def __init__(self, R):
"""
Initialize ``self``.
EXAMPLES::
sage: NCSymD1 = SymmetricFunctionsNonCommutingVariablesDual(FiniteField(23))
sage: NCSymD2 = SymmetricFunctionsNonCommutingVariablesDual(Integers(23))
sage: TestSuite(SymmetricFunctionsNonCommutingVariables(QQ).dual()).run()
"""
# change the line below to assert(R in Rings()) once MRO issues from #15536, #15475 are resolved
assert(R in Fields() or R in Rings()) # side effect of this statement assures MRO exists for R
self._base = R # Won't be needed once CategoryObject won't override base_ring
category = GradedHopfAlgebras(R) # TODO: .Commutative()
Parent.__init__(self, category=category.WithRealizations())
# Bases
w = self.w()
# Embedding of Sym in the homogeneous bases into DNCSym in the w basis
Sym = SymmetricFunctions(self.base_ring())
Sym_h_to_w = Sym.h().module_morphism(w.sum_of_partitions,
triangular='lower',
inverse_on_support=w._set_par_to_par,
codomain=w, category=category)
Sym_h_to_w.register_as_coercion()
self.to_symmetric_function = Sym_h_to_w.section()
def __init__(self, R):
"""
Initialize ``self``.
EXAMPLES::
sage: TestSuite(SymmetricFunctionsNonCommutingVariables(QQ).dual()).run()
"""
self._base = R # Won't be needed once CategoryObject won't override base_ring
category = GradedHopfAlgebras(R) # TODO: .Commutative()
Parent.__init__(self, category=category.WithRealizations())
# Bases
w = self.w()
# Embedding of Sym in the homogeneous bases into DNCSym in the w basis
Sym = SymmetricFunctions(self.base_ring())
Sym_h_to_w = Sym.h().module_morphism(
w.sum_of_partitions,
triangular='lower',
inverse_on_support=w._set_par_to_par,
codomain=w,
category=category)
Sym_h_to_w.register_as_coercion()
self.to_symmetric_function = Sym_h_to_w.section()
def K_k_Schur_non_commutative_variables(self,la):
r"""
Returns the K-`k`-Schur function, as embedded inside the affine zero Hecke algebra.
INPUT:
- ``la`` -- A `k`-bounded Partition
OUTPUT:
- An element of the affine zero Hecke algebra.
EXAMPLES::
sage: g = SymmetricFunctions(QQ).kBoundedSubspace(3,1).K_kschur()
sage: g.K_k_Schur_non_commutative_variables([2,1])
T3*T1*T0 + T1*T2*T0 + T3*T2*T0 - T2*T0 + T0*T1*T0 + T2*T0*T1 + T0*T3*T0 + T2*T0*T3 + T0*T3*T1 + T2*T3*T2 - T3*T1 + T2*T3*T1 + T3*T1*T2 + T1*T2*T1
sage: g.K_k_Schur_non_commutative_variables([])
1
sage: g.K_k_Schur_non_commutative_variables([4,1])
Traceback (most recent call last):
...
ValueError: Partition should be 3-bounded
"""
SF = SymmetricFunctions(self.base_ring())
h = SF.h()
S = h(self._g_to_kh_on_basis(la)).support()
return sum(h(self._g_to_kh_on_basis(la)).coefficient(x)*self.homogeneous_basis_noncommutative_variables_zero_Hecke(x) for x in S)
def __init__(self, base_ring=QQ['t'], prefix='S'):
r"""
Initialize ``self``.
EXAMPLES::
sage: S = ShiftingOperatorAlgebra(QQ['t'])
sage: TestSuite(S).run()
"""
indices = ShiftingSequenceSpace()
cat = Algebras(base_ring).WithBasis()
CombinatorialFreeModule.__init__(self,
base_ring,
indices,
prefix=prefix,
bracket=False,
category=cat)
# Setup default conversions
sym = SymmetricFunctions(base_ring)
self._sym_h = sym.h()
self._sym_s = sym.s()
self._sym_h.register_conversion(
self.module_morphism(self._supp_to_h, codomain=self._sym_h))
self._sym_s.register_conversion(
self.module_morphism(self._supp_to_s, codomain=self._sym_s))
def K_k_Schur_non_commutative_variables(self,la):
r"""
Returns the K-`k`-Schur function, as embedded inside the affine zero Hecke algebra.
INPUT:
- ``la`` -- A `k`-bounded Partition
OUTPUT:
- An element of the affine zero Hecke algebra.
EXAMPLES::
sage: g = SymmetricFunctions(QQ).kBoundedSubspace(3,1).K_kschur()
sage: g.K_k_Schur_non_commutative_variables([2,1])
T3*T1*T0 + T1*T2*T0 + T3*T2*T0 - T2*T0 + T0*T1*T0 + T2*T0*T1 + T0*T3*T0 + T2*T0*T3 + T0*T3*T1 + T2*T3*T2 - T3*T1 + T2*T3*T1 + T3*T1*T2 + T1*T2*T1
sage: g.K_k_Schur_non_commutative_variables([])
1
sage: g.K_k_Schur_non_commutative_variables([4,1])
Traceback (most recent call last):
...
ValueError: Partition should be 3-bounded
"""
SF = SymmetricFunctions(self.base_ring())
h = SF.h()
S = h(self._g_to_kh_on_basis(la)).support()
return sum(h(self._g_to_kh_on_basis(la)).coefficient(x)*self.homogeneous_basis_noncommutative_variables_zero_Hecke(x) for x in S)
def __init__(self, R):
"""
Initialize ``self``.
EXAMPLES::
sage: TestSuite(SymmetricFunctionsNonCommutingVariables(QQ).dual()).run()
"""
self._base = R # Won't be needed once CategoryObject won't override base_ring
category = GradedHopfAlgebras(R) # TODO: .Commutative()
Parent.__init__(self, category=category.WithRealizations())
# Bases
w = self.w()
# Embedding of Sym in the homogeneous bases into DNCSym in the w basis
Sym = SymmetricFunctions(self.base_ring())
Sym_h_to_w = Sym.h().module_morphism(
w.sum_of_partitions, triangular="lower", inverse_on_support=w._set_par_to_par, codomain=w, category=category
)
Sym_h_to_w.register_as_coercion()
self.to_symmetric_function = Sym_h_to_w.section()
