def __init__(self, alg, prefix="D"):
r"""
Initialize ``self``.
EXAMPLES::
sage: TestSuite(DescentAlgebra(QQ, 4).D()).run()
"""
self._prefix = prefix
self._basis_name = "standard"
p_set = subsets(range(1, alg._n))
CombinatorialFreeModule.__init__(self, alg.base_ring(),
map(tuple, p_set),
category=DescentAlgebraBases(alg),
bracket="", prefix=prefix)
# Change of basis:
B = alg.B()
self.module_morphism(self.to_B_basis,
codomain=B, category=self.category()
).register_as_coercion()
B.module_morphism(B.to_D_basis,
codomain=self, category=self.category()
).register_as_coercion()
# Coercion to symmetric group algebra
SGA = SymmetricGroupAlgebra(alg.base_ring(), alg._n)
self.module_morphism(self.to_symmetric_group_algebra,
codomain=SGA, category=Algebras(alg.base_ring())
).register_as_coercion()
def __init__(self, alg, prefix="D"):
r"""
Initialize ``self``.
EXAMPLES::
sage: TestSuite(DescentAlgebra(QQ, 4).D()).run()
"""
self._prefix = prefix
self._basis_name = "standard"
p_set = subsets(range(1, alg._n))
CombinatorialFreeModule.__init__(self,
alg.base_ring(),
map(tuple, p_set),
category=DescentAlgebraBases(alg),
bracket="",
prefix=prefix)
# Change of basis:
B = alg.B()
self.module_morphism(
self.to_B_basis, codomain=B,
category=self.category()).register_as_coercion()
B.module_morphism(B.to_D_basis,
codomain=self,
category=self.category()).register_as_coercion()
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[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 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 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[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 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 subsets(S)])
def nonempty_subsets(self):
r"""
EXAMPLES::
sage: from partitioner import ClassificationStrategy
sage: cs2 = ClassificationStrategy(2, make_disjoint=True)
sage: for cs in cs2.nonempty_subsets():
....: print(cs)
====================================
classification strategy with 1 trees
------------------------------------
classification tree 1 () (2 () ())
s2 >= 0
s1 >= 0
s0 >= 0
====================================
classification strategy with 1 trees
------------------------------------
classification tree 2 (1 () ()) ()
s2 >= 0
s1 >= 0
s0 >= 0
====================================
classification strategy with 2 trees
------------------------------------
classification tree 1 () (2 () ())
s2 >= 0
s1 >= 0
s0 >= s2
------------------------------------
classification tree 2 (1 () ()) ()
s2 > s0
s1 >= 0
s0 >= 0
"""
from sage.misc.misc import subsets
d = self.dimension()
return iter(ClassificationStrategy(d,
make_disjoint=self.is_disjoint(),
trees=srees)
for srees in subsets(self.trees)
if srees)
