Python Tableau.row_stabilizer方法代码示例

# 需要导入模块: from tableau import Tableau [as 别名]
# 或者: from tableau.Tableau import row_stabilizer [as 别名]
def a(tableau, star=0):
    r"""
    The row projection operator corresponding to the Young tableau
    ``tableau`` (which is supposed to contain every integer from
    `1` to its size precisely once, but may and may not be standard).

    This is the sum (in the group algebra of the relevant symmetric
    group over `\QQ`) of all the permutations which preserve
    the rows of ``tableau``. It is called `a_{\text{tableau}}` in
    [EtRT]_, Section 4.2.

    REFERENCES:

    .. [EtRT] Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai
       Liu, Alex Schwendner, Dmitry Vaintrob, Elena Yudovina,
       "Introduction to representation theory",
       :arXiv:`0901.0827v5`.

    EXAMPLES::

        sage: from sage.combinat.symmetric_group_algebra import a
        sage: a([[1,2]])
        [1, 2] + [2, 1]
        sage: a([[1],[2]])
        [1, 2]
        sage: a([])
        []
        sage: a([[1, 5], [2, 3], [4]])
        [1, 2, 3, 4, 5] + [1, 3, 2, 4, 5] + [5, 2, 3, 4, 1] + [5, 3, 2, 4, 1]
    """
    t = Tableau(tableau)
    if star:
        t = t.restrict(t.size()-star)

    rs = t.row_stabilizer().list()
    n = t.size()

    # This all should be over ZZ, not over QQ, but symmetric group
    # algebras don't seem to preserve coercion (the one over ZZ
    # doesn't coerce into the one over QQ even for the same n),
    # and the QQ version of this method is more important, so let
    # me stay with QQ.
    # TODO: Fix this.
    sgalg = SymmetricGroupAlgebra(QQ, n)
    one = QQ.one()
    P = permutation.Permutation

    # Ugly hack for the case of an empty tableau, due to the
    # annoyance of Permutation(Tableau([]).row_stabilizer()[0])
    # being [1] rather than [] (which seems to have its origins in
    # permutation group code).
    # TODO: Fix this.
    if len(tableau) == 0:
        return sgalg.one()

    rd = dict((P(h), one) for h in rs)
    return sgalg._from_dict(rd)

# 需要导入模块: from tableau import Tableau [as 别名]
# 或者: from tableau.Tableau import row_stabilizer [as 别名]
def e(tableau, star=0):
    """
    The unnormalized Young projection operator.

    EXAMPLES::
    
        sage: from sage.combinat.symmetric_group_algebra import e
        sage: e([[1,2]])
        [1, 2] + [2, 1]
        sage: e([[1],[2]])
        [1, 2] - [2, 1]

    There are differing conventions for the order of the symmetrizers
    and antisymmetrizers.  This example illustrates our conventions::

        sage: e([[1,2],[3]])
        [1, 2, 3] + [2, 1, 3] - [3, 1, 2] - [3, 2, 1]
    """
    t = Tableau(tableau)
    if star:
        t = t.restrict(t.size()-star)

    mult = permutation_options['mult']
    permutation_options['mult'] = 'l2r'
    
    if t in e_cache:
        res = e_cache[t]
    else:
        rs = t.row_stabilizer().list()
        cs = t.column_stabilizer().list()
        n = t.size()

        QSn = SymmetricGroupAlgebra(QQ, n)
        one = QQ(1)
        P = permutation.Permutation

        rd = dict((P(h), one) for h in rs)
        sym = QSn._from_dict(rd)

        cd = dict((P(v), v.sign()*one) for v in cs)
        antisym = QSn._from_dict(cd)

        res = antisym*sym

        e_cache[t] = res

    permutation_options['mult'] = mult    
    return res

# 需要导入模块: from tableau import Tableau [as 别名]
# 或者: from tableau.Tableau import row_stabilizer [as 别名]
def e(tableau, star=0):
    """
    The unnormalized Young projection operator corresponding to
    the Young tableau ``tableau`` (which is supposed to contain
    every integer from `1` to its size precisely once, but may
    and may not be standard).

    EXAMPLES::

        sage: from sage.combinat.symmetric_group_algebra import e
        sage: e([[1,2]])
        [1, 2] + [2, 1]
        sage: e([[1],[2]])
        [1, 2] - [2, 1]
        sage: e([])
        []

    There are differing conventions for the order of the symmetrizers
    and antisymmetrizers.  This example illustrates our conventions::

        sage: e([[1,2],[3]])
        [1, 2, 3] + [2, 1, 3] - [3, 1, 2] - [3, 2, 1]
    """
    t = Tableau(tableau)
    if star:
        t = t.restrict(t.size()-star)

    mult = permutation_options['mult']
    permutation_options['mult'] = 'l2r'

    if t in e_cache:
        res = e_cache[t]
    else:
        rs = t.row_stabilizer().list()
        cs = t.column_stabilizer().list()
        n = t.size()

        QSn = SymmetricGroupAlgebra(QQ, n)
        one = QQ.one()
        P = permutation.Permutation

        rd = dict((P(h), one) for h in rs)
        sym = QSn._from_dict(rd)

        cd = dict((P(v), v.sign()*one) for v in cs)
        antisym = QSn._from_dict(cd)

        res = antisym*sym

        # Ugly hack for the case of an empty tableau, due to the
        # annoyance of Permutation(Tableau([]).row_stabilizer()[0])
        # being [1] rather than [] (which seems to have its origins in
        # permutation group code).
        # TODO: Fix this.
        if len(tableau) == 0:
            res = QSn.one()

        e_cache[t] = res

    permutation_options['mult'] = mult

    return res