)\n",
      "\n",
      "v3^(-4,6) cup v3^(-7,11) = 2*v2^(-11,17)\n",
      "\n",
      "v3^(-4,6) cup v4^(-7,11) = 2*v3^(-11,17)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-4,6) cup v1^(-8,12) = -3*v1^(-12,18)\n",
      "\n",
      "v1^(-4,6) cup v2^(-8,12) = -3*v3^(-12,18)\n",
      "\n",
      "v1^(-4,6) cup v6^(-8,12) = 1*v2^(-12,18)\n",
      "\n",
      "v1^(-4,6) cup v7^(-8,12) = -1*v2^(-12,18)\n",
      "\n",
      "v1^(-4,6) cup v8^(-8,12) = 1*v4^(-12,18)\n",
      "\n",
      "v1^(-4,6) cup v9^(-8,12) = -1*v4^(-12,18)\n",
      "\n",
      "v2^(-4,6) cup v1^(-8,12) = 3*v2^(-12,18)\n",
      "\n",
      "v2^(-4,6) cup v3^(-8,12) = 3*v5^(-12,18)\n",
      "\n",
      "v2^(-4,6) cup v4^(-8,12) = 2*v1^(-12,18)\n",
      "\n",
      "v2^(-4,6) cup v5^(-8,12) = 1*v1^(-12,18)\n",
      "\n",
      "v2^(-4,6) cup v8^(-8,12) = -2*v6^(-12,18)\n",
      "\n",
      "v2^(-4,6) cup v9^(-8,12) = -1*v6^(-12,18)\n",
      "\n",
      "v3^(-4,6) cup v2^(-8,12) = 3*v4^(-12,18)\n",
      "\n",
      "v3^(-4,6) cup v3^(-8,12) = 3*v6^(-12,18)\n",
      "\n",
      "v3^(-4,6) cup v4^(-8,12) = 1*v3^(-12,18)\n",
      "\n",
      "v3^(-4,6) cup v5^(-8,12) = 2*v3^(-12,18)\n",
      "\n",
      "v3^(-4,6) cup v6^(-8,12) = 1*v5^(-12,18)\n",
      "\n",
      "v3^(-4,6) cup v7^(-8,12) = 2*v5^(-12,18)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-4,6) cup v3^(-10,15) = -2*v1^(-14,21)\n",
      "\n",
      "v1^(-4,6) cup v5^(-10,15) = -2*v2^(-14,21)\n",
      "\n",
      "v1^(-4,6) cup v6^(-10,15) = -1*v3^(-14,21)\n",
      "\n",
      "v2^(-4,6) cup v2^(-10,15) = -2*v1^(-14,21)\n",
      "\n",
      "v2^(-4,6) cup v4^(-10,15) = -1*v2^(-14,21)\n",
      "\n",
      "v2^(-4,6) cup v5^(-10,15) = 2*v3^(-14,21)\n",
      "\n",
      "v3^(-4,6) cup v1^(-10,15) = 1*v1^(-14,21)\n",
      "\n",
      "v3^(-4,6) cup v2^(-10,15) = -2*v2^(-14,21)\n",
      "\n",
      "v3^(-4,6) cup v3^(-10,15) = 2*v3^(-14,21)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-4,6) cup v3^(-11,17) = -1*v1^(-15,23)\n",
      "\n",
      "v2^(-4,6) cup v2^(-11,17) = -1*v1^(-15,23)\n",
      "\n",
      "v3^(-4,6) cup v1^(-11,17) = 1*v1^(-15,23)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-4,7) cup v3^(-11,16) = -1*v1^(-15,23)\n",
      "\n",
      "v2^(-4,7) cup v2^(-11,16) = -1*v1^(-15,23)\n",
      "\n",
      "v3^(-4,7) cup v1^(-11,16) = -1*v1^(-15,23)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-5,8) cup v3^(-6,9) = 3*v1^(-11,17)\n",
      "\n",
      "v1^(-5,8) cup v6^(-6,9) = -1*v2^(-11,17)\n",
      "\n",
      "v1^(-5,8) cup v7^(-6,9) = -1*v3^(-11,17)\n",
      "\n",
      "v2^(-5,8) cup v3^(-6,9) = 3*v2^(-11,17)\n",
      "\n",
      "v2^(-5,8) cup v4^(-6,9) = 1*v1^(-11,17)\n",
      "\n",
      "v2^(-5,8) cup v5^(-6,9) = -1*v3^(-11,17)\n",
      "\n",
      "v3^(-5,8) cup v1^(-6,9) = -1*v1^(-11,17)\n",
      "\n",
      "v3^(-5,8) cup v2^(-6,9) = -1*v2^(-11,17)\n",
      "\n",
      "v3^(-5,8) cup v3^(-6,9) = -3*v3^(-11,17)\n",
      "\n",
      "v4^(-5,8) cup v5^(-6,9) = -2*v1^(-11,17)\n",
      "\n",
      "v4^(-5,8) cup v7^(-6,9) = -2*v2^(-11,17)\n",
      "\n",
      "v5^(-5,8) cup v2^(-6,9) = -2*v1^(-11,17)\n",
      "\n",
      "v5^(-5,8) cup v6^(-6,9) = 2*v3^(-11,17)\n",
      "\n",
      "v6^(-5,8) cup v1^(-6,9) = 2*v2^(-11,17)\n",
      "\n",
      "v6^(-5,8) cup v4^(-6,9) = 2*v3^(-11,17)\n",
      "\n",
      "\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "v1^(-5,8) cup v5^(-10,15) = -1*v1^(-15,23)\n",
      "\n",
      "v2^(-5,8) cup v3^(-10,15) = 1*v1^(-15,23)\n",
      "\n",
      "v3^(-5,8) cup v2^(-10,15) = -1*v1^(-15,23)\n",
      "\n",
      "v4^(-5,8) cup v6^(-10,15) = 1*v1^(-15,23)\n",
      "\n",
      "v5^(-5,8) cup v4^(-10,15) = 1*v1^(-15,23)\n",
      "\n",
      "v6^(-5,8) cup v1^(-10,15) = 1*v1^(-15,23)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-6,9) cup v3^(-6,9) = 1*v1^(-12,18)\n",
      "\n",
      "v1^(-6,9) cup v5^(-6,9) = -1*v2^(-12,18)\n",
      "\n",
      "v1^(-6,9) cup v6^(-6,9) = 1*v3^(-12,18)\n",
      "\n",
      "v2^(-6,9) cup v3^(-6,9) = 1*v3^(-12,18)\n",
      "\n",
      "v2^(-6,9) cup v4^(-6,9) = -1*v1^(-12,18)\n",
      "\n",
      "v2^(-6,9) cup v7^(-6,9) = 1*v4^(-12,18)\n",
      "\n",
      "v3^(-6,9) cup v1^(-6,9) = -1*v1^(-12,18)\n",
      "\n",
      "v3^(-6,9) cup v2^(-6,9) = -1*v3^(-12,18)\n",
      "\n",
      "v3^(-6,9) cup v4^(-6,9) = -1*v2^(-12,18)\n",
      "\n",
      "v3^(-6,9) cup v5^(-6,9) = -1*v5^(-12,18)\n",
      "\n",
      "v3^(-6,9) cup v6^(-6,9) = -1*v4^(-12,18)\n",
      "\n",
      "v3^(-6,9) cup v7^(-6,9) = -1*v6^(-12,18)\n",
      "\n",
      "v4^(-6,9) cup v2^(-6,9) = 1*v1^(-12,18)\n",
      "\n",
      "v4^(-6,9) cup v3^(-6,9) = 1*v2^(-12,18)\n",
      "\n",
      "v4^(-6,9) cup v7^(-6,9) = -1*v5^(-12,18)\n",
      "\n",
      "v5^(-6,9) cup v1^(-6,9) = 1*v2^(-12,18)\n",
      "\n",
      "v5^(-6,9) cup v3^(-6,9) = 1*v5^(-12,18)\n",
      "\n",
      "v5^(-6,9) cup v6^(-6,9) = -1*v6^(-12,18)\n",
      "\n",
      "v6^(-6,9) cup v1^(-6,9) = -1*v3^(-12,18)\n",
      "\n",
      "v6^(-6,9) cup v3^(-6,9) = 1*v4^(-12,18)\n",
      "\n",
      "v6^(-6,9) cup v5^(-6,9) = 1*v6^(-12,18)\n",
      "\n",
      "v7^(-6,9) cup v2^(-6,9) = -1*v4^(-12,18)\n",
      "\n",
      "v7^(-6,9) cup v3^(-6,9) = 1*v6^(-12,18)\n",
      "\n",
      "v7^(-6,9) cup v4^(-6,9) = 1*v5^(-12,18)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-6,9) cup v3^(-8,12) = 1*v1^(-14,21)\n",
      "\n",
      "v1^(-6,9) cup v9^(-8,12) = 1*v2^(-14,21)\n",
      "\n",
      "v2^(-6,9) cup v3^(-8,12) = -1*v2^(-14,21)\n",
      "\n",
      "v2^(-6,9) cup v6^(-8,12) = 1*v1^(-14,21)\n",
      "\n",
      "v3^(-6,9) cup v1^(-8,12) = 3*v1^(-14,21)\n",
      "\n",
      "v3^(-6,9) cup v2^(-8,12) = 3*v2^(-14,21)\n",
      "\n",
      "v3^(-6,9) cup v3^(-8,12) = 3*v3^(-14,21)\n",
      "\n",
      "v4^(-6,9) cup v2^(-8,12) = -1*v1^(-14,21)\n",
      "\n",
      "v4^(-6,9) cup v8^(-8,12) = 1*v3^(-14,21)\n",
      "\n",
      "v4^(-6,9) cup v9^(-8,12) = 1*v3^(-14,21)\n",
      "\n",
      "v5^(-6,9) cup v2^(-8,12) = 1*v3^(-14,21)\n",
      "\n",
      "v5^(-6,9) cup v4^(-8,12) = 1*v1^(-14,21)\n",
      "\n",
      "v6^(-6,9) cup v1^(-8,12) = 1*v2^(-14,21)\n",
      "\n",
      "v6^(-6,9) cup v6^(-8,12) = 1*v3^(-14,21)\n",
      "\n",
      "v6^(-6,9) cup v7^(-8,12) = 1*v3^(-14,21)\n",
      "\n",
      "v7^(-6,9) cup v1^(-8,12) = -1*v3^(-14,21)\n",
      "\n",
      "v7^(-6,9) cup v5^(-8,12) = -1*v2^(-14,21)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-6,9) cup v7^(-9,14) = 1*v1^(-15,23)\n",
      "\n",
      "v2^(-6,9) cup v6^(-9,14) = 1*v1^(-15,23)\n",
      "\n",
      "v3^(-6,9) cup v1^(-9,14) = -1*v1^(-15,23)\n",
      "\n",
      "v4^(-6,9) cup v5^(-9,14) = -1*v1^(-15,23)\n",
      "\n",
      "v5^(-6,9) cup v4^(-9,14) = 1*v1^(-15,23)\n",
      "\n",
      "v6^(-6,9) cup v3^(-9,14) = -1*v1^(-15,23)\n",
      "\n",
      "v7^(-6,9) cup v2^(-9,14) = 1*v1^(-15,23)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-7,11) cup v9^(-8,12) = 1*v1^(-15,23)\n",
      "\n",
      "v2^(-7,11) cup v8^(-8,12) = -1*v1^(-15,23)\n",
      "\n",
      "v3^(-7,11) cup v3^(-8,12) = 1*v1^(-15,23)\n",
      "\n",
      "v4^(-7,11) cup v2^(-8,12) = -1*v1^(-15,23)\n",
      "\n",
      "v5^(-7,11) cup v7^(-8,12) = 1*v1^(-15,23)\n",
      "\n",
      "v6^(-7,11) cup v6^(-8,12) = -1*v1^(-15,23)\n",
      "\n",
      "v7^(-7,11) cup v1^(-8,12) = -1*v1^(-15,23)\n",
      "\n",
      "v8^(-7,11) cup v5^(-8,12) = 1*v1^(-15,23)\n",
      "\n",
      "v9^(-7,11) cup v4^(-8,12) = -1*v1^(-15,23)\n",
      "\n"
     ]
    }
   ],
   "source": [
    "print_cup_products(g_sl3, e_sort_sl3, sl3_H_s_t_fixed_2, sl3_bases)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "id": "100b686d",
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "        ===================================================\n",
      "        Non-zero cup products from H^(s,t) spaces:\n",
      "        =================================================== \n",
      "\n",
      "        \n",
      "NOTE: We are not printing any H^(0) cup products.\n",
      "\n",
      "We write vi^(s,t) for the i'th basis vector of H^(s,t).\n",
      "\n",
      "\n",
      "\n",
      "v1^(-1,2) cup v3^(-3,5) = (1, 0, 0) in H^(-4,7)\n",
      "\n",
      "v1^(-1,2) cup v5^(-3,5) = (0, 1, 0) in H^(-4,7)\n",
      "\n",
      "v2^(-1,2) cup v1^(-3,5) = (-1, 0, 0) in H^(-4,7)\n",
      "\n",
      "v2^(-1,2) cup v6^(-3,5) = (0, 0, 1) in H^(-4,7)\n",
      "\n",
      "v3^(-1,2) cup v2^(-3,5) = (0, -1, 0) in H^(-4,7)\n",
      "\n",
      "v3^(-1,2) cup v4^(-3,5) = (0, 0, -1) in H^(-4,7)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-1,2) cup v1^(-4,6) = (0, 0, 0, -1, 0, 0) in H^(-5,8)\n",
      "\n",
      "v1^(-1,2) cup v2^(-4,6) = (-2, 0, 0, 0, 0, 0) in H^(-5,8)\n",
      "\n",
      "v1^(-1,2) cup v3^(-4,6) = (0, 2, 0, 0, 0, 0) in H^(-5,8)\n",
      "\n",
      "v2^(-1,2) cup v1^(-4,6) = (-2, 0, 0, 0, 0, 0) in H^(-5,8)\n",
      "\n",
      "v2^(-1,2) cup v2^(-4,6) = (0, 0, 0, 0, 1, 0) in H^(-5,8)\n",
      "\n",
      "v2^(-1,2) cup v3^(-4,6) = (0, 0, -2, 0, 0, 0) in H^(-5,8)\n",
      "\n",
      "v3^(-1,2) cup v1^(-4,6) = (0, -2, 0, 0, 0, 0) in H^(-5,8)\n",
      "\n",
      "v3^(-1,2) cup v2^(-4,6) = (0, 0, 2, 0, 0, 0) in H^(-5,8)\n",
      "\n",
      "v3^(-1,2) cup v3^(-4,6) = (0, 0, 0, 0, 0, 1) in H^(-5,8)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-1,2) cup v3^(-6,9) = (0, 0, 3, 0, 0, 0, 0, 0, 0) in H^(-7,11)\n",
      "\n",
      "v1^(-1,2) cup v4^(-6,9) = (1, -1, 0, 0, 0, 0, 0, 0, 0) in H^(-7,11)\n",
      "\n",
      "v1^(-1,2) cup v5^(-6,9) = (0, 0, 0, -1, 0, 0, 0, 0, 0) in H^(-7,11)\n",
      "\n",
      "v1^(-1,2) cup v6^(-6,9) = (0, 0, 0, 0, 1, -1, 0, 0, 0) in H^(-7,11)\n",
      "\n",
      "v1^(-1,2) cup v7^(-6,9) = (0, 0, 0, 0, 0, 0, 1, 0, 0) in H^(-7,11)\n",
      "\n",
      "v2^(-1,2) cup v1^(-6,9) = (-1, 0, 0, 0, 0, 0, 0, 0, 0) in H^(-7,11)\n",
      "\n",
      "v2^(-1,2) cup v2^(-6,9) = (0, 0, 1, 0, 0, 0, 0, 0, 0) in H^(-7,11)\n",
      "\n",
      "v2^(-1,2) cup v3^(-6,9) = (0, 0, 0, 3, 0, 0, 0, 0, 0) in H^(-7,11)\n",
      "\n",
      "v2^(-1,2) cup v6^(-6,9) = (0, 0, 0, 0, 0, 0, 1, 0, 0) in H^(-7,11)\n",
      "\n",
      "v2^(-1,2) cup v7^(-6,9) = (0, 0, 0, 0, 0, 0, 0, 1, 0) in H^(-7,11)\n",
      "\n",
      "v3^(-1,2) cup v1^(-6,9) = (0, 0, 1, 0, 0, 0, 0, 0, 0) in H^(-7,11)\n",
      "\n",
      "v3^(-1,2) cup v2^(-6,9) = (0, 0, 0, 0, 0, -1, 0, 0, 0) in H^(-7,11)\n",
      "\n",
      "v3^(-1,2) cup v3^(-6,9) = (0, 0, 0, 0, 0, 0, -3, 0, 0) in H^(-7,11)\n",
      "\n",
      "v3^(-1,2) cup v4^(-6,9) = (0, 0, 0, 1, 0, 0, 0, 0, 0) in H^(-7,11)\n",
      "\n",
      "v3^(-1,2) cup v5^(-6,9) = (0, 0, 0, 0, 0, 0, 0, 0, -1) in H^(-7,11)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-1,2) cup v1^(-8,12) = (0, 1, 0, 0, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v1^(-1,2) cup v2^(-8,12) = (0, 0, 0, -1, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v1^(-1,2) cup v3^(-8,12) = (3, 0, 0, 0, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v1^(-1,2) cup v6^(-8,12) = (0, 0, 1, 0, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v1^(-1,2) cup v7^(-8,12) = (0, 0, 1, 0, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v1^(-1,2) cup v8^(-8,12) = (0, 0, 0, 0, 1, 0, 0) in H^(-9,14)\n",
      "\n",
      "v1^(-1,2) cup v9^(-8,12) = (0, 0, 0, 0, 1, 0, 0) in H^(-9,14)\n",
      "\n",
      "v2^(-1,2) cup v1^(-8,12) = (0, 0, -1, 0, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v2^(-1,2) cup v2^(-8,12) = (-3, 0, 0, 0, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v2^(-1,2) cup v3^(-8,12) = (0, 0, 0, 0, 0, -1, 0) in H^(-9,14)\n",
      "\n",
      "v2^(-1,2) cup v5^(-8,12) = (0, -1, 0, 0, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v2^(-1,2) cup v9^(-8,12) = (0, 0, 0, 0, 0, 0, 1) in H^(-9,14)\n",
      "\n",
      "v3^(-1,2) cup v1^(-8,12) = (3, 0, 0, 0, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v3^(-1,2) cup v2^(-8,12) = (0, 0, 0, 0, -1, 0, 0) in H^(-9,14)\n",
      "\n",
      "v3^(-1,2) cup v3^(-8,12) = (0, 0, 0, 0, 0, 0, -1) in H^(-9,14)\n",
      "\n",
      "v3^(-1,2) cup v4^(-8,12) = (0, 0, 0, -1, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v3^(-1,2) cup v6^(-8,12) = (0, 0, 0, 0, 0, -1, 0) in H^(-9,14)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-1,2) cup v3^(-10,15) = (2, 0, 0) in H^(-11,17)\n",
      "\n",
      "v1^(-1,2) cup v5^(-10,15) = (0, -2, 0) in H^(-11,17)\n",
      "\n",
      "v1^(-1,2) cup v6^(-10,15) = (0, 0, 1) in H^(-11,17)\n",
      "\n",
      "v2^(-1,2) cup v2^(-10,15) = (2, 0, 0) in H^(-11,17)\n",
      "\n",
      "v2^(-1,2) cup v4^(-10,15) = (0, -1, 0) in H^(-11,17)\n",
      "\n",
      "v2^(-1,2) cup v5^(-10,15) = (0, 0, -2) in H^(-11,17)\n",
      "\n",
      "v3^(-1,2) cup v1^(-10,15) = (1, 0, 0) in H^(-11,17)\n",
      "\n",
      "v3^(-1,2) cup v2^(-10,15) = (0, 2, 0) in H^(-11,17)\n",
      "\n",
      "v3^(-1,2) cup v3^(-10,15) = (0, 0, 2) in H^(-11,17)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-1,2) cup v2^(-11,16) = (0, 1, 0, 0, 0, 0) in H^(-12,18)\n",
      "\n",
      "v1^(-1,2) cup v3^(-11,16) = (0, 0, 0, 1, 0, 0) in H^(-12,18)\n",
      "\n",
      "v2^(-1,2) cup v1^(-11,16) = (-1, 0, 0, 0, 0, 0) in H^(-12,18)\n",
      "\n",
      "v2^(-1,2) cup v3^(-11,16) = (0, 0, 0, 0, 0, 1) in H^(-12,18)\n",
      "\n",
      "v3^(-1,2) cup v1^(-11,16) = (0, 0, -1, 0, 0, 0) in H^(-12,18)\n",
      "\n",
      "v3^(-1,2) cup v2^(-11,16) = (0, 0, 0, 0, -1, 0) in H^(-12,18)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-1,2) cup v3^(-14,21) = (1) in H^(-15,23)\n",
      "\n",
      "v2^(-1,2) cup v2^(-14,21) = (-1) in H^(-15,23)\n",
      "\n",
      "v3^(-1,2) cup v1^(-14,21) = (1) in H^(-15,23)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-3,5) cup v2^(-4,6) = (-1, 2, 0, 0, 0, 0, 0, 0, 0) in H^(-7,11)\n",
      "\n",
      "v1^(-3,5) cup v3^(-4,6) = (0, 0, 3, 0, 0, 0, 0, 0, 0) in H^(-7,11)\n",
      "\n",
      "v2^(-3,5) cup v2^(-4,6) = (0, 0, -3, 0, 0, 0, 0, 0, 0) in H^(-7,11)\n",
      "\n",
      "v2^(-3,5) cup v3^(-4,6) = (0, 0, 0, 0, -2, 1, 0, 0, 0) in H^(-7,11)\n",
      "\n",
      "v3^(-3,5) cup v1^(-4,6) = (1, 1, 0, 0, 0, 0, 0, 0, 0) in H^(-7,11)\n",
      "\n",
      "v3^(-3,5) cup v3^(-4,6) = (0, 0, 0, 3, 0, 0, 0, 0, 0) in H^(-7,11)\n",
      "\n",
      "v4^(-3,5) cup v1^(-4,6) = (0, 0, 0, -3, 0, 0, 0, 0, 0) in H^(-7,11)\n",
      "\n",
      "v4^(-3,5) cup v3^(-4,6) = (0, 0, 0, 0, 0, 0, 0, -2, 1) in H^(-7,11)\n",
      "\n",
      "v5^(-3,5) cup v1^(-4,6) = (0, 0, 0, 0, 1, 1, 0, 0, 0) in H^(-7,11)\n",
      "\n",
      "v5^(-3,5) cup v2^(-4,6) = (0, 0, 0, 0, 0, 0, 3, 0, 0) in H^(-7,11)\n",
      "\n",
      "v6^(-3,5) cup v1^(-4,6) = (0, 0, 0, 0, 0, 0, 3, 0, 0) in H^(-7,11)\n",
      "\n",
      "v6^(-3,5) cup v2^(-4,6) = (0, 0, 0, 0, 0, 0, 0, 1, -2) in H^(-7,11)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-3,5) cup v3^(-6,9) = (0, 1, 0, 0, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v1^(-3,5) cup v5^(-6,9) = (0, 0, -1, 0, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v1^(-3,5) cup v6^(-6,9) = (0, 0, 0, -1, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v1^(-3,5) cup v7^(-6,9) = (1, 0, 0, 0, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v2^(-3,5) cup v3^(-6,9) = (0, 0, 0, -1, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v2^(-3,5) cup v4^(-6,9) = (0, -1, 0, 0, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v2^(-3,5) cup v5^(-6,9) = (-1, 0, 0, 0, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v2^(-3,5) cup v7^(-6,9) = (0, 0, 0, 0, -1, 0, 0) in H^(-9,14)\n",
      "\n",
      "v3^(-3,5) cup v2^(-6,9) = (0, 1, 0, 0, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v3^(-3,5) cup v3^(-6,9) = (0, 0, 1, 0, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v3^(-3,5) cup v6^(-6,9) = (-1, 0, 0, 0, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v3^(-3,5) cup v7^(-6,9) = (0, 0, 0, 0, 0, -1, 0) in H^(-9,14)\n",
      "\n",
      "v4^(-3,5) cup v1^(-6,9) = (0, 0, -1, 0, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v4^(-3,5) cup v2^(-6,9) = (-1, 0, 0, 0, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v4^(-3,5) cup v3^(-6,9) = (0, 0, 0, 0, 0, -1, 0) in H^(-9,14)\n",
      "\n",
      "v4^(-3,5) cup v6^(-6,9) = (0, 0, 0, 0, 0, 0, -1) in H^(-9,14)\n",
      "\n",
      "v5^(-3,5) cup v1^(-6,9) = (0, 0, 0, -1, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v5^(-3,5) cup v3^(-6,9) = (0, 0, 0, 0, 1, 0, 0) in H^(-9,14)\n",
      "\n",
      "v5^(-3,5) cup v4^(-6,9) = (-1, 0, 0, 0, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v5^(-3,5) cup v5^(-6,9) = (0, 0, 0, 0, 0, 0, 1) in H^(-9,14)\n",
      "\n",
      "v6^(-3,5) cup v1^(-6,9) = (1, 0, 0, 0, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v6^(-3,5) cup v2^(-6,9) = (0, 0, 0, 0, -1, 0, 0) in H^(-9,14)\n",
      "\n",
      "v6^(-3,5) cup v3^(-6,9) = (0, 0, 0, 0, 0, 0, 1) in H^(-9,14)\n",
      "\n",
      "v6^(-3,5) cup v4^(-6,9) = (0, 0, 0, 0, 0, -1, 0) in H^(-9,14)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-3,5) cup v3^(-8,12) = (3, 0, 0) in H^(-11,17)\n",
      "\n",
      "v1^(-3,5) cup v8^(-8,12) = (0, 2, 0) in H^(-11,17)\n",
      "\n",
      "v1^(-3,5) cup v9^(-8,12) = (0, 1, 0) in H^(-11,17)\n",
      "\n",
      "v2^(-3,5) cup v3^(-8,12) = (0, 3, 0) in H^(-11,17)\n",
      "\n",
      "v2^(-3,5) cup v6^(-8,12) = (-1, 0, 0) in H^(-11,17)\n",
      "\n",
      "v2^(-3,5) cup v7^(-8,12) = (-2, 0, 0) in H^(-11,17)\n",
      "\n",
      "v3^(-3,5) cup v2^(-8,12) = (-3, 0, 0) in H^(-11,17)\n",
      "\n",
      "v3^(-3,5) cup v8^(-8,12) = (0, 0, 1) in H^(-11,17)\n",
      "\n",
      "v3^(-3,5) cup v9^(-8,12) = (0, 0, -1) in H^(-11,17)\n",
      "\n",
      "v4^(-3,5) cup v2^(-8,12) = (0, 0, -3) in H^(-11,17)\n",
      "\n",
      "v4^(-3,5) cup v4^(-8,12) = (-1, 0, 0) in H^(-11,17)\n",
      "\n",
      "v4^(-3,5) cup v5^(-8,12) = (-2, 0, 0) in H^(-11,17)\n",
      "\n",
      "v5^(-3,5) cup v1^(-8,12) = (0, 3, 0) in H^(-11,17)\n",
      "\n",
      "v5^(-3,5) cup v6^(-8,12) = (0, 0, 1) in H^(-11,17)\n",
      "\n",
      "v5^(-3,5) cup v7^(-8,12) = (0, 0, -1) in H^(-11,17)\n",
      "\n",
      "v6^(-3,5) cup v1^(-8,12) = (0, 0, 3) in H^(-11,17)\n",
      "\n",
      "v6^(-3,5) cup v4^(-8,12) = (0, -2, 0) in H^(-11,17)\n",
      "\n",
      "v6^(-3,5) cup v5^(-8,12) = (0, -1, 0) in H^(-11,17)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-3,5) cup v3^(-11,16) = (0, -1, 0) in H^(-14,21)\n",
      "\n",
      "v2^(-3,5) cup v2^(-11,16) = (1, 0, 0) in H^(-14,21)\n",
      "\n",
      "v3^(-3,5) cup v3^(-11,16) = (0, 0, -1) in H^(-14,21)\n",
      "\n",
      "v4^(-3,5) cup v1^(-11,16) = (1, 0, 0) in H^(-14,21)\n",
      "\n",
      "v5^(-3,5) cup v2^(-11,16) = (0, 0, -1) in H^(-14,21)\n",
      "\n",
      "v6^(-3,5) cup v1^(-11,16) = (0, 1, 0) in H^(-14,21)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-3,5) cup v6^(-12,18) = (1) in H^(-15,23)\n",
      "\n",
      "v2^(-3,5) cup v5^(-12,18) = (-1) in H^(-15,23)\n",
      "\n",
      "v3^(-3,5) cup v4^(-12,18) = (-1) in H^(-15,23)\n",
      "\n",
      "v4^(-3,5) cup v3^(-12,18) = (-1) in H^(-15,23)\n",
      "\n",
      "v5^(-3,5) cup v2^(-12,18) = (-1) in H^(-15,23)\n",
      "\n",
      "v6^(-3,5) cup v1^(-12,18) = (1) in H^(-15,23)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-4,6) cup v1^(-4,6) = (0, 0, 0, -2, -2, 0, 0, 0, 0) in H^(-8,12)\n",
      "\n",
      "v1^(-4,6) cup v2^(-4,6) = (2, 0, 0, 0, 0, 0, 0, 0, 0) in H^(-8,12)\n",
      "\n",
      "v1^(-4,6) cup v3^(-4,6) = (0, 2, 0, 0, 0, 0, 0, 0, 0) in H^(-8,12)\n",
      "\n",
      "v2^(-4,6) cup v1^(-4,6) = (2, 0, 0, 0, 0, 0, 0, 0, 0) in H^(-8,12)\n",
      "\n",
      "v2^(-4,6) cup v2^(-4,6) = (0, 0, 0, 0, 0, 2, -4, 0, 0) in H^(-8,12)\n",
      "\n",
      "v2^(-4,6) cup v3^(-4,6) = (0, 0, -2, 0, 0, 0, 0, 0, 0) in H^(-8,12)\n",
      "\n",
      "v3^(-4,6) cup v1^(-4,6) = (0, 2, 0, 0, 0, 0, 0, 0, 0) in H^(-8,12)\n",
      "\n",
      "v3^(-4,6) cup v2^(-4,6) = (0, 0, -2, 0, 0, 0, 0, 0, 0) in H^(-8,12)\n",
      "\n",
      "v3^(-4,6) cup v3^(-4,6) = (0, 0, 0, 0, 0, 0, 0, 4, -2) in H^(-8,12)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-4,6) cup v1^(-5,8) = (0, -1, 0, 0, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v1^(-4,6) cup v2^(-5,8) = (0, 0, 0, -1, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v1^(-4,6) cup v3^(-5,8) = (3, 0, 0, 0, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v1^(-4,6) cup v5^(-5,8) = (0, 0, -2, 0, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v1^(-4,6) cup v6^(-5,8) = (0, 0, 0, 0, -2, 0, 0) in H^(-9,14)\n",
      "\n",
      "v2^(-4,6) cup v1^(-5,8) = (0, 0, 1, 0, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v2^(-4,6) cup v2^(-5,8) = (-3, 0, 0, 0, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v2^(-4,6) cup v3^(-5,8) = (0, 0, 0, 0, 0, -1, 0) in H^(-9,14)\n",
      "\n",
      "v2^(-4,6) cup v4^(-5,8) = (0, -2, 0, 0, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v2^(-4,6) cup v6^(-5,8) = (0, 0, 0, 0, 0, 0, 2) in H^(-9,14)\n",
      "\n",
      "v3^(-4,6) cup v1^(-5,8) = (3, 0, 0, 0, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v3^(-4,6) cup v2^(-5,8) = (0, 0, 0, 0, 1, 0, 0) in H^(-9,14)\n",
      "\n",
      "v3^(-4,6) cup v3^(-5,8) = (0, 0, 0, 0, 0, 0, 1) in H^(-9,14)\n",
      "\n",
      "v3^(-4,6) cup v4^(-5,8) = (0, 0, 0, 2, 0, 0, 0) in H^(-9,14)\n",
      "\n",
      "v3^(-4,6) cup v5^(-5,8) = (0, 0, 0, 0, 0, 2, 0) in H^(-9,14)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-4,6) cup v3^(-6,9) = (0, -3, 0, 0, 0, 0) in H^(-10,15)\n",
      "\n",
      "v1^(-4,6) cup v4^(-6,9) = (-2, 0, 0, 0, 0, 0) in H^(-10,15)\n",
      "\n",
      "v1^(-4,6) cup v5^(-6,9) = (0, 0, 1, 0, 0, 0) in H^(-10,15)\n",
      "\n",
      "v1^(-4,6) cup v6^(-6,9) = (0, 0, 0, -2, 0, 0) in H^(-10,15)\n",
      "\n",
      "v1^(-4,6) cup v7^(-6,9) = (0, 0, 0, 0, -1, 0) in H^(-10,15)\n",
      "\n",
      "v2^(-4,6) cup v1^(-6,9) = (-2, 0, 0, 0, 0, 0) in H^(-10,15)\n",
      "\n",
      "v2^(-4,6) cup v2^(-6,9) = (0, -1, 0, 0, 0, 0) in H^(-10,15)\n",
      "\n",
      "v2^(-4,6) cup v3^(-6,9) = (0, 0, -3, 0, 0, 0) in H^(-10,15)\n",
      "\n",
      "v2^(-4,6) cup v6^(-6,9) = (0, 0, 0, 0, -1, 0) in H^(-10,15)\n",
      "\n",
      "v2^(-4,6) cup v7^(-6,9) = (0, 0, 0, 0, 0, 2) in H^(-10,15)\n",
      "\n",
      "v3^(-4,6) cup v1^(-6,9) = (0, 1, 0, 0, 0, 0) in H^(-10,15)\n",
      "\n",
      "v3^(-4,6) cup v2^(-6,9) = (0, 0, 0, -2, 0, 0) in H^(-10,15)\n",
      "\n",
      "v3^(-4,6) cup v3^(-6,9) = (0, 0, 0, 0, -3, 0) in H^(-10,15)\n",
      "\n",
      "v3^(-4,6) cup v4^(-6,9) = (0, 0, 1, 0, 0, 0) in H^(-10,15)\n",
      "\n",
      "v3^(-4,6) cup v5^(-6,9) = (0, 0, 0, 0, 0, -2) in H^(-10,15)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-4,6) cup v4^(-7,11) = (-2, 0, 0) in H^(-11,17)\n",
      "\n",
      "v1^(-4,6) cup v7^(-7,11) = (0, 2, 0) in H^(-11,17)\n",
      "\n",
      "v1^(-4,6) cup v8^(-7,11) = (0, 0, 2) in H^(-11,17)\n",
      "\n",
      "v1^(-4,6) cup v9^(-7,11) = (0, 0, -2) in H^(-11,17)\n",
      "\n",
      "v2^(-4,6) cup v3^(-7,11) = (-2, 0, 0) in H^(-11,17)\n",
      "\n",
      "v2^(-4,6) cup v5^(-7,11) = (0, 4, 0) in H^(-11,17)\n",
      "\n",
      "v2^(-4,6) cup v6^(-7,11) = (0, 2, 0) in H^(-11,17)\n",
      "\n",
      "v2^(-4,6) cup v7^(-7,11) = (0, 0, 2) in H^(-11,17)\n",
      "\n",
      "v3^(-4,6) cup v1^(-7,11) = (-2, 0, 0) in H^(-11,17)\n",
      "\n",
      "v3^(-4,6) cup v2^(-7,11) = (-4, 0, 0) in H^(-11,17)\n",
      "\n",
      "v3^(-4,6) cup v3^(-7,11) = (0, 2, 0) in H^(-11,17)\n",
      "\n",
      "v3^(-4,6) cup v4^(-7,11) = (0, 0, 2) in H^(-11,17)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-4,6) cup v1^(-8,12) = (-3, 0, 0, 0, 0, 0) in H^(-12,18)\n",
      "\n",
      "v1^(-4,6) cup v2^(-8,12) = (0, 0, -3, 0, 0, 0) in H^(-12,18)\n",
      "\n",
      "v1^(-4,6) cup v6^(-8,12) = (0, 1, 0, 0, 0, 0) in H^(-12,18)\n",
      "\n",
      "v1^(-4,6) cup v7^(-8,12) = (0, -1, 0, 0, 0, 0) in H^(-12,18)\n",
      "\n",
      "v1^(-4,6) cup v8^(-8,12) = (0, 0, 0, 1, 0, 0) in H^(-12,18)\n",
      "\n",
      "v1^(-4,6) cup v9^(-8,12) = (0, 0, 0, -1, 0, 0) in H^(-12,18)\n",
      "\n",
      "v2^(-4,6) cup v1^(-8,12) = (0, 3, 0, 0, 0, 0) in H^(-12,18)\n",
      "\n",
      "v2^(-4,6) cup v3^(-8,12) = (0, 0, 0, 0, 3, 0) in H^(-12,18)\n",
      "\n",
      "v2^(-4,6) cup v4^(-8,12) = (2, 0, 0, 0, 0, 0) in H^(-12,18)\n",
      "\n",
      "v2^(-4,6) cup v5^(-8,12) = (1, 0, 0, 0, 0, 0) in H^(-12,18)\n",
      "\n",
      "v2^(-4,6) cup v8^(-8,12) = (0, 0, 0, 0, 0, -2) in H^(-12,18)\n",
      "\n",
      "v2^(-4,6) cup v9^(-8,12) = (0, 0, 0, 0, 0, -1) in H^(-12,18)\n",
      "\n",
      "v3^(-4,6) cup v2^(-8,12) = (0, 0, 0, 3, 0, 0) in H^(-12,18)\n",
      "\n",
      "v3^(-4,6) cup v3^(-8,12) = (0, 0, 0, 0, 0, 3) in H^(-12,18)\n",
      "\n",
      "v3^(-4,6) cup v4^(-8,12) = (0, 0, 1, 0, 0, 0) in H^(-12,18)\n",
      "\n",
      "v3^(-4,6) cup v5^(-8,12) = (0, 0, 2, 0, 0, 0) in H^(-12,18)\n",
      "\n",
      "v3^(-4,6) cup v6^(-8,12) = (0, 0, 0, 0, 1, 0) in H^(-12,18)\n",
      "\n",
      "v3^(-4,6) cup v7^(-8,12) = (0, 0, 0, 0, 2, 0) in H^(-12,18)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-4,6) cup v3^(-10,15) = (-2, 0, 0) in H^(-14,21)\n",
      "\n",
      "v1^(-4,6) cup v5^(-10,15) = (0, -2, 0) in H^(-14,21)\n",
      "\n",
      "v1^(-4,6) cup v6^(-10,15) = (0, 0, -1) in H^(-14,21)\n",
      "\n",
      "v2^(-4,6) cup v2^(-10,15) = (-2, 0, 0) in H^(-14,21)\n",
      "\n",
      "v2^(-4,6) cup v4^(-10,15) = (0, -1, 0) in H^(-14,21)\n",
      "\n",
      "v2^(-4,6) cup v5^(-10,15) = (0, 0, 2) in H^(-14,21)\n",
      "\n",
      "v3^(-4,6) cup v1^(-10,15) = (1, 0, 0) in H^(-14,21)\n",
      "\n",
      "v3^(-4,6) cup v2^(-10,15) = (0, -2, 0) in H^(-14,21)\n",
      "\n",
      "v3^(-4,6) cup v3^(-10,15) = (0, 0, 2) in H^(-14,21)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-4,6) cup v3^(-11,17) = (-1) in H^(-15,23)\n",
      "\n",
      "v2^(-4,6) cup v2^(-11,17) = (-1) in H^(-15,23)\n",
      "\n",
      "v3^(-4,6) cup v1^(-11,17) = (1) in H^(-15,23)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-4,7) cup v3^(-11,16) = (-1) in H^(-15,23)\n",
      "\n",
      "v2^(-4,7) cup v2^(-11,16) = (-1) in H^(-15,23)\n",
      "\n",
      "v3^(-4,7) cup v1^(-11,16) = (-1) in H^(-15,23)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-5,8) cup v3^(-6,9) = (3, 0, 0) in H^(-11,17)\n",
      "\n",
      "v1^(-5,8) cup v6^(-6,9) = (0, -1, 0) in H^(-11,17)\n",
      "\n",
      "v1^(-5,8) cup v7^(-6,9) = (0, 0, -1) in H^(-11,17)\n",
      "\n",
      "v2^(-5,8) cup v3^(-6,9) = (0, 3, 0) in H^(-11,17)\n",
      "\n",
      "v2^(-5,8) cup v4^(-6,9) = (1, 0, 0) in H^(-11,17)\n",
      "\n",
      "v2^(-5,8) cup v5^(-6,9) = (0, 0, -1) in H^(-11,17)\n",
      "\n",
      "v3^(-5,8) cup v1^(-6,9) = (-1, 0, 0) in H^(-11,17)\n",
      "\n",
      "v3^(-5,8) cup v2^(-6,9) = (0, -1, 0) in H^(-11,17)\n",
      "\n",
      "v3^(-5,8) cup v3^(-6,9) = (0, 0, -3) in H^(-11,17)\n",
      "\n",
      "v4^(-5,8) cup v5^(-6,9) = (-2, 0, 0) in H^(-11,17)\n",
      "\n",
      "v4^(-5,8) cup v7^(-6,9) = (0, -2, 0) in H^(-11,17)\n",
      "\n",
      "v5^(-5,8) cup v2^(-6,9) = (-2, 0, 0) in H^(-11,17)\n",
      "\n",
      "v5^(-5,8) cup v6^(-6,9) = (0, 0, 2) in H^(-11,17)\n",
      "\n",
      "v6^(-5,8) cup v1^(-6,9) = (0, 2, 0) in H^(-11,17)\n",
      "\n",
      "v6^(-5,8) cup v4^(-6,9) = (0, 0, 2) in H^(-11,17)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-5,8) cup v5^(-10,15) = (-1) in H^(-15,23)\n",
      "\n",
      "v2^(-5,8) cup v3^(-10,15) = (1) in H^(-15,23)\n",
      "\n",
      "v3^(-5,8) cup v2^(-10,15) = (-1) in H^(-15,23)\n",
      "\n",
      "v4^(-5,8) cup v6^(-10,15) = (1) in H^(-15,23)\n",
      "\n",
      "v5^(-5,8) cup v4^(-10,15) = (1) in H^(-15,23)\n",
      "\n",
      "v6^(-5,8) cup v1^(-10,15) = (1) in H^(-15,23)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-6,9) cup v3^(-6,9) = (1, 0, 0, 0, 0, 0) in H^(-12,18)\n",
      "\n",
      "v1^(-6,9) cup v5^(-6,9) = (0, -1, 0, 0, 0, 0) in H^(-12,18)\n",
      "\n",
      "v1^(-6,9) cup v6^(-6,9) = (0, 0, 1, 0, 0, 0) in H^(-12,18)\n",
      "\n",
      "v2^(-6,9) cup v3^(-6,9) = (0, 0, 1, 0, 0, 0) in H^(-12,18)\n",
      "\n",
      "v2^(-6,9) cup v4^(-6,9) = (-1, 0, 0, 0, 0, 0) in H^(-12,18)\n",
      "\n",
      "v2^(-6,9) cup v7^(-6,9) = (0, 0, 0, 1, 0, 0) in H^(-12,18)\n",
      "\n",
      "v3^(-6,9) cup v1^(-6,9) = (-1, 0, 0, 0, 0, 0) in H^(-12,18)\n",
      "\n",
      "v3^(-6,9) cup v2^(-6,9) = (0, 0, -1, 0, 0, 0) in H^(-12,18)\n",
      "\n",
      "v3^(-6,9) cup v4^(-6,9) = (0, -1, 0, 0, 0, 0) in H^(-12,18)\n",
      "\n",
      "v3^(-6,9) cup v5^(-6,9) = (0, 0, 0, 0, -1, 0) in H^(-12,18)\n",
      "\n",
      "v3^(-6,9) cup v6^(-6,9) = (0, 0, 0, -1, 0, 0) in H^(-12,18)\n",
      "\n",
      "v3^(-6,9) cup v7^(-6,9) = (0, 0, 0, 0, 0, -1) in H^(-12,18)\n",
      "\n",
      "v4^(-6,9) cup v2^(-6,9) = (1, 0, 0, 0, 0, 0) in H^(-12,18)\n",
      "\n",
      "v4^(-6,9) cup v3^(-6,9) = (0, 1, 0, 0, 0, 0) in H^(-12,18)\n",
      "\n",
      "v4^(-6,9) cup v7^(-6,9) = (0, 0, 0, 0, -1, 0) in H^(-12,18)\n",
      "\n",
      "v5^(-6,9) cup v1^(-6,9) = (0, 1, 0, 0, 0, 0) in H^(-12,18)\n",
      "\n",
      "v5^(-6,9) cup v3^(-6,9) = (0, 0, 0, 0, 1, 0) in H^(-12,18)\n",
      "\n",
      "v5^(-6,9) cup v6^(-6,9) = (0, 0, 0, 0, 0, -1) in H^(-12,18)\n",
      "\n",
      "v6^(-6,9) cup v1^(-6,9) = (0, 0, -1, 0, 0, 0) in H^(-12,18)\n",
      "\n",
      "v6^(-6,9) cup v3^(-6,9) = (0, 0, 0, 1, 0, 0) in H^(-12,18)\n",
      "\n",
      "v6^(-6,9) cup v5^(-6,9) = (0, 0, 0, 0, 0, 1) in H^(-12,18)\n",
      "\n",
      "v7^(-6,9) cup v2^(-6,9) = (0, 0, 0, -1, 0, 0) in H^(-12,18)\n",
      "\n",
      "v7^(-6,9) cup v3^(-6,9) = (0, 0, 0, 0, 0, 1) in H^(-12,18)\n",
      "\n",
      "v7^(-6,9) cup v4^(-6,9) = (0, 0, 0, 0, 1, 0) in H^(-12,18)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-6,9) cup v3^(-8,12) = (1, 0, 0) in H^(-14,21)\n",
      "\n",
      "v1^(-6,9) cup v9^(-8,12) = (0, 1, 0) in H^(-14,21)\n",
      "\n",
      "v2^(-6,9) cup v3^(-8,12) = (0, -1, 0) in H^(-14,21)\n",
      "\n",
      "v2^(-6,9) cup v6^(-8,12) = (1, 0, 0) in H^(-14,21)\n",
      "\n",
      "v3^(-6,9) cup v1^(-8,12) = (3, 0, 0) in H^(-14,21)\n",
      "\n",
      "v3^(-6,9) cup v2^(-8,12) = (0, 3, 0) in H^(-14,21)\n",
      "\n",
      "v3^(-6,9) cup v3^(-8,12) = (0, 0, 3) in H^(-14,21)\n",
      "\n",
      "v4^(-6,9) cup v2^(-8,12) = (-1, 0, 0) in H^(-14,21)\n",
      "\n",
      "v4^(-6,9) cup v8^(-8,12) = (0, 0, 1) in H^(-14,21)\n",
      "\n",
      "v4^(-6,9) cup v9^(-8,12) = (0, 0, 1) in H^(-14,21)\n",
      "\n",
      "v5^(-6,9) cup v2^(-8,12) = (0, 0, 1) in H^(-14,21)\n",
      "\n",
      "v5^(-6,9) cup v4^(-8,12) = (1, 0, 0) in H^(-14,21)\n",
      "\n",
      "v6^(-6,9) cup v1^(-8,12) = (0, 1, 0) in H^(-14,21)\n",
      "\n",
      "v6^(-6,9) cup v6^(-8,12) = (0, 0, 1) in H^(-14,21)\n",
      "\n",
      "v6^(-6,9) cup v7^(-8,12) = (0, 0, 1) in H^(-14,21)\n",
      "\n",
      "v7^(-6,9) cup v1^(-8,12) = (0, 0, -1) in H^(-14,21)\n",
      "\n",
      "v7^(-6,9) cup v5^(-8,12) = (0, -1, 0) in H^(-14,21)\n",
      "\n",
      "\n",
      "\n",
      "v1^(-6,9) cup v7^(-9,14) = (1) in H^(-15,23)\n",
      "\n",
      "v2^(-6,9) cup v6^(-9,14) = (1) in H^(-15,23)\n",
      "\n",
      "v3^(-6,9) cup v1^(-9,14) = (-1) in H^(-15,23)\n",
      "\n",
      "v4^(-6,9) cup v5^(-9,14) = (-1) in H^(-15,23)\n",
      "\n",
      "v5^(-6,9) cup v4^(-9,14) = (1) in H^(-15,23)\n",
      "\n",
      "v6^(-6,9) cup v3^(-9,14) = (-1) in H^(-15,23)\n",
      "\n",
      "v7^(-6,9) cup v2^(-9,14) = (1) in H^(-15,23)\n",
      "\n",
      "\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "v1^(-7,11) cup v9^(-8,12) = (1) in H^(-15,23)\n",
      "\n",
      "v2^(-7,11) cup v8^(-8,12) = (-1) in H^(-15,23)\n",
      "\n",
      "v3^(-7,11) cup v3^(-8,12) = (1) in H^(-15,23)\n",
      "\n",
      "v4^(-7,11) cup v2^(-8,12) = (-1) in H^(-15,23)\n",
      "\n",
      "v5^(-7,11) cup v7^(-8,12) = (1) in H^(-15,23)\n",
      "\n",
      "v6^(-7,11) cup v6^(-8,12) = (-1) in H^(-15,23)\n",
      "\n",
      "v7^(-7,11) cup v1^(-8,12) = (-1) in H^(-15,23)\n",
      "\n",
      "v8^(-7,11) cup v5^(-8,12) = (1) in H^(-15,23)\n",
      "\n",
      "v9^(-7,11) cup v4^(-8,12) = (-1) in H^(-15,23)\n",
      "\n"
     ]
    }
   ],
   "source": [
    "print_cup_products_alt(g_sl3, e_sort_sl3, sl3_H_s_t_fixed_2, sl3_bases)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "1b07d990",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "6244741a",
   "metadata": {},
   "source": [
    "# Final note"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6173c2b7",
   "metadata": {},
   "source": [
    "We can use inbuilt sage functions to double check that some of our cohomology calculations (without considering grading) are correct."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "id": "d0e18956",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\verb|SL_3(Z_p)|\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\verb|SL_3(Z_p)|$"
      ],
      "text/plain": [
       "'SL_3(Z_p)'"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{0 : \\Bold{Q}^{1}, 1 : \\Bold{Q}^{3}, 2 : \\Bold{Q}^{9}, 3 : \\Bold{Q}^{16}, 4 : \\Bold{Q}^{18}, 5 : \\Bold{Q}^{16}, 6 : \\Bold{Q}^{9}, 7 : \\Bold{Q}^{3}, 8 : \\Bold{Q}^{1}\\right\\}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{0 : \\Bold{Q}^{1}, 1 : \\Bold{Q}^{3}, 2 : \\Bold{Q}^{9}, 3 : \\Bold{Q}^{16}, 4 : \\Bold{Q}^{18}, 5 : \\Bold{Q}^{16}, 6 : \\Bold{Q}^{9}, 7 : \\Bold{Q}^{3}, 8 : \\Bold{Q}^{1}\\right\\}$"
      ],
      "text/plain": [
       "{0: Vector space of dimension 1 over Rational Field,\n",
       " 1: Vector space of dimension 3 over Rational Field,\n",
       " 2: Vector space of dimension 9 over Rational Field,\n",
       " 3: Vector space of dimension 16 over Rational Field,\n",
       " 4: Vector space of dimension 18 over Rational Field,\n",
       " 5: Vector space of dimension 16 over Rational Field,\n",
       " 6: Vector space of dimension 9 over Rational Field,\n",
       " 7: Vector space of dimension 3 over Rational Field,\n",
       " 8: Vector space of dimension 1 over Rational Field}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\newcommand{\\ZZ}{\\Bold{Z}}\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{0 : \\ZZ, 1 : \\ZZ \\times \\ZZ \\times \\ZZ, 2 : \\ZZ^{9}, 3 : \\ZZ^{16}, 4 : \\ZZ^{18} \\times C_{2} \\times C_{2}, 5 : \\ZZ^{16} \\times C_{2} \\times C_{2}, 6 : \\ZZ^{9}, 7 : \\ZZ \\times \\ZZ \\times \\ZZ, 8 : \\ZZ\\right\\}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\newcommand{\\ZZ}{\\Bold{Z}}\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{0 : \\ZZ, 1 : \\ZZ \\times \\ZZ \\times \\ZZ, 2 : \\ZZ^{9}, 3 : \\ZZ^{16}, 4 : \\ZZ^{18} \\times C_{2} \\times C_{2}, 5 : \\ZZ^{16} \\times C_{2} \\times C_{2}, 6 : \\ZZ^{9}, 7 : \\ZZ \\times \\ZZ \\times \\ZZ, 8 : \\ZZ\\right\\}$"
      ],
      "text/plain": [
       "{0: Z,\n",
       " 1: Z x Z x Z,\n",
       " 2: Z^9,\n",
       " 3: Z^16,\n",
       " 4: Z^18 x C2 x C2,\n",
       " 5: Z^16 x C2 x C2,\n",
       " 6: Z^9,\n",
       " 7: Z x Z x Z,\n",
       " 8: Z}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{0 : \\Bold{F}_{2}^{1}, 1 : \\Bold{F}_{2}^{3}, 2 : \\Bold{F}_{2}^{9}, 3 : \\Bold{F}_{2}^{18}, 4 : \\Bold{F}_{2}^{22}, 5 : \\Bold{F}_{2}^{18}, 6 : \\Bold{F}_{2}^{9}, 7 : \\Bold{F}_{2}^{3}, 8 : \\Bold{F}_{2}^{1}\\right\\}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{0 : \\Bold{F}_{2}^{1}, 1 : \\Bold{F}_{2}^{3}, 2 : \\Bold{F}_{2}^{9}, 3 : \\Bold{F}_{2}^{18}, 4 : \\Bold{F}_{2}^{22}, 5 : \\Bold{F}_{2}^{18}, 6 : \\Bold{F}_{2}^{9}, 7 : \\Bold{F}_{2}^{3}, 8 : \\Bold{F}_{2}^{1}\\right\\}$"
      ],
      "text/plain": [
       "{0: Vector space of dimension 1 over Finite Field of size 2,\n",
       " 1: Vector space of dimension 3 over Finite Field of size 2,\n",
       " 2: Vector space of dimension 9 over Finite Field of size 2,\n",
       " 3: Vector space of dimension 18 over Finite Field of size 2,\n",
       " 4: Vector space of dimension 22 over Finite Field of size 2,\n",
       " 5: Vector space of dimension 18 over Finite Field of size 2,\n",
       " 6: Vector space of dimension 9 over Finite Field of size 2,\n",
       " 7: Vector space of dimension 3 over Finite Field of size 2,\n",
       " 8: Vector space of dimension 1 over Finite Field of size 2}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{0 : \\Bold{F}_{3}^{1}, 1 : \\Bold{F}_{3}^{3}, 2 : \\Bold{F}_{3}^{9}, 3 : \\Bold{F}_{3}^{16}, 4 : \\Bold{F}_{3}^{18}, 5 : \\Bold{F}_{3}^{16}, 6 : \\Bold{F}_{3}^{9}, 7 : \\Bold{F}_{3}^{3}, 8 : \\Bold{F}_{3}^{1}\\right\\}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{0 : \\Bold{F}_{3}^{1}, 1 : \\Bold{F}_{3}^{3}, 2 : \\Bold{F}_{3}^{9}, 3 : \\Bold{F}_{3}^{16}, 4 : \\Bold{F}_{3}^{18}, 5 : \\Bold{F}_{3}^{16}, 6 : \\Bold{F}_{3}^{9}, 7 : \\Bold{F}_{3}^{3}, 8 : \\Bold{F}_{3}^{1}\\right\\}$"
      ],
      "text/plain": [
       "{0: Vector space of dimension 1 over Finite Field of size 3,\n",
       " 1: Vector space of dimension 3 over Finite Field of size 3,\n",
       " 2: Vector space of dimension 9 over Finite Field of size 3,\n",
       " 3: Vector space of dimension 16 over Finite Field of size 3,\n",
       " 4: Vector space of dimension 18 over Finite Field of size 3,\n",
       " 5: Vector space of dimension 16 over Finite Field of size 3,\n",
       " 6: Vector space of dimension 9 over Finite Field of size 3,\n",
       " 7: Vector space of dimension 3 over Finite Field of size 3,\n",
       " 8: Vector space of dimension 1 over Finite Field of size 3}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{0 : \\Bold{F}_{5}^{1}, 1 : \\Bold{F}_{5}^{3}, 2 : \\Bold{F}_{5}^{9}, 3 : \\Bold{F}_{5}^{16}, 4 : \\Bold{F}_{5}^{18}, 5 : \\Bold{F}_{5}^{16}, 6 : \\Bold{F}_{5}^{9}, 7 : \\Bold{F}_{5}^{3}, 8 : \\Bold{F}_{5}^{1}\\right\\}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{0 : \\Bold{F}_{5}^{1}, 1 : \\Bold{F}_{5}^{3}, 2 : \\Bold{F}_{5}^{9}, 3 : \\Bold{F}_{5}^{16}, 4 : \\Bold{F}_{5}^{18}, 5 : \\Bold{F}_{5}^{16}, 6 : \\Bold{F}_{5}^{9}, 7 : \\Bold{F}_{5}^{3}, 8 : \\Bold{F}_{5}^{1}\\right\\}$"
      ],
      "text/plain": [
       "{0: Vector space of dimension 1 over Finite Field of size 5,\n",
       " 1: Vector space of dimension 3 over Finite Field of size 5,\n",
       " 2: Vector space of dimension 9 over Finite Field of size 5,\n",
       " 3: Vector space of dimension 16 over Finite Field of size 5,\n",
       " 4: Vector space of dimension 18 over Finite Field of size 5,\n",
       " 5: Vector space of dimension 16 over Finite Field of size 5,\n",
       " 6: Vector space of dimension 9 over Finite Field of size 5,\n",
       " 7: Vector space of dimension 3 over Finite Field of size 5,\n",
       " 8: Vector space of dimension 1 over Finite Field of size 5}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{0 : \\Bold{F}_{7}^{1}, 1 : \\Bold{F}_{7}^{3}, 2 : \\Bold{F}_{7}^{9}, 3 : \\Bold{F}_{7}^{16}, 4 : \\Bold{F}_{7}^{18}, 5 : \\Bold{F}_{7}^{16}, 6 : \\Bold{F}_{7}^{9}, 7 : \\Bold{F}_{7}^{3}, 8 : \\Bold{F}_{7}^{1}\\right\\}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{0 : \\Bold{F}_{7}^{1}, 1 : \\Bold{F}_{7}^{3}, 2 : \\Bold{F}_{7}^{9}, 3 : \\Bold{F}_{7}^{16}, 4 : \\Bold{F}_{7}^{18}, 5 : \\Bold{F}_{7}^{16}, 6 : \\Bold{F}_{7}^{9}, 7 : \\Bold{F}_{7}^{3}, 8 : \\Bold{F}_{7}^{1}\\right\\}$"
      ],
      "text/plain": [
       "{0: Vector space of dimension 1 over Finite Field of size 7,\n",
       " 1: Vector space of dimension 3 over Finite Field of size 7,\n",
       " 2: Vector space of dimension 9 over Finite Field of size 7,\n",
       " 3: Vector space of dimension 16 over Finite Field of size 7,\n",
       " 4: Vector space of dimension 18 over Finite Field of size 7,\n",
       " 5: Vector space of dimension 16 over Finite Field of size 7,\n",
       " 6: Vector space of dimension 9 over Finite Field of size 7,\n",
       " 7: Vector space of dimension 3 over Finite Field of size 7,\n",
       " 8: Vector space of dimension 1 over Finite Field of size 7}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{0 : \\Bold{F}_{11}^{1}, 1 : \\Bold{F}_{11}^{3}, 2 : \\Bold{F}_{11}^{9}, 3 : \\Bold{F}_{11}^{16}, 4 : \\Bold{F}_{11}^{18}, 5 : \\Bold{F}_{11}^{16}, 6 : \\Bold{F}_{11}^{9}, 7 : \\Bold{F}_{11}^{3}, 8 : \\Bold{F}_{11}^{1}\\right\\}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{0 : \\Bold{F}_{11}^{1}, 1 : \\Bold{F}_{11}^{3}, 2 : \\Bold{F}_{11}^{9}, 3 : \\Bold{F}_{11}^{16}, 4 : \\Bold{F}_{11}^{18}, 5 : \\Bold{F}_{11}^{16}, 6 : \\Bold{F}_{11}^{9}, 7 : \\Bold{F}_{11}^{3}, 8 : \\Bold{F}_{11}^{1}\\right\\}$"
      ],
      "text/plain": [
       "{0: Vector space of dimension 1 over Finite Field of size 11,\n",
       " 1: Vector space of dimension 3 over Finite Field of size 11,\n",
       " 2: Vector space of dimension 9 over Finite Field of size 11,\n",
       " 3: Vector space of dimension 16 over Finite Field of size 11,\n",
       " 4: Vector space of dimension 18 over Finite Field of size 11,\n",
       " 5: Vector space of dimension 16 over Finite Field of size 11,\n",
       " 6: Vector space of dimension 9 over Finite Field of size 11,\n",
       " 7: Vector space of dimension 3 over Finite Field of size 11,\n",
       " 8: Vector space of dimension 1 over Finite Field of size 11}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{0 : \\Bold{F}_{13}^{1}, 1 : \\Bold{F}_{13}^{3}, 2 : \\Bold{F}_{13}^{9}, 3 : \\Bold{F}_{13}^{16}, 4 : \\Bold{F}_{13}^{18}, 5 : \\Bold{F}_{13}^{16}, 6 : \\Bold{F}_{13}^{9}, 7 : \\Bold{F}_{13}^{3}, 8 : \\Bold{F}_{13}^{1}\\right\\}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{0 : \\Bold{F}_{13}^{1}, 1 : \\Bold{F}_{13}^{3}, 2 : \\Bold{F}_{13}^{9}, 3 : \\Bold{F}_{13}^{16}, 4 : \\Bold{F}_{13}^{18}, 5 : \\Bold{F}_{13}^{16}, 6 : \\Bold{F}_{13}^{9}, 7 : \\Bold{F}_{13}^{3}, 8 : \\Bold{F}_{13}^{1}\\right\\}$"
      ],
      "text/plain": [
       "{0: Vector space of dimension 1 over Finite Field of size 13,\n",
       " 1: Vector space of dimension 3 over Finite Field of size 13,\n",
       " 2: Vector space of dimension 9 over Finite Field of size 13,\n",
       " 3: Vector space of dimension 16 over Finite Field of size 13,\n",
       " 4: Vector space of dimension 18 over Finite Field of size 13,\n",
       " 5: Vector space of dimension 16 over Finite Field of size 13,\n",
       " 6: Vector space of dimension 9 over Finite Field of size 13,\n",
       " 7: Vector space of dimension 3 over Finite Field of size 13,\n",
       " 8: Vector space of dimension 1 over Finite Field of size 13}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "show(\"SL_3(Z_p)\")\n",
    "for ring in [QQ,ZZ,GF(2),GF(3),GF(5),GF(7),GF(11),GF(13)]:\n",
    "    tmpL = LieAlgebra(ring, {\n",
    "            ('x1','x6'): {'x2':-1},\n",
    "            ('x1','x7'): {'x3':1},\n",
    "            ('x1','x8'): {'x4':-1, 'x5':-1},\n",
    "            ('x2','x7'): {'x4':-1},\n",
    "            ('x3','x6'): {'x5':-1},\n",
    "            ('x6','x7'): {'x8':-1},\n",
    "        }, names='x1,x2,x3,x4,x5,x6,x7,x8')\n",
    "    show(tmpL.cohomology())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a9d20867",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "1276951d",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "SageMath 9.7",
   "language": "sage",
   "name": "sagemath-9.7"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.10.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}