def to_B_basis(self, S):
r"""
Return `D_S` as a linear combination of `B_p`-basis elements.
EXAMPLES::
sage: DA = DescentAlgebra(QQ, 4)
sage: D = DA.D()
sage: B = DA.B()
sage: for b in D.basis(): B(b) # indirect doctest
B[4]
B[1, 3] - B[4]
B[2, 2] - B[4]
B[1, 1, 2] - B[1, 3] - B[2, 2] + B[4]
B[3, 1] - B[4]
B[1, 2, 1] - B[1, 3] - B[3, 1] + B[4]
B[2, 1, 1] - B[2, 2] - B[3, 1] + B[4]
B[1, 1, 1, 1] - B[1, 1, 2] - B[1, 2, 1] + B[1, 3] - B[2, 1, 1] + B[2, 2] + B[3, 1] - B[4]
"""
B = self.realization_of().B()
if len(S) == 0:
return B.one()
n = self.realization_of()._n
C = Compositions(n)
return B.sum_of_terms([(C.from_subset(T, n), (-1)**(len(S)-len(T))) for T in subsets(S)])
def to_B_basis(self, S):
r"""
Return `D_S` as a linear combination of `B_p`-basis elements.
EXAMPLES::
sage: DA = DescentAlgebra(QQ, 4)
sage: D = DA.D()
sage: B = DA.B()
sage: map(B, D.basis()) # indirect doctest
[B[4],
B[1, 3] - B[4],
B[2, 2] - B[4],
B[3, 1] - B[4],
B[1, 1, 2] - B[1, 3] - B[2, 2] + B[4],
B[1, 2, 1] - B[1, 3] - B[3, 1] + B[4],
B[2, 1, 1] - B[2, 2] - B[3, 1] + B[4],
B[1, 1, 1, 1] - B[1, 1, 2] - B[1, 2, 1] + B[1, 3]
- B[2, 1, 1] + B[2, 2] + B[3, 1] - B[4]]
"""
B = self.realization_of().B()
if not S:
return B.one()
n = self.realization_of()._n
C = Compositions(n)
return B.sum_of_terms([(C.from_subset(T,
n), (-1)**(len(S) - len(T)))
for T in SubsetsSorted(S)])
