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 __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):
"""
The Hopf algebra of quasi-symmetric functions.
See ``QuasiSymmetricFunctions`` for full documentation.
EXAMPLES::
sage: QuasiSymmetricFunctions(QQ)
Quasisymmetric functions over the Rational Field
sage: TestSuite(QuasiSymmetricFunctions(QQ)).run()
"""
assert R in Rings()
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
Monomial = self.Monomial()
Fundamental = self.Fundamental()
dualImmaculate = self.dualImmaculate()
# Change of bases
Fundamental.module_morphism(Monomial.sum_of_finer_compositions,
codomain=Monomial,
category=category).register_as_coercion()
Monomial.module_morphism(
Fundamental.alternating_sum_of_finer_compositions,
codomain=Fundamental,
category=category).register_as_coercion()
#This changes dualImmaculate into Monomial
dualImmaculate.module_morphism(
dualImmaculate._to_Monomial_on_basis,
codomain=Monomial,
category=category).register_as_coercion()
#This changes Monomial into dualImmaculate
Monomial.module_morphism(dualImmaculate._from_Monomial_on_basis,
codomain=dualImmaculate,
category=category).register_as_coercion()
# Embedding of Sym into QSym in the monomial bases
Sym = SymmetricFunctions(self.base_ring())
Sym_m_to_M = Sym.m().module_morphism(
Monomial.sum_of_partition_rearrangements,
triangular='upper',
inverse_on_support=Monomial._comp_to_par,
codomain=Monomial,
category=category)
Sym_m_to_M.register_as_coercion()
self.to_symmetric_function = Sym_m_to_M.section()