def _homogeneous_generators_noncommutative_variables_zero_Hecke(self, r):
r"""
Returns the ``r^{th}`` homogeneous generator, viewed as an element inside the
affine zero Hecke algebra. This is the sum of all cyclicly decreasing elements
of order ``r``.
INPUT:
- ``r`` -- A positive integer
OUTPUT:
- An element of the affine zero Hecke algebra.
EXAMPLES::
sage: g = SymmetricFunctions(QQ).kBoundedSubspace(3,1).K_kschur()
sage: g._homogeneous_generators_noncommutative_variables_zero_Hecke(2)
T1*T0 + T2*T0 + T0*T3 + T3*T2 + T3*T1 + T2*T1
sage: g._homogeneous_generators_noncommutative_variables_zero_Hecke(0)
1
"""
from sage.combinat.root_system.weyl_group import WeylGroup
from sage.algebras.iwahori_hecke_algebra import IwahoriHeckeAlgebraT
W = WeylGroup(['A',self.k,1])
H = IwahoriHeckeAlgebraT(W, 0, base_ring = self.base_ring())
Hgens = H.algebra_generators()
S = [w.reduced_word() for w in W.pieri_factors() if w.length() == r]
return sum( (prod((Hgens[i] for i in w), 1) for w in S), 0 )
def _homogeneous_generators_noncommutative_variables_zero_Hecke(self, r):
r"""
Returns the ``r^{th}`` homogeneous generator, viewed as an element inside the
affine zero Hecke algebra. This is the sum of all cyclicly decreasing elements
of order ``r``.
INPUT:
- ``r`` -- A positive integer
OUTPUT:
- An element of the affine zero Hecke algebra.
EXAMPLES::
sage: g = SymmetricFunctions(QQ).kBoundedSubspace(3,1).K_kschur()
sage: g._homogeneous_generators_noncommutative_variables_zero_Hecke(2)
T1*T0 + T2*T0 + T0*T3 + T3*T2 + T3*T1 + T2*T1
sage: g._homogeneous_generators_noncommutative_variables_zero_Hecke(0)
1
"""
from sage.combinat.root_system.weyl_group import WeylGroup
from sage.algebras.iwahori_hecke_algebra import IwahoriHeckeAlgebraT
W = WeylGroup(['A',self.k,1])
H = IwahoriHeckeAlgebraT(W, 0, base_ring = self.base_ring())
Hgens = H.algebra_generators()
S = [w.reduced_word() for w in W.pieri_factors() if w.length() == r]
return sum( (prod((Hgens[i] for i in w), 1) for w in S), 0 )
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()
IwahoriHeckeAlgebraT.__init__(self, W, 0, 0, base_ring, prefix=prefix)
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()
IwahoriHeckeAlgebraT.__init__(self, W, 0, 0, base_ring, prefix=prefix)
