def to_symmetric_group_algebra_on_basis(self, S):
"""
Return `D_S` as a linear combination of basis elements in the
symmetric group algebra.
EXAMPLES::
sage: D = DescentAlgebra(QQ, 4).D()
sage: [D.to_symmetric_group_algebra_on_basis(tuple(b))
....: for b in Subsets(3)]
[[1, 2, 3, 4],
[2, 1, 3, 4] + [3, 1, 2, 4] + [4, 1, 2, 3],
[1, 3, 2, 4] + [1, 4, 2, 3] + [2, 3, 1, 4]
+ [2, 4, 1, 3] + [3, 4, 1, 2],
[1, 2, 4, 3] + [1, 3, 4, 2] + [2, 3, 4, 1],
[3, 2, 1, 4] + [4, 2, 1, 3] + [4, 3, 1, 2],
[2, 1, 4, 3] + [3, 1, 4, 2] + [3, 2, 4, 1]
+ [4, 1, 3, 2] + [4, 2, 3, 1],
[1, 4, 3, 2] + [2, 4, 3, 1] + [3, 4, 2, 1],
[4, 3, 2, 1]]
"""
n = self.realization_of()._n
SGA = SymmetricGroupAlgebra(self.base_ring(), n)
# Need to convert S to a list of positions by -1 for indexing
P = Permutations(descents=([x-1 for x in S], n))
return SGA.sum_of_terms([(p, 1) for p in P])
def to_symmetric_group_algebra_on_basis(self, S):
"""
Return `D_S` as a linear combination of basis elements in the
symmetric group algebra.
EXAMPLES::
sage: D = DescentAlgebra(QQ, 4).D()
sage: [D.to_symmetric_group_algebra_on_basis(tuple(b))
....: for b in Subsets(3)]
[[1, 2, 3, 4],
[2, 1, 3, 4] + [3, 1, 2, 4] + [4, 1, 2, 3],
[1, 3, 2, 4] + [1, 4, 2, 3] + [2, 3, 1, 4]
+ [2, 4, 1, 3] + [3, 4, 1, 2],
[1, 2, 4, 3] + [1, 3, 4, 2] + [2, 3, 4, 1],
[3, 2, 1, 4] + [4, 2, 1, 3] + [4, 3, 1, 2],
[2, 1, 4, 3] + [3, 1, 4, 2] + [3, 2, 4, 1]
+ [4, 1, 3, 2] + [4, 2, 3, 1],
[1, 4, 3, 2] + [2, 4, 3, 1] + [3, 4, 2, 1],
[4, 3, 2, 1]]
"""
n = self.realization_of()._n
SGA = SymmetricGroupAlgebra(self.base_ring(), n)
# Need to convert S to a list of positions by -1 for indexing
P = Permutations(descents=([x - 1 for x in S], n))
return SGA.sum_of_terms([(p, 1) for p in P])
def example(self, gens=None):
"""
Return an example of a Lie algebra as per
:meth:`Category.example <sage.categories.category.Category.example>`.
EXAMPLES::
sage: LieAlgebras(QQ).example()
An example of a Lie algebra: the Lie algebra from the associative algebra
Symmetric group algebra of order 3 over Rational Field
generated by ([2, 1, 3], [2, 3, 1])
Another set of generators can be specified as an optional argument::
sage: F.<x,y,z> = FreeAlgebra(QQ)
sage: LieAlgebras(QQ).example(F.gens())
An example of a Lie algebra: the Lie algebra from the associative algebra
Free Algebra on 3 generators (x, y, z) over Rational Field
generated by (x, y, z)
"""
if gens is None:
from sage.combinat.symmetric_group_algebra import SymmetricGroupAlgebra
from sage.rings.rational_field import QQ
gens = SymmetricGroupAlgebra(QQ, 3).algebra_generators()
from sage.categories.examples.lie_algebras import Example
return Example(gens)
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 to_symmetric_group_algebra(self, n=None):
"""
Return the element of a symmetric group algebra
corresponding to the element ``self`` of `FQSym`.
INPUT:
- ``n`` -- integer (default: the maximal degree of ``self``);
the rank of the target symmetric group algebra
EXAMPLES::
sage: A = algebras.FQSym(QQ).F()
sage: x = A([1,3,2,4]) + 5/2 * A([1,2,4,3])
sage: x.to_symmetric_group_algebra()
5/2*[1, 2, 4, 3] + [1, 3, 2, 4]
sage: x.to_symmetric_group_algebra(n=7)
5/2*[1, 2, 4, 3, 5, 6, 7] + [1, 3, 2, 4, 5, 6, 7]
sage: a = A.zero().to_symmetric_group_algebra(); a
0
sage: parent(a)
Symmetric group algebra of order 0 over Rational Field
sage: y = A([1,3,2,4]) + 5/2 * A([2,1])
sage: y.to_symmetric_group_algebra()
[1, 3, 2, 4] + 5/2*[2, 1, 3, 4]
sage: y.to_symmetric_group_algebra(6)
[1, 3, 2, 4, 5, 6] + 5/2*[2, 1, 3, 4, 5, 6]
"""
if not self:
if n is None:
n = 0
return SymmetricGroupAlgebra(self.base_ring(), n).zero()
m = self.maximal_degree()
if n is None:
n = m
elif n < m:
raise ValueError("n must be at least the maximal degree")
SGA = SymmetricGroupAlgebra(self.base_ring(), n)
return SGA._from_dict(
{Permutations(n)(key): c
for (key, c) in self})
def to_symmetric_group_algebra(self, n=None):
"""
Return the element of a symmetric group algebra
corresponding to the element ``self`` of `FQSym`.
INPUT:
- ``n`` -- integer (default: the maximal degree of ``self``);
the rank of the target symmetric group algebra
EXAMPLES::
sage: A = algebras.FQSym(QQ).F()
sage: x = A([1,3,2,4]) + 5/2 * A([1,2,4,3])
sage: x.to_symmetric_group_algebra()
5/2*[1, 2, 4, 3] + [1, 3, 2, 4]
sage: x.to_symmetric_group_algebra(n=7)
5/2*[1, 2, 4, 3, 5, 6, 7] + [1, 3, 2, 4, 5, 6, 7]
sage: a = A.zero().to_symmetric_group_algebra(); a
0
sage: parent(a)
Symmetric group algebra of order 0 over Rational Field
sage: y = A([1,3,2,4]) + 5/2 * A([2,1])
sage: y.to_symmetric_group_algebra()
[1, 3, 2, 4] + 5/2*[2, 1, 3, 4]
sage: y.to_symmetric_group_algebra(6)
[1, 3, 2, 4, 5, 6] + 5/2*[2, 1, 3, 4, 5, 6]
"""
if not self:
if n is None:
n = 0
return SymmetricGroupAlgebra(self.base_ring(), n).zero()
m = self.maximal_degree()
if n is None:
n = m
elif n < m:
raise ValueError("n must be at least the maximal degree")
SGA = SymmetricGroupAlgebra(self.base_ring(), n)
return SGA._from_dict({Permutations(n)(key): c for (key, c) in self})
def __init__(self, R, n, r):
"""
Initialize ``self``.
TESTS::
sage: T = SchurTensorModule(QQ, 2, 3)
sage: TestSuite(T).run()
"""
C = CombinatorialFreeModule(R, range(1, n + 1))
self._n = n
self._r = r
self._sga = SymmetricGroupAlgebra(R, r)
self._schur = SchurAlgebra(R, n, r)
CombinatorialFreeModule_Tensor.__init__(self, tuple([C] * r))
g = self._schur.module_morphism(self._monomial_product, codomain=self)
self._schur_action = self.module_morphism(g, codomain=self, position=1)
def to_symmetric_group_algebra(self):
"""
Morphism from ``self`` to the symmetric group algebra.
EXAMPLES::
sage: D = DescentAlgebra(QQ, 4).D()
sage: D.to_symmetric_group_algebra(D[1,3])
[2, 1, 4, 3] + [3, 1, 4, 2] + [3, 2, 4, 1] + [4, 1, 3, 2] + [4, 2, 3, 1]
sage: B = DescentAlgebra(QQ, 4).B()
sage: B.to_symmetric_group_algebra(B[1,2,1])
[1, 2, 3, 4] + [1, 2, 4, 3] + [1, 3, 4, 2] + [2, 1, 3, 4]
+ [2, 1, 4, 3] + [2, 3, 4, 1] + [3, 1, 2, 4] + [3, 1, 4, 2]
+ [3, 2, 4, 1] + [4, 1, 2, 3] + [4, 1, 3, 2] + [4, 2, 3, 1]
"""
SGA = SymmetricGroupAlgebra(self.base_ring(), self.realization_of()._n)
return self.module_morphism(self.to_symmetric_group_algebra_on_basis, codomain=SGA)
def __init__(self, R, n, r):
"""
Initialize ``self``.
TESTS::
sage: T = SchurTensorModule(QQ, 2, 3)
sage: TestSuite(T).run()
"""
C = CombinatorialFreeModule(R, list(range(1, n + 1)))
self._n = n
self._r = r
self._sga = SymmetricGroupAlgebra(R, r)
self._schur = SchurAlgebra(R, n, r)
cat = ModulesWithBasis(R).TensorProducts().FiniteDimensional()
CombinatorialFreeModule_Tensor.__init__(self, tuple([C] * r), category=cat)
g = self._schur.module_morphism(self._monomial_product, codomain=self)
self._schur_action = self.module_morphism(g, codomain=self, position=1)
