nil_coxeter_algebra.py

"""
Nil-Coxeter Algebra
"""
#*****************************************************************************
#  Copyright (C) 2011 Chris Berg <cberg at fields.utoronto.ca>
#                     Anne Schilling <anne at math.ucdavis.edu>
#
#  Distributed under the terms of the GNU General Public License (GPL)
#                  http://www.gnu.org/licenses/
#*****************************************************************************
from sage.algebras.iwahori_hecke_algebra import IwahoriHeckeAlgebra
from sage.combinat.sf.sf import SymmetricFunctions
from sage.combinat.sf.kschur import kSchurFunctions
from sage.misc.misc_c import prod
from sage.rings.rational_field import QQ
from sage.combinat.partition import Partitions

class NilCoxeterAlgebra(IwahoriHeckeAlgebra.T):
    r"""
    Construct the Nil-Coxeter algebra of given type. This is the algebra
    with generators `u_i` for every node `i` of the corresponding Dynkin
    diagram. It has the usual braid relations (from the Weyl group) as well
    as the quadratic relation `u_i^2 = 0`.

    INPUT:

    - ``W`` -- a Weyl group

    OPTIONAL ARGUEMENTS:

    - ``base_ring`` -- a ring (default is the rational numbers)
    - ``prefix`` -- a label for the generators (default "u")

    EXAMPLES::

        sage: U = NilCoxeterAlgebra(WeylGroup(['A',3,1]))
        sage: u0, u1, u2, u3 = U.algebra_generators()
        sage: u1*u1
        0
        sage: u2*u1*u2 == u1*u2*u1
        True
        sage: U.an_element()
        u[0,1,2,3] + 3*u[0,1] + 2*u[0] + 1
    """

    def __init__(self, W, base_ring = QQ, prefix='u'):
        r"""
        Initiate the affine nil-Coxeter algebra corresponding to the Weyl
        group `W` over the base ring.

        EXAMPLES::

            sage: U = NilCoxeterAlgebra(WeylGroup(['A',3,1])); U
            The Nil-Coxeter Algebra of Type A3~ over Rational Field
            sage: TestSuite(U).run()

            sage: U = NilCoxeterAlgebra(WeylGroup(['C',3]), ZZ); U
            The Nil-Coxeter Algebra of Type C3 over Integer Ring
            sage: TestSuite(U).run()
        """

        self._W = W
        self._n = W.n
        self._base_ring = base_ring
        self._cartan_type = W.cartan_type()
        H = IwahoriHeckeAlgebra(W, 0, 0, base_ring=base_ring)
        super(IwahoriHeckeAlgebra.T,self).__init__(H, prefix=prefix)

    def _repr_(self):
        r"""
        EXAMPLES ::

            sage: NilCoxeterAlgebra(WeylGroup(['A',3,1])) # indirect doctest
            The Nil-Coxeter Algebra of Type A3~ over Rational Field

        """

        return "The Nil-Coxeter Algebra of Type %s over %s"%(self._cartan_type._repr_(compact=True), self.base_ring())

    def homogeneous_generator_noncommutative_variables(self, r):
        r"""
        Give the `r^{th}` homogeneous function inside the Nil-Coxeter algebra.
        In finite type `A` this is the sum of all decreasing elements of length `r`.
        In affine type `A` this is the sum of all cyclically decreasing elements of length `r`.
        This is only defined in finite type `A`, `B` and affine types `A^{(1)}`, `B^{(1)}`, `C^{(1)}`, `D^{(1)}`.

        INPUT:

        - ``r`` -- a positive integer at most the rank of the Weyl group

        EXAMPLES::

            sage: U = NilCoxeterAlgebra(WeylGroup(['A',3,1]))
            sage: U.homogeneous_generator_noncommutative_variables(2)
            u[1,0] + u[2,0] + u[0,3] + u[3,2] + u[3,1] + u[2,1]

            sage: U = NilCoxeterAlgebra(WeylGroup(['B',4]))
            sage: U.homogeneous_generator_noncommutative_variables(2)
            u[1,2] + u[2,1] + u[3,1] + u[4,1] + u[2,3] + u[3,2] + u[4,2] + u[3,4] + u[4,3]

            sage: U = NilCoxeterAlgebra(WeylGroup(['C',3]))
            sage: U.homogeneous_generator_noncommutative_variables(2)
            Traceback (most recent call last):
            ...
            AssertionError: Analogue of symmetric functions in noncommutative variables is not defined in type ['C', 3]

        TESTS::

            sage: U = NilCoxeterAlgebra(WeylGroup(['B',3,1]))
            sage: U.homogeneous_generator_noncommutative_variables(-1)
            0
            sage: U.homogeneous_generator_noncommutative_variables(0)
            1

        """
        assert (len(self._cartan_type) == 2 and self._cartan_type[0] in ['A','B']) or (len(self._cartan_type) == 3 and self._cartan_type[2] == 1), "Analogue of symmetric functions in noncommutative variables is not defined in type %s"%(self._cartan_type)
        if r >= self._n:
            return self.zero()
        return self.sum_of_monomials(w for w in self._W.pieri_factors() if w.length() == r)

    def homogeneous_noncommutative_variables(self,la):
        r"""
        Give the homogeneous function indexed by `la`, viewed inside the Nil-Coxeter algebra.
        This is only defined in finite type `A`, `B` and affine types `A^{(1)}`, `B^{(1)}`, `C^{(1)}`, `D^{(1)}`.

        INPUT:

        - ``la`` -- a partition with first part bounded by the rank of the Weyl group

        EXAMPLES::

            sage: U = NilCoxeterAlgebra(WeylGroup(['B',2,1]))
            sage: U.homogeneous_noncommutative_variables([2,1])
            u[1,2,0] + 2*u[2,1,0] + u[0,2,0] + u[0,2,1] + u[1,2,1] + u[2,1,2] + u[2,0,2] + u[1,0,2]

        TESTS::

            sage: U = NilCoxeterAlgebra(WeylGroup(['B',2,1]))
            sage: U.homogeneous_noncommutative_variables([])
            1

        """
        return prod(self.homogeneous_generator_noncommutative_variables(p) for p in la)

    def k_schur_noncommutative_variables(self, la):
        r"""
        In type `A^{(1)}` this is the `k`-Schur function in noncommutative variables defined by Thomas Lam.

        REFERENCES:

           .. [Lam2005] T. Lam, Affine Stanley symmetric functions, Amer. J. Math.  128  (2006),  no. 6, 1553--1586.

        This function is currently only defined in type `A^{(1)}`.

        INPUT:

        - ``la`` -- a partition with first part bounded by the rank of the Weyl group

        EXAMPLES::

            sage: A = NilCoxeterAlgebra(WeylGroup(['A',3,1]))
            sage: A.k_schur_noncommutative_variables([2,2])
            u[0,3,1,0] + u[3,1,2,0] + u[1,2,0,1] + u[3,2,0,3] + u[2,0,3,1] + u[2,3,1,2]

        TESTS::

            sage: A = NilCoxeterAlgebra(WeylGroup(['A',3,1]))
            sage: A.k_schur_noncommutative_variables([])
            1

            sage: A.k_schur_noncommutative_variables([1,2])
            Traceback (most recent call last):
            ...
            AssertionError: [1, 2] is not a partition.

            sage: A.k_schur_noncommutative_variables([4,2])
            Traceback (most recent call last):
            ...
            AssertionError: [4, 2] is not a 3-bounded partition.

            sage: C = NilCoxeterAlgebra(WeylGroup(['C',3,1]))
            sage: C.k_schur_noncommutative_variables([2,2])
            Traceback (most recent call last):
            ...
            AssertionError: Weyl Group of type ['C', 3, 1] (as a matrix group acting on the root space) is not affine type A.


        """
        assert self._cartan_type[0] == 'A' and len(self._cartan_type) == 3 and self._cartan_type[2] == 1, "%s is not affine type A."%(self._W)
        assert la in Partitions(), "%s is not a partition."%(la)
        assert (len(la) == 0 or la[0] < self._W.n), "%s is not a %s-bounded partition."%(la, self._W.n-1)
        Sym = SymmetricFunctions(self._base_ring)
        h = Sym.homogeneous()
        ks = Sym.kschur(self._n-1,1)
        f = h(ks[la])
        return sum(f.coefficient(x)*self.homogeneous_noncommutative_variables(x) for x in f.support())