def _matrix_monomial_to_dual_immaculate(self, n):
r"""
This function caches the change of basis matrix from the quasisymmetric monomial basis to the dual immaculate basis.
INPUT:
- ``J`` -- a composition
OUTPUT:
- A list. Each entry in the list is a row in the change of basis matrix.
EXAMPLES::
sage: dI = QuasiSymmetricFunctions(QQ).dI()
sage: dI._matrix_monomial_to_dual_immaculate(3)
[[1, -1, -1, 1], [0, 1, -1, 0], [0, 0, 1, -1], [0, 0, 0, 1]]
sage: dI._matrix_monomial_to_dual_immaculate(0)
[[1]]
"""
N = NonCommutativeSymmetricFunctions(self.base_ring())
I = N.I()
S = N.S()
mat = []
C = Compositions()
for alp in Compositions(n):
row = []
expansion = S(I(C(alp)))
for bet in Compositions(n):
row.append(expansion.coefficient(C(bet)))
mat.append(row)
return mat
def to_nsym(self, p):
"""
Return `B_p` as an element in `NSym`, the non-commutative
symmetric functions.
This maps `B_p` to `S_p` where `S` denotes the Complete basis of
`NSym`.
EXAMPLES::
sage: B = DescentAlgebra(QQ, 4).B()
sage: S = NonCommutativeSymmetricFunctions(QQ).Complete()
sage: for b in B.basis(): S(b) # indirect doctest
S[1, 1, 1, 1]
S[1, 1, 2]
S[1, 2, 1]
S[1, 3]
S[2, 1, 1]
S[2, 2]
S[3, 1]
S[4]
"""
S = NonCommutativeSymmetricFunctions(self.base_ring()).Complete()
return S.monomial(p)
def to_nsym(self, p):
"""
Return `B_p` as an element in `NSym`, the non-commutative
symmetric functions.
This maps `B_p` to `S_p` where `S` denotes the Complete basis of
`NSym`.
EXAMPLES::
sage: B = DescentAlgebra(QQ, 4).B()
sage: S = NonCommutativeSymmetricFunctions(QQ).Complete()
sage: map(S, B.basis()) # indirect doctest
[S[1, 1, 1, 1],
S[1, 1, 2],
S[1, 2, 1],
S[1, 3],
S[2, 1, 1],
S[2, 2],
S[3, 1],
S[4]]
"""
S = NonCommutativeSymmetricFunctions(self.base_ring()).Complete()
return S.monomial(p)
def dual(self):
r"""
Returns the dual Hopf algebra of the quasi-symmetric functions, which is the
non-commutative symmetric functions.
OUTPUT:
- The non-commutative symmetric functions.
EXAMPLES::
sage: QSym = QuasiSymmetricFunctions(QQ)
sage: QSym.dual()
Non-Commutative Symmetric Functions over the Rational Field
"""
from sage.combinat.ncsf_qsym.ncsf import NonCommutativeSymmetricFunctions
return NonCommutativeSymmetricFunctions(self.base_ring())
def __init__(self, alg, prefix="B"):
r"""
Initialize ``self``.
EXAMPLES::
sage: TestSuite(DescentAlgebra(QQ, 4).B()).run()
"""
self._prefix = prefix
self._basis_name = "subset"
CombinatorialFreeModule.__init__(self, alg.base_ring(),
Compositions(alg._n),
category=DescentAlgebraBases(alg),
bracket="", prefix=prefix)
S = NonCommutativeSymmetricFunctions(alg.base_ring()).Complete()
self.module_morphism(self.to_nsym,
codomain=S, category=Algebras(alg.base_ring())
).register_as_coercion()