Python Matrix.nullity方法代码示例

# 需要导入模块: from sage.matrix.constructor import Matrix [as 别名]
# 或者: from sage.matrix.constructor.Matrix import nullity [as 别名]

#.........这里部分代码省略.........
        sage: from sage.algebras.schur_algebra import GL_irreducible_character
        sage: GL_irreducible_character(2, [7], GF(3))
        m[4, 3] + m[6, 1] + m[7]
    """
    mbasis = SymmetricFunctions(QQ).m()
    r = sum(mu)
    M = SchurTensorModule(KK, n, r)
    A = M._schur
    SGA = M._sga

    # make ST the superstandard tableau of shape mu
    from sage.combinat.tableau import from_shape_and_word

    ST = from_shape_and_word(mu, range(1, r + 1), convention="English")

    # make ell the reading word of the highest weight tableau of shape mu
    ell = [i + 1 for i, l in enumerate(mu) for dummy in range(l)]

    e = M.basis()[tuple(ell)]  # the element e_l

    # This is the notation `\{X\}` from just before (5.3a) of [GreenPoly]_.
    S = SGA._indices
    BracC = SGA._from_dict({S(x.tuple()): x.sign() for x in ST.column_stabilizer()}, remove_zeros=False)
    f = e * BracC  # M.action_by_symmetric_group_algebra(e, BracC)

    # [Green, Theorem 5.3b] says that a basis of the Carter-Lusztig
    # module V_\mu is given by taking this f, and multiplying by all
    # xi_{i,ell} with ell as above and i semistandard.

    carter_lusztig = []
    for T in SemistandardTableaux(mu, max_entry=n):
        i = tuple(flatten(T))
        schur_rep = schur_representative_from_index(i, tuple(ell))
        y = A.basis()[schur_rep] * e  # M.action_by_Schur_alg(A.basis()[schur_rep], e)
        carter_lusztig.append(y.to_vector())

    # Therefore, we now have carter_lusztig as a list giving the basis
    # of `V_\mu`

    # We want to think of expressing this character as a sum of monomial
    # symmetric functions.

    # We will determine a basis element for each m_\lambda in the
    # character, and we want to keep track of them by \lambda.

    # That means that we only want to pick out the basis elements above for
    # those semistandard words whose content is a partition.

    contents = Partitions(r, max_length=n).list()
    # all partitions of r, length at most n

    # JJ will consist of a list for each element of `contents`,
    # recording the list
    # of semistandard tableaux words with that content

    # graded_basis will consist of the a corresponding basis element
    graded_basis = []
    JJ = []
    for i in range(len(contents)):
        graded_basis.append([])
        JJ.append([])
    for T in SemistandardTableaux(mu, max_entry=n):
        i = tuple(flatten(T))
        # Get the content of T
        con = [0] * n
        for a in i:
            con[a - 1] += 1
        try:
            P = Partition(con)
            P_index = contents.index(P)
            JJ[P_index].append(i)
            schur_rep = schur_representative_from_index(i, tuple(ell))
            x = A.basis()[schur_rep] * f  # M.action_by_Schur_alg(A.basis()[schur_rep], f)
            graded_basis[P_index].append(x.to_vector())
        except ValueError:
            pass

    # There is an inner product on the Carter-Lusztig module V_\mu; its
    # maximal submodule is exactly the kernel of the inner product.

    # Now, for each possible partition content, we look at the graded piece of
    # that degree, and we record how these elements pair with each of the
    # elements of carter_lusztig.

    # The kernel of this pairing is the part of this graded piece which is
    # not in the irreducible module for \mu.

    length = len(carter_lusztig)

    phi = mbasis.zero()
    for aa in range(len(contents)):
        mat = []
        for kk in range(len(JJ[aa])):
            temp = []
            for j in range(length):
                temp.append(graded_basis[aa][kk].inner_product(carter_lusztig[j]))
            mat.append(temp)
        angle = Matrix(mat)
        phi += (len(JJ[aa]) - angle.nullity()) * mbasis(contents[aa])
    return phi