# Symmetric Group Algebra#

sage.combinat.symmetric_group_algebra.HeckeAlgebraSymmetricGroupT(R, n, q=None)[source]#

Return the Hecke algebra of the symmetric group $$S_n$$ on the T-basis with quantum parameter q over the ring $$R$$.

If $$R$$ is a commutative ring and $$q$$ is an invertible element of $$R$$, and if $$n$$ is a nonnegative integer, then the Hecke algebra of the symmetric group $$S_n$$ over $$R$$ with quantum parameter $$q$$ is defined as the algebra generated by the generators $$T_1, T_2, \ldots, T_{n-1}$$ with relations

$T_i T_{i+1} T_i = T_{i+1} T_i T_{i+1}$

for all $$i < n-1$$ (“braid relations”),

$T_i T_j = T_j T_i$

for all $$i$$ and $$j$$ such that $$| i-j | > 1$$ (“locality relations”), and

$T_i^2 = q + (q-1) T_i$

for all $$i$$ (the “quadratic relations”, also known in the form $$(T_i + 1) (T_i - q) = 0$$). (This is only one of several existing definitions in literature, not all of which are fully equivalent. We are following the conventions of [Go1993].) For any permutation $$w \in S_n$$, we can define an element $$T_w$$ of this Hecke algebra by setting $$T_w = T_{i_1} T_{i_2} \cdots T_{i_k}$$, where $$w = s_{i_1} s_{i_2} \cdots s_{i_k}$$ is a reduced word for $$w$$ (with $$s_i$$ meaning the transposition $$(i, i+1)$$, and the product of permutations being evaluated by first applying $$s_{i_k}$$, then $$s_{i_{k-1}}$$, etc.). This element is independent of the choice of the reduced decomposition, and can be computed in Sage by calling H[w] where H is the Hecke algebra and w is the permutation.

The Hecke algebra of the symmetric group $$S_n$$ with quantum parameter $$q$$ over $$R$$ can be seen as a deformation of the group algebra $$R S_n$$; indeed, it becomes $$R S_n$$ when $$q = 1$$.

Warning

The multiplication on the Hecke algebra of the symmetric group does not follow the global option mult of the Permutations class (see options()). It is always as defined above. It does not match the default option (mult=l2r) of the symmetric group algebra!

EXAMPLES:

sage: HeckeAlgebraSymmetricGroupT(QQ, 3)
Hecke algebra of the symmetric group of order 3 on the T basis over Univariate Polynomial Ring in q over Rational Field

>>> from sage.all import *
>>> HeckeAlgebraSymmetricGroupT(QQ, Integer(3))
Hecke algebra of the symmetric group of order 3 on the T basis over Univariate Polynomial Ring in q over Rational Field

sage: HeckeAlgebraSymmetricGroupT(QQ, 3, 2)
Hecke algebra of the symmetric group of order 3 with q=2 on the T basis over Rational Field

>>> from sage.all import *
>>> HeckeAlgebraSymmetricGroupT(QQ, Integer(3), Integer(2))
Hecke algebra of the symmetric group of order 3 with q=2 on the T basis over Rational Field


The multiplication on the Hecke algebra follows a different convention than the one on the symmetric group algebra does by default:

sage: H3 = HeckeAlgebraSymmetricGroupT(QQ, 3)
sage: H3([1,3,2]) * H3([2,1,3])
T[3, 1, 2]
sage: S3 = SymmetricGroupAlgebra(QQ, 3)
sage: S3([1,3,2]) * S3([2,1,3])
[2, 3, 1]

sage: TestSuite(H3).run()

>>> from sage.all import *
>>> H3 = HeckeAlgebraSymmetricGroupT(QQ, Integer(3))
>>> H3([Integer(1),Integer(3),Integer(2)]) * H3([Integer(2),Integer(1),Integer(3)])
T[3, 1, 2]
>>> S3 = SymmetricGroupAlgebra(QQ, Integer(3))
>>> S3([Integer(1),Integer(3),Integer(2)]) * S3([Integer(2),Integer(1),Integer(3)])
[2, 3, 1]

>>> TestSuite(H3).run()


Note

IwahoriHeckeAlgebra gives a different implementation of the Iwahori-Hecke algebras of a Coxeter system $$(W,S)$$. This includes the Hecke algebras of the symmetric group as special case.

class sage.combinat.symmetric_group_algebra.HeckeAlgebraSymmetricGroup_generic(R, n, q=None)[source]#
one_basis()[source]#

Return the identity permutation.

EXAMPLES:

sage: HeckeAlgebraSymmetricGroupT(QQ, 3).one()  # indirect doctest
T[1, 2, 3]

>>> from sage.all import *
>>> HeckeAlgebraSymmetricGroupT(QQ, Integer(3)).one()  # indirect doctest
T[1, 2, 3]

q()[source]#

Return the variable or parameter $$q$$.

EXAMPLES:

sage: HeckeAlgebraSymmetricGroupT(QQ, 3).q()
q
sage: HeckeAlgebraSymmetricGroupT(QQ, 3, 2).q()
2

>>> from sage.all import *
>>> HeckeAlgebraSymmetricGroupT(QQ, Integer(3)).q()
q
>>> HeckeAlgebraSymmetricGroupT(QQ, Integer(3), Integer(2)).q()
2

class sage.combinat.symmetric_group_algebra.HeckeAlgebraSymmetricGroup_t(R, n, q=None)[source]#
algebra_generators()[source]#

Return the generators of the algebra.

EXAMPLES:

sage: HeckeAlgebraSymmetricGroupT(QQ,3).algebra_generators()
[T[2, 1, 3], T[1, 3, 2]]

>>> from sage.all import *
>>> HeckeAlgebraSymmetricGroupT(QQ,Integer(3)).algebra_generators()
[T[2, 1, 3], T[1, 3, 2]]

jucys_murphy(k)[source]#

Return the Jucys-Murphy element $$J_k$$ of the Hecke algebra.

These Jucys-Murphy elements are defined by

$J_k = (T_{k-1} T_{k-2} \cdots T_1) (T_1 T_2 \cdots T_{k-1}).$

More explicitly,

$J_k = q^{k-1} + \sum_{l=1}^{k-1} (q^l - q^{l-1}) T_{(l, k)}.$

For generic $$q$$, the $$J_k$$ generate a maximal commutative sub-algebra of the Hecke algebra.

Warning

The specialization $$q = 1$$ does not map these elements $$J_k$$ to the Young-Jucys-Murphy elements of the group algebra $$R S_n$$. (Instead, it maps the “reduced” Jucys-Murphy elements $$(J_k - q^{k-1}) / (q - 1)$$ to the Young-Jucys-Murphy elements of $$R S_n$$.)

EXAMPLES:

sage: H3 = HeckeAlgebraSymmetricGroupT(QQ,3)
sage: j2 = H3.jucys_murphy(2); j2
q*T[1, 2, 3] + (q-1)*T[2, 1, 3]
sage: j3 = H3.jucys_murphy(3); j3
q^2*T[1, 2, 3] + (q^2-q)*T[1, 3, 2] + (q-1)*T[3, 2, 1]
sage: j2*j3 == j3*j2
True
sage: j0 = H3.jucys_murphy(1); j0 == H3.one()
True
sage: H3.jucys_murphy(0)
Traceback (most recent call last):
...
ValueError: k (= 0) must be between 1 and n (= 3)

>>> from sage.all import *
>>> H3 = HeckeAlgebraSymmetricGroupT(QQ,Integer(3))
>>> j2 = H3.jucys_murphy(Integer(2)); j2
q*T[1, 2, 3] + (q-1)*T[2, 1, 3]
>>> j3 = H3.jucys_murphy(Integer(3)); j3
q^2*T[1, 2, 3] + (q^2-q)*T[1, 3, 2] + (q-1)*T[3, 2, 1]
>>> j2*j3 == j3*j2
True
>>> j0 = H3.jucys_murphy(Integer(1)); j0 == H3.one()
True
>>> H3.jucys_murphy(Integer(0))
Traceback (most recent call last):
...
ValueError: k (= 0) must be between 1 and n (= 3)

product_on_basis(perm1, perm2)[source]#

EXAMPLES:

sage: H3 = HeckeAlgebraSymmetricGroupT(QQ, 3, 1)
sage: a = H3([2,1,3])+2*H3([1,2,3])-H3([3,2,1])
sage: a^2 #indirect doctest
6*T[1, 2, 3] + 4*T[2, 1, 3] - T[2, 3, 1] - T[3, 1, 2] - 4*T[3, 2, 1]

>>> from sage.all import *
>>> H3 = HeckeAlgebraSymmetricGroupT(QQ, Integer(3), Integer(1))
>>> a = H3([Integer(2),Integer(1),Integer(3)])+Integer(2)*H3([Integer(1),Integer(2),Integer(3)])-H3([Integer(3),Integer(2),Integer(1)])
>>> a**Integer(2) #indirect doctest
6*T[1, 2, 3] + 4*T[2, 1, 3] - T[2, 3, 1] - T[3, 1, 2] - 4*T[3, 2, 1]

sage: QS3 = SymmetricGroupAlgebra(QQ, 3)
sage: a = QS3([2,1,3])+2*QS3([1,2,3])-QS3([3,2,1])
sage: a^2
6*[1, 2, 3] + 4*[2, 1, 3] - [2, 3, 1] - [3, 1, 2] - 4*[3, 2, 1]

>>> from sage.all import *
>>> QS3 = SymmetricGroupAlgebra(QQ, Integer(3))
>>> a = QS3([Integer(2),Integer(1),Integer(3)])+Integer(2)*QS3([Integer(1),Integer(2),Integer(3)])-QS3([Integer(3),Integer(2),Integer(1)])
>>> a**Integer(2)
6*[1, 2, 3] + 4*[2, 1, 3] - [2, 3, 1] - [3, 1, 2] - 4*[3, 2, 1]

t(i)[source]#

Return the element $$T_i$$ of the Hecke algebra self.

EXAMPLES:

sage: H3 = HeckeAlgebraSymmetricGroupT(QQ,3)
sage: H3.t(1)
T[2, 1, 3]
sage: H3.t(2)
T[1, 3, 2]
sage: H3.t(0)
Traceback (most recent call last):
...
ValueError: i (= 0) must be between 1 and n-1 (= 2)

>>> from sage.all import *
>>> H3 = HeckeAlgebraSymmetricGroupT(QQ,Integer(3))
>>> H3.t(Integer(1))
T[2, 1, 3]
>>> H3.t(Integer(2))
T[1, 3, 2]
>>> H3.t(Integer(0))
Traceback (most recent call last):
...
ValueError: i (= 0) must be between 1 and n-1 (= 2)

t_action(a, i)[source]#

Return the product $$T_i \cdot a$$.

EXAMPLES:

sage: H3 = HeckeAlgebraSymmetricGroupT(QQ, 3)
sage: a = H3([2,1,3])+2*H3([1,2,3])
sage: H3.t_action(a, 1)
q*T[1, 2, 3] + (q+1)*T[2, 1, 3]
sage: H3.t(1)*a
q*T[1, 2, 3] + (q+1)*T[2, 1, 3]

>>> from sage.all import *
>>> H3 = HeckeAlgebraSymmetricGroupT(QQ, Integer(3))
>>> a = H3([Integer(2),Integer(1),Integer(3)])+Integer(2)*H3([Integer(1),Integer(2),Integer(3)])
>>> H3.t_action(a, Integer(1))
q*T[1, 2, 3] + (q+1)*T[2, 1, 3]
>>> H3.t(Integer(1))*a
q*T[1, 2, 3] + (q+1)*T[2, 1, 3]

t_action_on_basis(perm, i)[source]#

Return the product $$T_i \cdot T_{perm}$$, where perm is a permutation in the symmetric group $$S_n$$.

EXAMPLES:

sage: H3 = HeckeAlgebraSymmetricGroupT(QQ, 3)
sage: H3.t_action_on_basis(Permutation([2,1,3]), 1)
q*T[1, 2, 3] + (q-1)*T[2, 1, 3]
sage: H3.t_action_on_basis(Permutation([1,2,3]), 1)
T[2, 1, 3]
sage: H3 = HeckeAlgebraSymmetricGroupT(QQ, 3, 1)
sage: H3.t_action_on_basis(Permutation([2,1,3]), 1)
T[1, 2, 3]
sage: H3.t_action_on_basis(Permutation([1,3,2]), 2)
T[1, 2, 3]

>>> from sage.all import *
>>> H3 = HeckeAlgebraSymmetricGroupT(QQ, Integer(3))
>>> H3.t_action_on_basis(Permutation([Integer(2),Integer(1),Integer(3)]), Integer(1))
q*T[1, 2, 3] + (q-1)*T[2, 1, 3]
>>> H3.t_action_on_basis(Permutation([Integer(1),Integer(2),Integer(3)]), Integer(1))
T[2, 1, 3]
>>> H3 = HeckeAlgebraSymmetricGroupT(QQ, Integer(3), Integer(1))
>>> H3.t_action_on_basis(Permutation([Integer(2),Integer(1),Integer(3)]), Integer(1))
T[1, 2, 3]
>>> H3.t_action_on_basis(Permutation([Integer(1),Integer(3),Integer(2)]), Integer(2))
T[1, 2, 3]

class sage.combinat.symmetric_group_algebra.KLCellularBasis(SGA)[source]#

The Kazhdan-Lusztig $$C'$$ basis (at $$q = 1$$) of the symmetric group algebra realized as a cellular basis

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(GF(3), 5)
sage: KL = SGA.kazhdan_lusztig_cellular_basis()
sage: for la in KL.simple_module_parameterization():
....:     CM = KL.cell_module(la)
....:     print(la, CM.dimension(), CM.simple_module().dimension())
[2, 2, 1] 5 4
[3, 1, 1] 6 6
[3, 2] 5 1
[4, 1] 4 4
[5] 1 1

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(GF(Integer(3)), Integer(5))
>>> KL = SGA.kazhdan_lusztig_cellular_basis()
>>> for la in KL.simple_module_parameterization():
...     CM = KL.cell_module(la)
...     print(la, CM.dimension(), CM.simple_module().dimension())
[2, 2, 1] 5 4
[3, 1, 1] 6 6
[3, 2] 5 1
[4, 1] 4 4
[5] 1 1

class sage.combinat.symmetric_group_algebra.MurphyBasis(SGA)[source]#

The Murphy basis of a symmetric group algebra.

Let $$R$$ be a commutative ring, and let $$A = R[S_n]$$ denote the group algebra (over $$R$$) of $$S_n$$. The Murphy basis is the basis of $$A$$ defined as follows. Let $$S, T$$ be standard tableaux of shape $$\lambda$$. Define $$T^{\lambda}$$ as the standard tableau of shape $$\lambda$$ with the first row filled with $$1, \ldots, \lambda_1$$, the second row $$\lambda_1+1, \ldots, \lambda_1+\lambda_2$$, and so on. Let $$d(S)$$ be the unique permutation such that $$S = T^{\lambda} d(S)$$ under the natural action. Then the Murphy basis element indexed by $$S$$ and $$T$$ is

$M_{S'T'} = d(S)^{-1} R_{\lambda} d(T),$

where $$S'$$ denotes the conjugate tableau. The Murphy basis is a cellular basis of $$A$$.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(GF(3), 5)
sage: M = SGA.murphy_basis()
sage: for la in M.simple_module_parameterization():
....:     CM = M.cell_module(la)
....:     print(la, CM.dimension(), CM.simple_module().dimension())
[2, 2, 1] 5 4
[3, 1, 1] 6 6
[3, 2] 5 1
[4, 1] 4 4
[5] 1 1

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(GF(Integer(3)), Integer(5))
>>> M = SGA.murphy_basis()
>>> for la in M.simple_module_parameterization():
...     CM = M.cell_module(la)
...     print(la, CM.dimension(), CM.simple_module().dimension())
[2, 2, 1] 5 4
[3, 1, 1] 6 6
[3, 2] 5 1
[4, 1] 4 4
[5] 1 1


REFERENCES:

class sage.combinat.symmetric_group_algebra.SGACellularBasis(SGA)[source]#

Bases: CellularBasis

A cellular basis of the symmetric group algebra.

one()[source]#

Return the element $$1$$ in self.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(GF(3), 4)
sage: M = SGA.murphy_basis()
sage: M.one()
C([4], [[1, 2, 3, 4]], [[1, 2, 3, 4]])

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(GF(Integer(3)), Integer(4))
>>> M = SGA.murphy_basis()
>>> M.one()
C([4], [[1, 2, 3, 4]], [[1, 2, 3, 4]])

one_basis()[source]#

Return the index of the basis element for the multiplicative identity.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(GF(3), 4)
sage: M = SGA.murphy_basis()
sage: M.one_basis()
([4], [[1, 2, 3, 4]], [[1, 2, 3, 4]])

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(GF(Integer(3)), Integer(4))
>>> M = SGA.murphy_basis()
>>> M.one_basis()
([4], [[1, 2, 3, 4]], [[1, 2, 3, 4]])

sage.combinat.symmetric_group_algebra.SymmetricGroupAlgebra(R, W, category=None)[source]#

Return the symmetric group algebra of order W over the ring R.

INPUT:

• W – a symmetric group; alternatively an integer $$n$$ can be provided, as shorthand for Permutations(n).

• R – a base ring

• category – a category (default: the category of W)

This supports several implementations of the symmetric group. At this point this has been tested with W=Permutations(n) and W=SymmetricGroup(n).

Warning

Some features are failing in the latter case, in particular if the domain of the symmetric group is not $$1,\ldots,n$$.

Note

The brave can also try setting W=WeylGroup(['A',n-1]), but little support for this currently exists.

EXAMPLES:

sage: QS3 = SymmetricGroupAlgebra(QQ, 3); QS3
Symmetric group algebra of order 3 over Rational Field
sage: QS3(1)
[1, 2, 3]
sage: QS3(2)
2*[1, 2, 3]
sage: basis = [QS3(p) for p in Permutations(3)]
sage: a = sum(basis); a
[1, 2, 3] + [1, 3, 2] + [2, 1, 3] + [2, 3, 1] + [3, 1, 2] + [3, 2, 1]
sage: a^2
6*[1, 2, 3] + 6*[1, 3, 2] + 6*[2, 1, 3] + 6*[2, 3, 1] + 6*[3, 1, 2] + 6*[3, 2, 1]
sage: a^2 == 6*a
True
sage: b = QS3([3, 1, 2])
sage: b
[3, 1, 2]
sage: b*a
[1, 2, 3] + [1, 3, 2] + [2, 1, 3] + [2, 3, 1] + [3, 1, 2] + [3, 2, 1]
sage: b*a == a
True

>>> from sage.all import *
>>> QS3 = SymmetricGroupAlgebra(QQ, Integer(3)); QS3
Symmetric group algebra of order 3 over Rational Field
>>> QS3(Integer(1))
[1, 2, 3]
>>> QS3(Integer(2))
2*[1, 2, 3]
>>> basis = [QS3(p) for p in Permutations(Integer(3))]
>>> a = sum(basis); a
[1, 2, 3] + [1, 3, 2] + [2, 1, 3] + [2, 3, 1] + [3, 1, 2] + [3, 2, 1]
>>> a**Integer(2)
6*[1, 2, 3] + 6*[1, 3, 2] + 6*[2, 1, 3] + 6*[2, 3, 1] + 6*[3, 1, 2] + 6*[3, 2, 1]
>>> a**Integer(2) == Integer(6)*a
True
>>> b = QS3([Integer(3), Integer(1), Integer(2)])
>>> b
[3, 1, 2]
>>> b*a
[1, 2, 3] + [1, 3, 2] + [2, 1, 3] + [2, 3, 1] + [3, 1, 2] + [3, 2, 1]
>>> b*a == a
True


We now construct the symmetric group algebra by providing explicitly the underlying group:

sage: SGA = SymmetricGroupAlgebra(QQ, Permutations(4)); SGA
Symmetric group algebra of order 4 over Rational Field
sage: SGA.group()
Standard permutations of 4
sage: SGA.an_element()
[1, 2, 3, 4] + 2*[1, 2, 4, 3] + 3*[1, 3, 2, 4] + [4, 1, 2, 3]

sage: SGA = SymmetricGroupAlgebra(QQ, SymmetricGroup(4)); SGA
Symmetric group algebra of order 4 over Rational Field
sage: SGA.group()
Symmetric group of order 4! as a permutation group
sage: SGA.an_element()
() + (2,3,4) + 2*(1,3)(2,4) + 3*(1,4)(2,3)

sage: SGA = SymmetricGroupAlgebra(QQ, WeylGroup(["A",3], prefix='s')); SGA
Symmetric group algebra of order 4 over Rational Field
sage: SGA.group()
Weyl Group of type ['A', 3] (as a matrix group acting on the ambient space)
sage: SGA.an_element()
s1*s2*s3 + 3*s2*s3*s1*s2 + 2*s3*s1 + 1

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(QQ, Permutations(Integer(4))); SGA
Symmetric group algebra of order 4 over Rational Field
>>> SGA.group()
Standard permutations of 4
>>> SGA.an_element()
[1, 2, 3, 4] + 2*[1, 2, 4, 3] + 3*[1, 3, 2, 4] + [4, 1, 2, 3]

>>> SGA = SymmetricGroupAlgebra(QQ, SymmetricGroup(Integer(4))); SGA
Symmetric group algebra of order 4 over Rational Field
>>> SGA.group()
Symmetric group of order 4! as a permutation group
>>> SGA.an_element()
() + (2,3,4) + 2*(1,3)(2,4) + 3*(1,4)(2,3)

>>> SGA = SymmetricGroupAlgebra(QQ, WeylGroup(["A",Integer(3)], prefix='s')); SGA
Symmetric group algebra of order 4 over Rational Field
>>> SGA.group()
Weyl Group of type ['A', 3] (as a matrix group acting on the ambient space)
>>> SGA.an_element()
s1*s2*s3 + 3*s2*s3*s1*s2 + 2*s3*s1 + 1


The preferred way to construct the symmetric group algebra is to go through the usual algebra method:

sage: SGA = Permutations(3).algebra(QQ); SGA
Symmetric group algebra of order 3 over Rational Field
sage: SGA.group()
Standard permutations of 3

sage: SGA = SymmetricGroup(3).algebra(QQ); SGA
Symmetric group algebra of order 3 over Rational Field
sage: SGA.group()
Symmetric group of order 3! as a permutation group

>>> from sage.all import *
>>> SGA = Permutations(Integer(3)).algebra(QQ); SGA
Symmetric group algebra of order 3 over Rational Field
>>> SGA.group()
Standard permutations of 3

>>> SGA = SymmetricGroup(Integer(3)).algebra(QQ); SGA
Symmetric group algebra of order 3 over Rational Field
>>> SGA.group()
Symmetric group of order 3! as a permutation group


The canonical embedding from the symmetric group algebra of order $$n$$ to the symmetric group algebra of order $$p > n$$ is available as a coercion:

sage: QS3 = SymmetricGroupAlgebra(QQ, 3)
sage: QS4 = SymmetricGroupAlgebra(QQ, 4)
sage: QS4.coerce_map_from(QS3)
Generic morphism:
From: Symmetric group algebra of order 3 over Rational Field
To:   Symmetric group algebra of order 4 over Rational Field

sage: x3  = QS3([3,1,2]) + 2 * QS3([2,3,1]); x3
2*[2, 3, 1] + [3, 1, 2]
sage: QS4(x3)
2*[2, 3, 1, 4] + [3, 1, 2, 4]

>>> from sage.all import *
>>> QS3 = SymmetricGroupAlgebra(QQ, Integer(3))
>>> QS4 = SymmetricGroupAlgebra(QQ, Integer(4))
>>> QS4.coerce_map_from(QS3)
Generic morphism:
From: Symmetric group algebra of order 3 over Rational Field
To:   Symmetric group algebra of order 4 over Rational Field

>>> x3  = QS3([Integer(3),Integer(1),Integer(2)]) + Integer(2) * QS3([Integer(2),Integer(3),Integer(1)]); x3
2*[2, 3, 1] + [3, 1, 2]
>>> QS4(x3)
2*[2, 3, 1, 4] + [3, 1, 2, 4]


This allows for mixed expressions:

sage: x4  = 3 * QS4([3, 1, 4, 2])
sage: x3 + x4
2*[2, 3, 1, 4] + [3, 1, 2, 4] + 3*[3, 1, 4, 2]

sage: QS0 = SymmetricGroupAlgebra(QQ, 0)
sage: QS1 = SymmetricGroupAlgebra(QQ, 1)
sage: x0 = QS0([])
sage: x1 = QS1([1])
sage: x0 * x1
[1]
sage: x3 - (2*x0 + x1) - x4
-3*[1, 2, 3, 4] + 2*[2, 3, 1, 4] + [3, 1, 2, 4] - 3*[3, 1, 4, 2]

>>> from sage.all import *
>>> x4  = Integer(3) * QS4([Integer(3), Integer(1), Integer(4), Integer(2)])
>>> x3 + x4
2*[2, 3, 1, 4] + [3, 1, 2, 4] + 3*[3, 1, 4, 2]

>>> QS0 = SymmetricGroupAlgebra(QQ, Integer(0))
>>> QS1 = SymmetricGroupAlgebra(QQ, Integer(1))
>>> x0 = QS0([])
>>> x1 = QS1([Integer(1)])
>>> x0 * x1
[1]
>>> x3 - (Integer(2)*x0 + x1) - x4
-3*[1, 2, 3, 4] + 2*[2, 3, 1, 4] + [3, 1, 2, 4] - 3*[3, 1, 4, 2]


Caveat: to achieve this, constructing SymmetricGroupAlgebra(QQ, 10) currently triggers the construction of all symmetric group algebras of smaller order. Is this a feature we really want to have?

Warning

The semantics of multiplication in symmetric group algebras with index set Permutations(n) is determined by the order in which permutations are multiplied, which currently defaults to “in such a way that multiplication is associative with permutations acting on integers from the right”, but can be changed to the opposite order at runtime by setting the global variable Permutations.options['mult'] (see sage.combinat.permutation.Permutations.options() ). On the other hand, the semantics of multiplication in symmetric group algebras with index set SymmetricGroup(n) does not depend on this global variable. (This has the awkward consequence that the coercions between these two sorts of symmetric group algebras do not respect multiplication when this global variable is set to 'r2l'.) In view of this, it is recommended that code not rely on the usual multiplication function, but rather use the methods left_action_product() and right_action_product() for multiplying permutations (these methods don’t depend on the setting). See Issue #14885 for more information.

We conclude by constructing the algebra of the symmetric group as a monoid algebra:

sage: QS3 = SymmetricGroupAlgebra(QQ, 3, category=Monoids())
sage: QS3.category()
Category of finite dimensional cellular monoid algebras over Rational Field
sage: TestSuite(QS3).run(skip=['_test_construction'])

>>> from sage.all import *
>>> QS3 = SymmetricGroupAlgebra(QQ, Integer(3), category=Monoids())
>>> QS3.category()
Category of finite dimensional cellular monoid algebras over Rational Field
>>> TestSuite(QS3).run(skip=['_test_construction'])

class sage.combinat.symmetric_group_algebra.SymmetricGroupAlgebra_n(R, W, category)[source]#
algebra_generators()[source]#

Return generators of this group algebra (as algebra) as a list of permutations.

The generators used for the group algebra of $$S_n$$ are the transposition $$(2, 1)$$ and the $$n$$-cycle $$(1, 2, \ldots, n)$$, unless $$n \leq 1$$ (in which case no generators are needed).

EXAMPLES:

sage: SymmetricGroupAlgebra(ZZ,5).algebra_generators()
Family ([2, 1, 3, 4, 5], [2, 3, 4, 5, 1])

sage: SymmetricGroupAlgebra(QQ,0).algebra_generators()
Family ()

sage: SymmetricGroupAlgebra(QQ,1).algebra_generators()
Family ()

>>> from sage.all import *
>>> SymmetricGroupAlgebra(ZZ,Integer(5)).algebra_generators()
Family ([2, 1, 3, 4, 5], [2, 3, 4, 5, 1])

>>> SymmetricGroupAlgebra(QQ,Integer(0)).algebra_generators()
Family ()

>>> SymmetricGroupAlgebra(QQ,Integer(1)).algebra_generators()
Family ()

antipode(x)[source]#

Return the image of the element x of self under the antipode of the Hopf algebra self (where the comultiplication is the usual one on a group algebra).

Explicitly, this is obtained by replacing each permutation $$\sigma$$ by $$\sigma^{-1}$$ in x while keeping all coefficients as they are.

EXAMPLES:

sage: QS4 = SymmetricGroupAlgebra(QQ, 4)
sage: QS4.antipode(2 * QS4([1, 3, 4, 2]) - 1/2 * QS4([1, 4, 2, 3]))
-1/2*[1, 3, 4, 2] + 2*[1, 4, 2, 3]
sage: all( QS4.antipode(QS4(p)) == QS4(p.inverse())
....:      for p in Permutations(4) )
True

sage: ZS3 = SymmetricGroupAlgebra(ZZ, 3)
sage: ZS3.antipode(ZS3.zero())
0
sage: ZS3.antipode(-ZS3(Permutation([2, 3, 1])))
-[3, 1, 2]

>>> from sage.all import *
>>> QS4 = SymmetricGroupAlgebra(QQ, Integer(4))
>>> QS4.antipode(Integer(2) * QS4([Integer(1), Integer(3), Integer(4), Integer(2)]) - Integer(1)/Integer(2) * QS4([Integer(1), Integer(4), Integer(2), Integer(3)]))
-1/2*[1, 3, 4, 2] + 2*[1, 4, 2, 3]
>>> all( QS4.antipode(QS4(p)) == QS4(p.inverse())
...      for p in Permutations(Integer(4)) )
True

>>> ZS3 = SymmetricGroupAlgebra(ZZ, Integer(3))
>>> ZS3.antipode(ZS3.zero())
0
>>> ZS3.antipode(-ZS3(Permutation([Integer(2), Integer(3), Integer(1)])))
-[3, 1, 2]

binary_unshuffle_sum(k)[source]#

Return the $$k$$-th binary unshuffle sum in the group algebra self.

The $$k$$-th binary unshuffle sum in the symmetric group algebra $$R S_n$$ over a ring $$R$$ is defined as the sum of all permutations $$\sigma \in S_n$$ satisfying $$\sigma(1) < \sigma(2) < \cdots < \sigma(k)$$ and $$\sigma(k+1) < \sigma(k+2) < \cdots < \sigma(n)$$.

This element has the property that, if it is denoted by $$t_k$$, and if the $$k$$-th semi-RSW element (see semi_rsw_element()) is denoted by $$s_k$$, then $$s_k S(t_k)$$ and $$t_k S(s_k)$$ both equal the $$k$$-th Reiner-Saliola-Welker shuffling element of $$R S_n$$ (see rsw_shuffling_element()).

The $$k$$-th binary unshuffle sum is the image of the complete non-commutative symmetric function $$S^{(k, n-k)}$$ in the ring of non-commutative symmetric functions under the canonical projection on the symmetric group algebra (through the descent algebra).

EXAMPLES:

The binary unshuffle sums on $$\QQ S_3$$:

sage: QS3 = SymmetricGroupAlgebra(QQ, 3)
sage: QS3.binary_unshuffle_sum(0)
[1, 2, 3]
sage: QS3.binary_unshuffle_sum(1)
[1, 2, 3] + [2, 1, 3] + [3, 1, 2]
sage: QS3.binary_unshuffle_sum(2)
[1, 2, 3] + [1, 3, 2] + [2, 3, 1]
sage: QS3.binary_unshuffle_sum(3)
[1, 2, 3]
sage: QS3.binary_unshuffle_sum(4)
0

>>> from sage.all import *
>>> QS3 = SymmetricGroupAlgebra(QQ, Integer(3))
>>> QS3.binary_unshuffle_sum(Integer(0))
[1, 2, 3]
>>> QS3.binary_unshuffle_sum(Integer(1))
[1, 2, 3] + [2, 1, 3] + [3, 1, 2]
>>> QS3.binary_unshuffle_sum(Integer(2))
[1, 2, 3] + [1, 3, 2] + [2, 3, 1]
>>> QS3.binary_unshuffle_sum(Integer(3))
[1, 2, 3]
>>> QS3.binary_unshuffle_sum(Integer(4))
0


Let us check the relation with the $$k$$-th Reiner-Saliola-Welker shuffling element stated in the docstring:

sage: def test_rsw(n):
....:     ZSn = SymmetricGroupAlgebra(ZZ, n)
....:     for k in range(1, n):
....:         a = ZSn.semi_rsw_element(k)
....:         b = ZSn.binary_unshuffle_sum(k)
....:         c = ZSn.left_action_product(a, ZSn.antipode(b))
....:         d = ZSn.left_action_product(b, ZSn.antipode(a))
....:         e = ZSn.rsw_shuffling_element(k)
....:         if c != e or d != e:
....:             return False
....:     return True
sage: test_rsw(3)
True
sage: test_rsw(4)  # long time
True
sage: test_rsw(5)  # long time
True

>>> from sage.all import *
>>> def test_rsw(n):
...     ZSn = SymmetricGroupAlgebra(ZZ, n)
...     for k in range(Integer(1), n):
...         a = ZSn.semi_rsw_element(k)
...         b = ZSn.binary_unshuffle_sum(k)
...         c = ZSn.left_action_product(a, ZSn.antipode(b))
...         d = ZSn.left_action_product(b, ZSn.antipode(a))
...         e = ZSn.rsw_shuffling_element(k)
...         if c != e or d != e:
...             return False
...     return True
>>> test_rsw(Integer(3))
True
>>> test_rsw(Integer(4))  # long time
True
>>> test_rsw(Integer(5))  # long time
True


Let us also check the statement about the complete non-commutative symmetric function:

sage: def test_rsw_ncsf(n):
....:     ZSn = SymmetricGroupAlgebra(ZZ, n)
....:     NSym = NonCommutativeSymmetricFunctions(ZZ)
....:     S = NSym.S()
....:     for k in range(1, n):
....:         a = S(Composition([k, n-k])).to_symmetric_group_algebra()
....:         if a != ZSn.binary_unshuffle_sum(k):
....:             return False
....:     return True
sage: test_rsw_ncsf(3)
True
sage: test_rsw_ncsf(4)
True
sage: test_rsw_ncsf(5)  # long time
True

>>> from sage.all import *
>>> def test_rsw_ncsf(n):
...     ZSn = SymmetricGroupAlgebra(ZZ, n)
...     NSym = NonCommutativeSymmetricFunctions(ZZ)
...     S = NSym.S()
...     for k in range(Integer(1), n):
...         a = S(Composition([k, n-k])).to_symmetric_group_algebra()
...         if a != ZSn.binary_unshuffle_sum(k):
...             return False
...     return True
>>> test_rsw_ncsf(Integer(3))
True
>>> test_rsw_ncsf(Integer(4))
True
>>> test_rsw_ncsf(Integer(5))  # long time
True

canonical_embedding(other)[source]#

Return the canonical coercion of self into a symmetric group algebra other.

INPUT:

• other – a symmetric group algebra with order $$p$$ satisfying $$p \geq n$$, where $$n$$ is the order of self, over a ground ring into which the ground ring of self coerces.

EXAMPLES:

sage: QS2 = SymmetricGroupAlgebra(QQ, 2)
sage: QS4 = SymmetricGroupAlgebra(QQ, 4)
sage: phi = QS2.canonical_embedding(QS4); phi
Generic morphism:
From: Symmetric group algebra of order 2 over Rational Field
To:   Symmetric group algebra of order 4 over Rational Field

sage: x = QS2([2,1]) + 2 * QS2([1,2])
sage: phi(x)
2*[1, 2, 3, 4] + [2, 1, 3, 4]

Generic morphism:
From: Symmetric group algebra of order 2 over Rational Field
To:   Symmetric group algebra of order 4 over Rational Field

sage: ZS2 = SymmetricGroupAlgebra(ZZ, 2)
sage: phi = ZS2.canonical_embedding(QS4); phi
Generic morphism:
From: Symmetric group algebra of order 2 over Integer Ring
To:   Symmetric group algebra of order 4 over Rational Field

sage: phi = ZS2.canonical_embedding(QS2); phi
Generic morphism:
From: Symmetric group algebra of order 2 over Integer Ring
To:   Symmetric group algebra of order 2 over Rational Field

sage: QS4.canonical_embedding(QS2)
Traceback (most recent call last):
...
ValueError: There is no canonical embedding from Symmetric group
algebra of order 2 over Rational Field to Symmetric group
algebra of order 4 over Rational Field

sage: QS4g = SymmetricGroup(4).algebra(QQ)
sage: QS4.canonical_embedding(QS4g)(QS4([1,3,2,4]))
(2,3)
sage: QS4g.canonical_embedding(QS4)(QS4g((2,3)))
[1, 3, 2, 4]
sage: ZS2.canonical_embedding(QS4g)(ZS2([2,1]))
(1,2)
sage: ZS2g = SymmetricGroup(2).algebra(ZZ)
sage: ZS2g.canonical_embedding(QS4)(ZS2g((1,2)))
[2, 1, 3, 4]

>>> from sage.all import *
>>> QS2 = SymmetricGroupAlgebra(QQ, Integer(2))
>>> QS4 = SymmetricGroupAlgebra(QQ, Integer(4))
>>> phi = QS2.canonical_embedding(QS4); phi
Generic morphism:
From: Symmetric group algebra of order 2 over Rational Field
To:   Symmetric group algebra of order 4 over Rational Field

>>> x = QS2([Integer(2),Integer(1)]) + Integer(2) * QS2([Integer(1),Integer(2)])
>>> phi(x)
2*[1, 2, 3, 4] + [2, 1, 3, 4]

Generic morphism:
From: Symmetric group algebra of order 2 over Rational Field
To:   Symmetric group algebra of order 4 over Rational Field

>>> ZS2 = SymmetricGroupAlgebra(ZZ, Integer(2))
>>> phi = ZS2.canonical_embedding(QS4); phi
Generic morphism:
From: Symmetric group algebra of order 2 over Integer Ring
To:   Symmetric group algebra of order 4 over Rational Field

>>> phi = ZS2.canonical_embedding(QS2); phi
Generic morphism:
From: Symmetric group algebra of order 2 over Integer Ring
To:   Symmetric group algebra of order 2 over Rational Field

>>> QS4.canonical_embedding(QS2)
Traceback (most recent call last):
...
ValueError: There is no canonical embedding from Symmetric group
algebra of order 2 over Rational Field to Symmetric group
algebra of order 4 over Rational Field

>>> QS4g = SymmetricGroup(Integer(4)).algebra(QQ)
>>> QS4.canonical_embedding(QS4g)(QS4([Integer(1),Integer(3),Integer(2),Integer(4)]))
(2,3)
>>> QS4g.canonical_embedding(QS4)(QS4g((Integer(2),Integer(3))))
[1, 3, 2, 4]
>>> ZS2.canonical_embedding(QS4g)(ZS2([Integer(2),Integer(1)]))
(1,2)
>>> ZS2g = SymmetricGroup(Integer(2)).algebra(ZZ)
>>> ZS2g.canonical_embedding(QS4)(ZS2g((Integer(1),Integer(2))))
[2, 1, 3, 4]

cell_module(la, **kwds)[source]#

Return the cell module indexed by la.

EXAMPLES:

sage: S = SymmetricGroupAlgebra(QQ, 3)
sage: M = S.cell_module(Partition([2,1])); M
Cell module indexed by [2, 1] of Cellular basis of
Symmetric group algebra of order 3 over Rational Field

>>> from sage.all import *
>>> S = SymmetricGroupAlgebra(QQ, Integer(3))
>>> M = S.cell_module(Partition([Integer(2),Integer(1)])); M
Cell module indexed by [2, 1] of Cellular basis of
Symmetric group algebra of order 3 over Rational Field


We check that the input la is standardized:

sage: N = S.cell_module([2,1])
sage: M is N
True

>>> from sage.all import *
>>> N = S.cell_module([Integer(2),Integer(1)])
>>> M is N
True

cell_module_indices(la)[source]#

Return the indices of the cell module of self indexed by la .

This is the finite set $$M(\lambda)$$.

EXAMPLES:

sage: S = SymmetricGroupAlgebra(QQ, 4)
sage: S.cell_module_indices([3,1])
Standard tableaux of shape [3, 1]

>>> from sage.all import *
>>> S = SymmetricGroupAlgebra(QQ, Integer(4))
>>> S.cell_module_indices([Integer(3),Integer(1)])
Standard tableaux of shape [3, 1]

cell_poset()[source]#

Return the cell poset of self.

EXAMPLES:

sage: S = SymmetricGroupAlgebra(QQ, 4)
sage: S.cell_poset()
Finite poset containing 5 elements

>>> from sage.all import *
>>> S = SymmetricGroupAlgebra(QQ, Integer(4))
>>> S.cell_poset()
Finite poset containing 5 elements

central_orthogonal_idempotent(la, block=True)[source]#

Return the central idempotent for the symmetric group of order $$n$$ corresponding to the indecomposable block to which the partition la is associated.

If self.base_ring() contains $$\QQ$$, this corresponds to the classical central idempotent corresponding to the irreducible representation indexed by la.

Alternatively, if self.base_ring() has characteristic $$p > 0$$, then Theorem 2.8 in [Mur1983] provides that la is associated to an idempotent $$f_\mu$$, where $$\mu$$ is the $$p$$-core of la. This $$f_\mu$$ is a sum of classical idempotents,

$f_\mu = \sum_{c(\lambda)=\mu} e_\lambda,$

where the sum ranges over the partitions $$\lambda$$ of $$n$$ with $$p$$-core equal to $$\mu$$.

INPUT:

• la – a partition of self.n or a self.base_ring().characteristic()-core of such a partition

• block – boolean (default: True); when False, this returns the classical idempotent associated to la (defined over $$\QQ$$)

OUTPUT:

If block=False and the corresponding coefficients are not defined over self.base_ring(), then return None. Otherwise return an element of self.

EXAMPLES:

Asking for block idempotents in any characteristic, by passing a partition of self.n:

sage: S0 = SymmetricGroup(4).algebra(QQ)
sage: S2 = SymmetricGroup(4).algebra(GF(2))
sage: S3 = SymmetricGroup(4).algebra(GF(3))
sage: S0.central_orthogonal_idempotent([2,1,1])
3/8*() - 1/8*(3,4) - 1/8*(2,3) - 1/8*(2,4) - 1/8*(1,2)
- 1/8*(1,2)(3,4) + 1/8*(1,2,3,4) + 1/8*(1,2,4,3)
+ 1/8*(1,3,4,2) - 1/8*(1,3) - 1/8*(1,3)(2,4)
+ 1/8*(1,3,2,4) + 1/8*(1,4,3,2) - 1/8*(1,4)
+ 1/8*(1,4,2,3) - 1/8*(1,4)(2,3)
sage: S2.central_orthogonal_idempotent([2,1,1])
()
sage: idem = S3.central_orthogonal_idempotent([4]); idem
() + (1,2)(3,4) + (1,3)(2,4) + (1,4)(2,3)
sage: idem == S3.central_orthogonal_idempotent([1,1,1,1])
True
sage: S3.central_orthogonal_idempotent([2,2])
() + (1,2)(3,4) + (1,3)(2,4) + (1,4)(2,3)

>>> from sage.all import *
>>> S0 = SymmetricGroup(Integer(4)).algebra(QQ)
>>> S2 = SymmetricGroup(Integer(4)).algebra(GF(Integer(2)))
>>> S3 = SymmetricGroup(Integer(4)).algebra(GF(Integer(3)))
>>> S0.central_orthogonal_idempotent([Integer(2),Integer(1),Integer(1)])
3/8*() - 1/8*(3,4) - 1/8*(2,3) - 1/8*(2,4) - 1/8*(1,2)
- 1/8*(1,2)(3,4) + 1/8*(1,2,3,4) + 1/8*(1,2,4,3)
+ 1/8*(1,3,4,2) - 1/8*(1,3) - 1/8*(1,3)(2,4)
+ 1/8*(1,3,2,4) + 1/8*(1,4,3,2) - 1/8*(1,4)
+ 1/8*(1,4,2,3) - 1/8*(1,4)(2,3)
>>> S2.central_orthogonal_idempotent([Integer(2),Integer(1),Integer(1)])
()
>>> idem = S3.central_orthogonal_idempotent([Integer(4)]); idem
() + (1,2)(3,4) + (1,3)(2,4) + (1,4)(2,3)
>>> idem == S3.central_orthogonal_idempotent([Integer(1),Integer(1),Integer(1),Integer(1)])
True
>>> S3.central_orthogonal_idempotent([Integer(2),Integer(2)])
() + (1,2)(3,4) + (1,3)(2,4) + (1,4)(2,3)


Asking for block idempotents in any characteristic, by passing $$p$$-cores:

sage: S0.central_orthogonal_idempotent([1,1])
Traceback (most recent call last):
...
ValueError: [1, 1] is not a partition of integer 4
sage: S2.central_orthogonal_idempotent([])
()
sage: S2.central_orthogonal_idempotent([1])
Traceback (most recent call last):
...
ValueError: the 2-core of [1] is not a 2-core of a partition of 4
sage: S3.central_orthogonal_idempotent([1])
() + (1,2)(3,4) + (1,3)(2,4) + (1,4)(2,3)
sage: S3.central_orthogonal_idempotent([7])
() + (1,2)(3,4) + (1,3)(2,4) + (1,4)(2,3)

>>> from sage.all import *
>>> S0.central_orthogonal_idempotent([Integer(1),Integer(1)])
Traceback (most recent call last):
...
ValueError: [1, 1] is not a partition of integer 4
>>> S2.central_orthogonal_idempotent([])
()
>>> S2.central_orthogonal_idempotent([Integer(1)])
Traceback (most recent call last):
...
ValueError: the 2-core of [1] is not a 2-core of a partition of 4
>>> S3.central_orthogonal_idempotent([Integer(1)])
() + (1,2)(3,4) + (1,3)(2,4) + (1,4)(2,3)
>>> S3.central_orthogonal_idempotent([Integer(7)])
() + (1,2)(3,4) + (1,3)(2,4) + (1,4)(2,3)


sage: S3.central_orthogonal_idempotent([2,2], block=False) is None
True
sage: S3.central_orthogonal_idempotent([2,1,1], block=False)
(3,4) + (2,3) + (2,4) + (1,2) + (1,2)(3,4) + 2*(1,2,3,4)
+ 2*(1,2,4,3) + 2*(1,3,4,2) + (1,3) + (1,3)(2,4)
+ 2*(1,3,2,4) + 2*(1,4,3,2) + (1,4) + 2*(1,4,2,3)
+ (1,4)(2,3)

>>> from sage.all import *
>>> S3.central_orthogonal_idempotent([Integer(2),Integer(2)], block=False) is None
True
>>> S3.central_orthogonal_idempotent([Integer(2),Integer(1),Integer(1)], block=False)
(3,4) + (2,3) + (2,4) + (1,2) + (1,2)(3,4) + 2*(1,2,3,4)
+ 2*(1,2,4,3) + 2*(1,3,4,2) + (1,3) + (1,3)(2,4)
+ 2*(1,3,2,4) + 2*(1,4,3,2) + (1,4) + 2*(1,4,2,3)
+ (1,4)(2,3)

central_orthogonal_idempotents()[source]#

Return a maximal list of central orthogonal idempotents for self.

This method does not require that self be semisimple, relying on Nakayama’s Conjecture whenever self.base_ring() has positive characteristic.

EXAMPLES:

sage: QS3 = SymmetricGroupAlgebra(QQ,3)
sage: a = QS3.central_orthogonal_idempotents()
sage: a[0]  # [3]
1/6*[1, 2, 3] + 1/6*[1, 3, 2] + 1/6*[2, 1, 3] + 1/6*[2, 3, 1]
+ 1/6*[3, 1, 2] + 1/6*[3, 2, 1]
sage: a[1]  # [2, 1]
2/3*[1, 2, 3] - 1/3*[2, 3, 1] - 1/3*[3, 1, 2]

>>> from sage.all import *
>>> QS3 = SymmetricGroupAlgebra(QQ,Integer(3))
>>> a = QS3.central_orthogonal_idempotents()
>>> a[Integer(0)]  # [3]
1/6*[1, 2, 3] + 1/6*[1, 3, 2] + 1/6*[2, 1, 3] + 1/6*[2, 3, 1]
+ 1/6*[3, 1, 2] + 1/6*[3, 2, 1]
>>> a[Integer(1)]  # [2, 1]
2/3*[1, 2, 3] - 1/3*[2, 3, 1] - 1/3*[3, 1, 2]

dft(form=None, mult='l2r')[source]#

Return the discrete Fourier transform for self.

See [Mur1983] for the construction of central primitive orthogonal idempotents. For each idempotent $$e_i$$ we have a homomorphic projection $$v \mapsto v e_i$$. Choose a basis for each submodule spanned by $$\{\sigma e_i | \sigma \in S_n\}$$. The change-of-basis from the standard basis $$\{\sigma\}_\sigma$$ is returned.

INPUT:

• mult – string (default: $$l2r$$). If set to $$r2l$$, this causes the method to use the antipodes (antipode()) of the seminormal basis instead of the seminormal basis.

EXAMPLES:

sage: QS3 = SymmetricGroupAlgebra(QQ, 3)
sage: QS3.dft()
[   1    1    1    1    1    1]
[   1  1/2   -1 -1/2 -1/2  1/2]
[   0  3/4    0  3/4 -3/4 -3/4]
[   0    1    0   -1    1   -1]
[   1 -1/2    1 -1/2 -1/2 -1/2]
[   1   -1   -1    1    1   -1]

>>> from sage.all import *
>>> QS3 = SymmetricGroupAlgebra(QQ, Integer(3))
>>> QS3.dft()
[   1    1    1    1    1    1]
[   1  1/2   -1 -1/2 -1/2  1/2]
[   0  3/4    0  3/4 -3/4 -3/4]
[   0    1    0   -1    1   -1]
[   1 -1/2    1 -1/2 -1/2 -1/2]
[   1   -1   -1    1    1   -1]


Over fields of characteristic $$p > 0$$ such that $$p \mid n!$$, we use the modular Fourier transform (Issue #37751):

sage: GF2S3 = SymmetricGroupAlgebra(GF(2), 3)
sage: GF2S3.dft()
[1 0 0 0 1 0]
[0 1 0 0 0 1]
[0 0 1 0 0 1]
[0 0 0 1 1 0]
[1 0 0 1 1 0]
[0 1 1 0 0 1]

>>> from sage.all import *
>>> GF2S3 = SymmetricGroupAlgebra(GF(Integer(2)), Integer(3))
>>> GF2S3.dft()
[1 0 0 0 1 0]
[0 1 0 0 0 1]
[0 0 1 0 0 1]
[0 0 0 1 1 0]
[1 0 0 1 1 0]
[0 1 1 0 0 1]

epsilon_ik(itab, ktab, star=0, mult='l2r')[source]#

Return the seminormal basis element of self corresponding to the pair of tableaux itab and ktab (or restrictions of these tableaux, if the optional variable star is set).

INPUT:

• itab, ktab – two standard tableaux of size $$n$$.

• star – integer (default: $$0$$).

• mult – string (default: $$l2r$$). If set to $$r2l$$, this causes the method to return the antipode (antipode()) of $$\epsilon(I, K)$$ instead of $$\epsilon(I, K)$$ itself.

OUTPUT:

The element $$\epsilon(I, K)$$, where $$I$$ and $$K$$ are the tableaux obtained by removing all entries higher than $$n - \mathrm{star}$$ from itab and ktab, respectively. Here, we are using the notations from seminormal_basis().

EXAMPLES:

sage: QS3 = SymmetricGroupAlgebra(QQ, 3)
sage: a = QS3.epsilon_ik([[1,2,3]], [[1,2,3]]); a
1/6*[1, 2, 3] + 1/6*[1, 3, 2] + 1/6*[2, 1, 3] + 1/6*[2, 3, 1] + 1/6*[3, 1, 2] + 1/6*[3, 2, 1]
sage: QS3.dft()*vector(a)
(1, 0, 0, 0, 0, 0)
sage: a = QS3.epsilon_ik([[1,2],[3]], [[1,2],[3]]); a
1/3*[1, 2, 3] - 1/6*[1, 3, 2] + 1/3*[2, 1, 3] - 1/6*[2, 3, 1] - 1/6*[3, 1, 2] - 1/6*[3, 2, 1]
sage: QS3.dft()*vector(a)
(0, 0, 0, 0, 1, 0)

>>> from sage.all import *
>>> QS3 = SymmetricGroupAlgebra(QQ, Integer(3))
>>> a = QS3.epsilon_ik([[Integer(1),Integer(2),Integer(3)]], [[Integer(1),Integer(2),Integer(3)]]); a
1/6*[1, 2, 3] + 1/6*[1, 3, 2] + 1/6*[2, 1, 3] + 1/6*[2, 3, 1] + 1/6*[3, 1, 2] + 1/6*[3, 2, 1]
>>> QS3.dft()*vector(a)
(1, 0, 0, 0, 0, 0)
>>> a = QS3.epsilon_ik([[Integer(1),Integer(2)],[Integer(3)]], [[Integer(1),Integer(2)],[Integer(3)]]); a
1/3*[1, 2, 3] - 1/6*[1, 3, 2] + 1/3*[2, 1, 3] - 1/6*[2, 3, 1] - 1/6*[3, 1, 2] - 1/6*[3, 2, 1]
>>> QS3.dft()*vector(a)
(0, 0, 0, 0, 1, 0)


Let us take some properties of the seminormal basis listed in the docstring of seminormal_basis(), and verify them on the situation of $$S_3$$.

First, check the formula

$\epsilon(T) = \frac{1}{\kappa_{\mathrm{sh}(T)}} \epsilon(\overline{T}) e(T) \epsilon(\overline{T}).$

In fact:

sage: from sage.combinat.symmetric_group_algebra import e
sage: def test_sn1(n):
....:     QSn = SymmetricGroupAlgebra(QQ, n)
....:     QSn1 = SymmetricGroupAlgebra(QQ, n - 1)
....:     for T in StandardTableaux(n):
....:         TT = T.restrict(n-1)
....:         eTT = QSn1.epsilon_ik(TT, TT)
....:         eT = QSn.epsilon_ik(T, T)
....:         kT = prod(T.shape().hooks())
....:         if kT * eT != eTT * e(T) * eTT:
....:             return False
....:     return True
sage: test_sn1(3)
True
sage: test_sn1(4)   # long time
True

>>> from sage.all import *
>>> from sage.combinat.symmetric_group_algebra import e
>>> def test_sn1(n):
...     QSn = SymmetricGroupAlgebra(QQ, n)
...     QSn1 = SymmetricGroupAlgebra(QQ, n - Integer(1))
...     for T in StandardTableaux(n):
...         TT = T.restrict(n-Integer(1))
...         eTT = QSn1.epsilon_ik(TT, TT)
...         eT = QSn.epsilon_ik(T, T)
...         kT = prod(T.shape().hooks())
...         if kT * eT != eTT * e(T) * eTT:
...             return False
...     return True
>>> test_sn1(Integer(3))
True
>>> test_sn1(Integer(4))   # long time
True


Next, we check the identity

$\epsilon(T, S) = \frac{1}{\kappa_{\mathrm{sh}(T)}} \epsilon(\overline S) \pi_{T, S} e(T) \epsilon(\overline T)$

which we used to define $$\epsilon(T, S)$$. In fact:

sage: from sage.combinat.symmetric_group_algebra import e
sage: def test_sn2(n):
....:     QSn = SymmetricGroupAlgebra(QQ, n)
....:     mul = QSn.left_action_product
....:     QSn1 = SymmetricGroupAlgebra(QQ, n - 1)
....:     for lam in Partitions(n):
....:         k = prod(lam.hooks())
....:         for T in StandardTableaux(lam):
....:             for S in StandardTableaux(lam):
....:                 TT = T.restrict(n-1)
....:                 SS = S.restrict(n-1)
....:                 eTT = QSn1.epsilon_ik(TT, TT)
....:                 eSS = QSn1.epsilon_ik(SS, SS)
....:                 eTS = QSn.epsilon_ik(T, S)
....:                 piTS = [0] * n
....:                 for (i, j) in T.cells():
....:                     piTS[T[i][j] - 1] = S[i][j]
....:                 piTS = QSn(Permutation(piTS))
....:                 if k * eTS != mul(mul(eSS, piTS), mul(e(T), eTT)):
....:                     return False
....:     return True
sage: test_sn2(3)
True
sage: test_sn2(4)   # long time
True

>>> from sage.all import *
>>> from sage.combinat.symmetric_group_algebra import e
>>> def test_sn2(n):
...     QSn = SymmetricGroupAlgebra(QQ, n)
...     mul = QSn.left_action_product
...     QSn1 = SymmetricGroupAlgebra(QQ, n - Integer(1))
...     for lam in Partitions(n):
...         k = prod(lam.hooks())
...         for T in StandardTableaux(lam):
...             for S in StandardTableaux(lam):
...                 TT = T.restrict(n-Integer(1))
...                 SS = S.restrict(n-Integer(1))
...                 eTT = QSn1.epsilon_ik(TT, TT)
...                 eSS = QSn1.epsilon_ik(SS, SS)
...                 eTS = QSn.epsilon_ik(T, S)
...                 piTS = [Integer(0)] * n
...                 for (i, j) in T.cells():
...                     piTS[T[i][j] - Integer(1)] = S[i][j]
...                 piTS = QSn(Permutation(piTS))
...                 if k * eTS != mul(mul(eSS, piTS), mul(e(T), eTT)):
...                     return False
...     return True
>>> test_sn2(Integer(3))
True
>>> test_sn2(Integer(4))   # long time
True


Let us finally check the identity

$\epsilon(T, S) \epsilon(U, V) = \delta_{T, V} \epsilon(U, S)$

In fact:

sage: def test_sn3(lam):
....:     n = lam.size()
....:     QSn = SymmetricGroupAlgebra(QQ, n)
....:     mul = QSn.left_action_product
....:     for T in StandardTableaux(lam):
....:         for S in StandardTableaux(lam):
....:             for U in StandardTableaux(lam):
....:                 for V in StandardTableaux(lam):
....:                     lhs = mul(QSn.epsilon_ik(T, S), QSn.epsilon_ik(U, V))
....:                     if T == V:
....:                         rhs = QSn.epsilon_ik(U, S)
....:                     else:
....:                         rhs = QSn.zero()
....:                     if rhs != lhs:
....:                         return False
....:     return True
sage: all( test_sn3(lam) for lam in Partitions(3) )
True
sage: all( test_sn3(lam) for lam in Partitions(4) )   # long time
True

>>> from sage.all import *
>>> def test_sn3(lam):
...     n = lam.size()
...     QSn = SymmetricGroupAlgebra(QQ, n)
...     mul = QSn.left_action_product
...     for T in StandardTableaux(lam):
...         for S in StandardTableaux(lam):
...             for U in StandardTableaux(lam):
...                 for V in StandardTableaux(lam):
...                     lhs = mul(QSn.epsilon_ik(T, S), QSn.epsilon_ik(U, V))
...                     if T == V:
...                         rhs = QSn.epsilon_ik(U, S)
...                     else:
...                         rhs = QSn.zero()
...                     if rhs != lhs:
...                         return False
...     return True
>>> all( test_sn3(lam) for lam in Partitions(Integer(3)) )
True
>>> all( test_sn3(lam) for lam in Partitions(Integer(4)) )   # long time
True

garsia_procesi_module(la)[source]#

Return the Garsia-Procesi module of self indexed by la.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(GF(2), 6)
sage: SGA.garsia_procesi_module(Partition([2,2,1,1]))
Garsia-Procesi module of shape [2, 2, 1, 1] over Finite Field of size 2

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(GF(Integer(2)), Integer(6))
>>> SGA.garsia_procesi_module(Partition([Integer(2),Integer(2),Integer(1),Integer(1)]))
Garsia-Procesi module of shape [2, 2, 1, 1] over Finite Field of size 2

jucys_murphy(k)[source]#

Return the Jucys-Murphy element $$J_k$$ (also known as a Young-Jucys-Murphy element) for the symmetric group algebra self.

The Jucys-Murphy element $$J_k$$ in the symmetric group algebra $$R S_n$$ is defined for every $$k \in \{ 1, 2, \ldots, n \}$$ by

$J_k = (1, k) + (2, k) + \cdots + (k-1, k) \in R S_n,$

where the addends are transpositions in $$S_n$$ (regarded as elements of $$R S_n$$). We note that there is not a dependence on $$n$$, so it is often suppressed in the notation.

EXAMPLES:

sage: QS3 = SymmetricGroupAlgebra(QQ, 3)
sage: QS3.jucys_murphy(1)
0
sage: QS3.jucys_murphy(2)
[2, 1, 3]
sage: QS3.jucys_murphy(3)
[1, 3, 2] + [3, 2, 1]

sage: QS4 = SymmetricGroupAlgebra(QQ, 4)
sage: j3 = QS4.jucys_murphy(3); j3
[1, 3, 2, 4] + [3, 2, 1, 4]
sage: j4 = QS4.jucys_murphy(4); j4
[1, 2, 4, 3] + [1, 4, 3, 2] + [4, 2, 3, 1]
sage: j3*j4 == j4*j3
True

sage: QS5 = SymmetricGroupAlgebra(QQ, 5)
sage: QS5.jucys_murphy(4)
[1, 2, 4, 3, 5] + [1, 4, 3, 2, 5] + [4, 2, 3, 1, 5]

>>> from sage.all import *
>>> QS3 = SymmetricGroupAlgebra(QQ, Integer(3))
>>> QS3.jucys_murphy(Integer(1))
0
>>> QS3.jucys_murphy(Integer(2))
[2, 1, 3]
>>> QS3.jucys_murphy(Integer(3))
[1, 3, 2] + [3, 2, 1]

>>> QS4 = SymmetricGroupAlgebra(QQ, Integer(4))
>>> j3 = QS4.jucys_murphy(Integer(3)); j3
[1, 3, 2, 4] + [3, 2, 1, 4]
>>> j4 = QS4.jucys_murphy(Integer(4)); j4
[1, 2, 4, 3] + [1, 4, 3, 2] + [4, 2, 3, 1]
>>> j3*j4 == j4*j3
True

>>> QS5 = SymmetricGroupAlgebra(QQ, Integer(5))
>>> QS5.jucys_murphy(Integer(4))
[1, 2, 4, 3, 5] + [1, 4, 3, 2, 5] + [4, 2, 3, 1, 5]

kazhdan_lusztig_basis_element(w)[source]#

Return the Kazhdan-Lusztig $$C'_w$$ basis element at $$q = 1$$.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(QQ, 3)
sage: for w in SGA.group():
....:     print(w, SGA.kazhdan_lusztig_basis_element(w))
[1, 2, 3] [1, 2, 3]
[1, 3, 2] [1, 2, 3] + [1, 3, 2]
[2, 1, 3] [1, 2, 3] + [2, 1, 3]
[2, 3, 1] [1, 2, 3] + [1, 3, 2] + [2, 1, 3] + [2, 3, 1]
[3, 1, 2] [1, 2, 3] + [1, 3, 2] + [2, 1, 3] + [3, 1, 2]
[3, 2, 1] [1, 2, 3] + [1, 3, 2] + [2, 1, 3] + [2, 3, 1] + [3, 1, 2] + [3, 2, 1]

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(QQ, Integer(3))
>>> for w in SGA.group():
...     print(w, SGA.kazhdan_lusztig_basis_element(w))
[1, 2, 3] [1, 2, 3]
[1, 3, 2] [1, 2, 3] + [1, 3, 2]
[2, 1, 3] [1, 2, 3] + [2, 1, 3]
[2, 3, 1] [1, 2, 3] + [1, 3, 2] + [2, 1, 3] + [2, 3, 1]
[3, 1, 2] [1, 2, 3] + [1, 3, 2] + [2, 1, 3] + [3, 1, 2]
[3, 2, 1] [1, 2, 3] + [1, 3, 2] + [2, 1, 3] + [2, 3, 1] + [3, 1, 2] + [3, 2, 1]

kazhdan_lusztig_cellular_basis()[source]#

Return the Kazhdan-Lusztig basis (at $$q = 1$$) of self as a cellular basis.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(QQ, 3)
sage: KL = SGA.kazhdan_lusztig_cellular_basis()
sage: KL(SGA.an_element())
C([2, 1], [[1, 2], [3]], [[1, 2], [3]])
+ C([2, 1], [[1, 3], [2]], [[1, 2], [3]])
+ 2*C([2, 1], [[1, 3], [2]], [[1, 3], [2]])
- 3*C([3], [[1, 2, 3]], [[1, 2, 3]])

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(QQ, Integer(3))
>>> KL = SGA.kazhdan_lusztig_cellular_basis()
>>> KL(SGA.an_element())
C([2, 1], [[1, 2], [3]], [[1, 2], [3]])
+ C([2, 1], [[1, 3], [2]], [[1, 2], [3]])
+ 2*C([2, 1], [[1, 3], [2]], [[1, 3], [2]])
- 3*C([3], [[1, 2, 3]], [[1, 2, 3]])


Return the ladder idempontent of self.

Let $$F$$ be a field of characteristic $$p$$. The ladder idempotent of shape $$\lambda$$ is the idempotent of $$F[S_n]$$ defined as follows. Let $$T$$ be the ladder tableau of shape $$\lambda$$. Let $$[T]$$ be the set of standard tableaux whose residue sequence is the same as for $$T$$. Let $$\alpha$$ be the sizes of the ladders of $$\lambda$$. Then the ladder idempontent is constructed as

$\widetilde{e}_{\lambda} := \frac{1}{\alpha!} \left( \sum_{\sigma \in S_{\alpha}} \sigma \right) \left( \overline{\sum_{U \in [T]} E_U} \right),$

where $$E_{UU}$$ is the seminormal_basis() element over $$\QQ$$ and we project the sum to $$F$$, $$S_{\alpha}$$ is the Young subgroup corresponding to $$\alpha$$, and $$\alpha! = \alpha_1! \cdots \alpha_k!$$.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(GF(3), 4)
sage: for la in Partitions(SGA.n):
....:     print(la)
....:     print(idem)
....:     assert idem^2 == idem
[4]
2*[1, 2, 3, 4] + 2*[1, 2, 4, 3] + 2*[2, 1, 3, 4] + 2*[2, 1, 4, 3]
+ 2*[3, 4, 1, 2] + 2*[3, 4, 2, 1] + 2*[4, 3, 1, 2] + 2*[4, 3, 2, 1]
[3, 1]
2*[1, 2, 3, 4] + 2*[1, 2, 4, 3] + 2*[2, 1, 3, 4] + 2*[2, 1, 4, 3]
+ [3, 4, 1, 2] + [3, 4, 2, 1] + [4, 3, 1, 2] + [4, 3, 2, 1]
[2, 2]
2*[1, 2, 3, 4] + 2*[1, 2, 4, 3] + 2*[2, 1, 3, 4] + 2*[2, 1, 4, 3]
+ 2*[3, 4, 1, 2] + 2*[3, 4, 2, 1] + 2*[4, 3, 1, 2] + 2*[4, 3, 2, 1]
[2, 1, 1]
2*[1, 2, 3, 4] + [1, 2, 4, 3] + 2*[1, 3, 2, 4] + [1, 3, 4, 2]
+ [1, 4, 2, 3] + 2*[1, 4, 3, 2] + 2*[2, 1, 3, 4] + [2, 1, 4, 3]
+ 2*[2, 3, 1, 4] + [2, 3, 4, 1] + [2, 4, 1, 3] + 2*[2, 4, 3, 1]
+ 2*[3, 1, 2, 4] + [3, 1, 4, 2] + 2*[3, 2, 1, 4] + [3, 2, 4, 1]
+ [4, 1, 2, 3] + 2*[4, 1, 3, 2] + [4, 2, 1, 3] + 2*[4, 2, 3, 1]
[1, 1, 1, 1]
2*[1, 2, 3, 4] + [1, 2, 4, 3] + [2, 1, 3, 4] + 2*[2, 1, 4, 3]
+ 2*[3, 4, 1, 2] + [3, 4, 2, 1] + [4, 3, 1, 2] + 2*[4, 3, 2, 1]

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(GF(Integer(3)), Integer(4))
>>> for la in Partitions(SGA.n):
...     print(la)
...     print(idem)
...     assert idem**Integer(2) == idem
[4]
2*[1, 2, 3, 4] + 2*[1, 2, 4, 3] + 2*[2, 1, 3, 4] + 2*[2, 1, 4, 3]
+ 2*[3, 4, 1, 2] + 2*[3, 4, 2, 1] + 2*[4, 3, 1, 2] + 2*[4, 3, 2, 1]
[3, 1]
2*[1, 2, 3, 4] + 2*[1, 2, 4, 3] + 2*[2, 1, 3, 4] + 2*[2, 1, 4, 3]
+ [3, 4, 1, 2] + [3, 4, 2, 1] + [4, 3, 1, 2] + [4, 3, 2, 1]
[2, 2]
2*[1, 2, 3, 4] + 2*[1, 2, 4, 3] + 2*[2, 1, 3, 4] + 2*[2, 1, 4, 3]
+ 2*[3, 4, 1, 2] + 2*[3, 4, 2, 1] + 2*[4, 3, 1, 2] + 2*[4, 3, 2, 1]
[2, 1, 1]
2*[1, 2, 3, 4] + [1, 2, 4, 3] + 2*[1, 3, 2, 4] + [1, 3, 4, 2]
+ [1, 4, 2, 3] + 2*[1, 4, 3, 2] + 2*[2, 1, 3, 4] + [2, 1, 4, 3]
+ 2*[2, 3, 1, 4] + [2, 3, 4, 1] + [2, 4, 1, 3] + 2*[2, 4, 3, 1]
+ 2*[3, 1, 2, 4] + [3, 1, 4, 2] + 2*[3, 2, 1, 4] + [3, 2, 4, 1]
+ [4, 1, 2, 3] + 2*[4, 1, 3, 2] + [4, 2, 1, 3] + 2*[4, 2, 3, 1]
[1, 1, 1, 1]
2*[1, 2, 3, 4] + [1, 2, 4, 3] + [2, 1, 3, 4] + 2*[2, 1, 4, 3]
+ 2*[3, 4, 1, 2] + [3, 4, 2, 1] + [4, 3, 1, 2] + 2*[4, 3, 2, 1]


When $$p = 0$$, these idempotents will generate all of the simple modules (which are the Specht modules and also projective modules):

sage: SGA = SymmetricGroupAlgebra(QQ, 5)
sage: for la in Partitions(SGA.n):
....:     assert idem^2 == idem
....:     print(la, SGA.principal_ideal(idem).dimension())
[5] 1
[4, 1] 4
[3, 2] 5
[3, 1, 1] 6
[2, 2, 1] 5
[2, 1, 1, 1] 4
[1, 1, 1, 1, 1] 1
sage: [StandardTableaux(la).cardinality() for la in Partitions(SGA.n)]
[1, 4, 5, 6, 5, 4, 1]

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(QQ, Integer(5))
>>> for la in Partitions(SGA.n):
...     assert idem**Integer(2) == idem
...     print(la, SGA.principal_ideal(idem).dimension())
[5] 1
[4, 1] 4
[3, 2] 5
[3, 1, 1] 6
[2, 2, 1] 5
[2, 1, 1, 1] 4
[1, 1, 1, 1, 1] 1
>>> [StandardTableaux(la).cardinality() for la in Partitions(SGA.n)]
[1, 4, 5, 6, 5, 4, 1]


REFERENCES:

left_action_product(left, right)[source]#

Return the product of two elements left and right of self, where multiplication is defined in such a way that for two permutations $$p$$ and $$q$$, the product $$pq$$ is the permutation obtained by first applying $$q$$ and then applying $$p$$. This definition of multiplication is tailored to make multiplication of permutations associative with their action on numbers if permutations are to act on numbers from the left.

EXAMPLES:

sage: QS3 = SymmetricGroupAlgebra(QQ, 3)
sage: p1 = Permutation([2, 1, 3])
sage: p2 = Permutation([3, 1, 2])
sage: QS3.left_action_product(QS3(p1), QS3(p2))
[3, 2, 1]
sage: x = QS3([1, 2, 3]) - 2*QS3([1, 3, 2])
sage: y = 1/2 * QS3([3, 1, 2]) + 3*QS3([1, 2, 3])
sage: QS3.left_action_product(x, y)
3*[1, 2, 3] - 6*[1, 3, 2] - [2, 1, 3] + 1/2*[3, 1, 2]
sage: QS3.left_action_product(0, x)
0

>>> from sage.all import *
>>> QS3 = SymmetricGroupAlgebra(QQ, Integer(3))
>>> p1 = Permutation([Integer(2), Integer(1), Integer(3)])
>>> p2 = Permutation([Integer(3), Integer(1), Integer(2)])
>>> QS3.left_action_product(QS3(p1), QS3(p2))
[3, 2, 1]
>>> x = QS3([Integer(1), Integer(2), Integer(3)]) - Integer(2)*QS3([Integer(1), Integer(3), Integer(2)])
>>> y = Integer(1)/Integer(2) * QS3([Integer(3), Integer(1), Integer(2)]) + Integer(3)*QS3([Integer(1), Integer(2), Integer(3)])
>>> QS3.left_action_product(x, y)
3*[1, 2, 3] - 6*[1, 3, 2] - [2, 1, 3] + 1/2*[3, 1, 2]
>>> QS3.left_action_product(Integer(0), x)
0


The method coerces its input into the algebra self:

sage: QS4 = SymmetricGroupAlgebra(QQ, 4)
sage: QS4.left_action_product(QS3([1, 2, 3]), QS3([2, 1, 3]))
[2, 1, 3, 4]
sage: QS4.left_action_product(1, Permutation([4, 1, 2, 3]))
[4, 1, 2, 3]

>>> from sage.all import *
>>> QS4 = SymmetricGroupAlgebra(QQ, Integer(4))
>>> QS4.left_action_product(QS3([Integer(1), Integer(2), Integer(3)]), QS3([Integer(2), Integer(1), Integer(3)]))
[2, 1, 3, 4]
>>> QS4.left_action_product(Integer(1), Permutation([Integer(4), Integer(1), Integer(2), Integer(3)]))
[4, 1, 2, 3]


Warning

Note that coercion presently works from permutations of n into the n-th symmetric group algebra, and also from all smaller symmetric group algebras into the n-th symmetric group algebra, but not from permutations of integers smaller than n into the n-th symmetric group algebra.

monomial_from_smaller_permutation(permutation)[source]#

Convert permutation into a permutation, possibly extending it to the appropriate size, and return the corresponding basis element of self.

EXAMPLES:

sage: QS5 = SymmetricGroupAlgebra(QQ, 5)
sage: QS5.monomial_from_smaller_permutation([])
[1, 2, 3, 4, 5]
sage: QS5.monomial_from_smaller_permutation(Permutation([3,1,2]))
[3, 1, 2, 4, 5]
sage: QS5.monomial_from_smaller_permutation([5,3,4,1,2])
[5, 3, 4, 1, 2]
sage: QS5.monomial_from_smaller_permutation(SymmetricGroup(2)((1,2)))
[2, 1, 3, 4, 5]

sage: QS5g = SymmetricGroup(5).algebra(QQ)
sage: QS5g.monomial_from_smaller_permutation([2,1])
(1,2)

>>> from sage.all import *
>>> QS5 = SymmetricGroupAlgebra(QQ, Integer(5))
>>> QS5.monomial_from_smaller_permutation([])
[1, 2, 3, 4, 5]
>>> QS5.monomial_from_smaller_permutation(Permutation([Integer(3),Integer(1),Integer(2)]))
[3, 1, 2, 4, 5]
>>> QS5.monomial_from_smaller_permutation([Integer(5),Integer(3),Integer(4),Integer(1),Integer(2)])
[5, 3, 4, 1, 2]
>>> QS5.monomial_from_smaller_permutation(SymmetricGroup(Integer(2))((Integer(1),Integer(2))))
[2, 1, 3, 4, 5]

>>> QS5g = SymmetricGroup(Integer(5)).algebra(QQ)
>>> QS5g.monomial_from_smaller_permutation([Integer(2),Integer(1)])
(1,2)

murphy_basis()[source]#

Return the Murphy basis of self.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(QQ, 3)
sage: M = SGA.murphy_basis()
sage: M(SGA.an_element())
-C([1, 1, 1], [[1], [2], [3]], [[1], [2], [3]])
+ C([2, 1], [[1, 2], [3]], [[1, 2], [3]])
+ C([2, 1], [[1, 2], [3]], [[1, 3], [2]])
+ 2*C([2, 1], [[1, 3], [2]], [[1, 2], [3]])
+ 4*C([2, 1], [[1, 3], [2]], [[1, 3], [2]])
- 3*C([3], [[1, 2, 3]], [[1, 2, 3]])

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(QQ, Integer(3))
>>> M = SGA.murphy_basis()
>>> M(SGA.an_element())
-C([1, 1, 1], [[1], [2], [3]], [[1], [2], [3]])
+ C([2, 1], [[1, 2], [3]], [[1, 2], [3]])
+ C([2, 1], [[1, 2], [3]], [[1, 3], [2]])
+ 2*C([2, 1], [[1, 3], [2]], [[1, 2], [3]])
+ 4*C([2, 1], [[1, 3], [2]], [[1, 3], [2]])
- 3*C([3], [[1, 2, 3]], [[1, 2, 3]])

murphy_basis_element(S, T)[source]#

Return the Murphy basis element indexed by S and T.

EXAMPLES:

sage: import itertools
sage: SGA = SymmetricGroupAlgebra(QQ, 3)
sage: for S, T in itertools.product(StandardTableaux([2,1]), repeat=2):
....:     print(S, T, SGA.murphy_basis_element(S, T))
[[1, 3], [2]] [[1, 3], [2]] [1, 2, 3] + [2, 1, 3]
[[1, 3], [2]] [[1, 2], [3]] [1, 3, 2] + [3, 1, 2]
[[1, 2], [3]] [[1, 3], [2]] [1, 3, 2] + [2, 3, 1]
[[1, 2], [3]] [[1, 2], [3]] [1, 2, 3] + [3, 2, 1]

>>> from sage.all import *
>>> import itertools
>>> SGA = SymmetricGroupAlgebra(QQ, Integer(3))
>>> for S, T in itertools.product(StandardTableaux([Integer(2),Integer(1)]), repeat=Integer(2)):
...     print(S, T, SGA.murphy_basis_element(S, T))
[[1, 3], [2]] [[1, 3], [2]] [1, 2, 3] + [2, 1, 3]
[[1, 3], [2]] [[1, 2], [3]] [1, 3, 2] + [3, 1, 2]
[[1, 2], [3]] [[1, 3], [2]] [1, 3, 2] + [2, 3, 1]
[[1, 2], [3]] [[1, 2], [3]] [1, 2, 3] + [3, 2, 1]

retract_direct_product(f, m)[source]#

Return the direct-product retract of the element $$f \in R S_n$$ to $$R S_m$$, where $$m \leq n$$ (and where $$R S_n$$ is self).

If $$m$$ is a nonnegative integer less or equal to $$n$$, then the direct-product retract from $$S_n$$ to $$S_m$$ is defined as an $$R$$-linear map $$S_n \to S_m$$ which sends every permutation $$p \in S_n$$ to

$\begin{split}\begin{cases} \mbox{dret}(p) &\mbox{if } \mbox{dret}(p)\mbox{ is defined;} \\ 0 & \mbox{otherwise} \end{cases}.\end{split}$

Here $$\mbox{dret}(p)$$ denotes the direct-product retract of the permutation $$p$$ to $$S_m$$, which is defined in retract_direct_product().

EXAMPLES:

sage: SGA3 = SymmetricGroupAlgebra(QQ, 3)
sage: SGA3.retract_direct_product(2*SGA3([1,2,3]) - 4*SGA3([2,1,3]) + 7*SGA3([1,3,2]), 2)
2*[1, 2] - 4*[2, 1]
sage: SGA3.retract_direct_product(2*SGA3([1,3,2]) - 5*SGA3([2,3,1]), 2)
0

sage: SGA5 = SymmetricGroupAlgebra(QQ, 5)
sage: SGA5.retract_direct_product(8*SGA5([1,4,2,5,3]) - 6*SGA5([1,3,2,5,4]) + 11*SGA5([3,2,1,4,5]), 4)
11*[3, 2, 1, 4]
sage: SGA5.retract_direct_product(8*SGA5([1,4,2,5,3]) - 6*SGA5([1,3,2,5,4]) + 11*SGA5([3,2,1,4,5]), 3)
-6*[1, 3, 2] + 11*[3, 2, 1]
sage: SGA5.retract_direct_product(8*SGA5([1,4,2,5,3]) - 6*SGA5([1,3,2,5,4]) + 11*SGA5([3,2,1,4,5]), 2)
0
sage: SGA5.retract_direct_product(8*SGA5([1,4,2,5,3]) - 6*SGA5([1,3,2,5,4]) + 11*SGA5([3,2,1,4,5]), 1)
2*[1]

sage: SGA5.retract_direct_product(8*SGA5([1,2,3,4,5]) - 6*SGA5([1,3,2,4,5]), 3)
8*[1, 2, 3] - 6*[1, 3, 2]
sage: SGA5.retract_direct_product(8*SGA5([1,2,3,4,5]) - 6*SGA5([1,3,2,4,5]), 1)
2*[1]
sage: SGA5.retract_direct_product(8*SGA5([1,2,3,4,5]) - 6*SGA5([1,3,2,4,5]), 0)
2*[]

>>> from sage.all import *
>>> SGA3 = SymmetricGroupAlgebra(QQ, Integer(3))
>>> SGA3.retract_direct_product(Integer(2)*SGA3([Integer(1),Integer(2),Integer(3)]) - Integer(4)*SGA3([Integer(2),Integer(1),Integer(3)]) + Integer(7)*SGA3([Integer(1),Integer(3),Integer(2)]), Integer(2))
2*[1, 2] - 4*[2, 1]
>>> SGA3.retract_direct_product(Integer(2)*SGA3([Integer(1),Integer(3),Integer(2)]) - Integer(5)*SGA3([Integer(2),Integer(3),Integer(1)]), Integer(2))
0

>>> SGA5 = SymmetricGroupAlgebra(QQ, Integer(5))
>>> SGA5.retract_direct_product(Integer(8)*SGA5([Integer(1),Integer(4),Integer(2),Integer(5),Integer(3)]) - Integer(6)*SGA5([Integer(1),Integer(3),Integer(2),Integer(5),Integer(4)]) + Integer(11)*SGA5([Integer(3),Integer(2),Integer(1),Integer(4),Integer(5)]), Integer(4))
11*[3, 2, 1, 4]
>>> SGA5.retract_direct_product(Integer(8)*SGA5([Integer(1),Integer(4),Integer(2),Integer(5),Integer(3)]) - Integer(6)*SGA5([Integer(1),Integer(3),Integer(2),Integer(5),Integer(4)]) + Integer(11)*SGA5([Integer(3),Integer(2),Integer(1),Integer(4),Integer(5)]), Integer(3))
-6*[1, 3, 2] + 11*[3, 2, 1]
>>> SGA5.retract_direct_product(Integer(8)*SGA5([Integer(1),Integer(4),Integer(2),Integer(5),Integer(3)]) - Integer(6)*SGA5([Integer(1),Integer(3),Integer(2),Integer(5),Integer(4)]) + Integer(11)*SGA5([Integer(3),Integer(2),Integer(1),Integer(4),Integer(5)]), Integer(2))
0
>>> SGA5.retract_direct_product(Integer(8)*SGA5([Integer(1),Integer(4),Integer(2),Integer(5),Integer(3)]) - Integer(6)*SGA5([Integer(1),Integer(3),Integer(2),Integer(5),Integer(4)]) + Integer(11)*SGA5([Integer(3),Integer(2),Integer(1),Integer(4),Integer(5)]), Integer(1))
2*[1]

>>> SGA5.retract_direct_product(Integer(8)*SGA5([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]) - Integer(6)*SGA5([Integer(1),Integer(3),Integer(2),Integer(4),Integer(5)]), Integer(3))
8*[1, 2, 3] - 6*[1, 3, 2]
>>> SGA5.retract_direct_product(Integer(8)*SGA5([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]) - Integer(6)*SGA5([Integer(1),Integer(3),Integer(2),Integer(4),Integer(5)]), Integer(1))
2*[1]
>>> SGA5.retract_direct_product(Integer(8)*SGA5([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]) - Integer(6)*SGA5([Integer(1),Integer(3),Integer(2),Integer(4),Integer(5)]), Integer(0))
2*[]

retract_okounkov_vershik(f, m)[source]#

Return the Okounkov-Vershik retract of the element $$f \in R S_n$$ to $$R S_m$$, where $$m \leq n$$ (and where $$R S_n$$ is self).

If $$m$$ is a nonnegative integer less or equal to $$n$$, then the Okounkov-Vershik retract from $$S_n$$ to $$S_m$$ is defined as an $$R$$-linear map $$S_n \to S_m$$ which sends every permutation $$p \in S_n$$ to the Okounkov-Vershik retract of the permutation $$p$$ to $$S_m$$, which is defined in retract_okounkov_vershik().

EXAMPLES:

sage: SGA3 = SymmetricGroupAlgebra(QQ, 3)
sage: SGA3.retract_okounkov_vershik(2*SGA3([1,2,3]) - 4*SGA3([2,1,3]) + 7*SGA3([1,3,2]), 2)
9*[1, 2] - 4*[2, 1]
sage: SGA3.retract_okounkov_vershik(2*SGA3([1,3,2]) - 5*SGA3([2,3,1]), 2)
2*[1, 2] - 5*[2, 1]

sage: SGA5 = SymmetricGroupAlgebra(QQ, 5)
sage: SGA5.retract_okounkov_vershik(8*SGA5([1,4,2,5,3]) - 6*SGA5([1,3,2,5,4]) + 11*SGA5([3,2,1,4,5]), 4)
-6*[1, 3, 2, 4] + 8*[1, 4, 2, 3] + 11*[3, 2, 1, 4]
sage: SGA5.retract_okounkov_vershik(8*SGA5([1,4,2,5,3]) - 6*SGA5([1,3,2,5,4]) + 11*SGA5([3,2,1,4,5]), 3)
2*[1, 3, 2] + 11*[3, 2, 1]
sage: SGA5.retract_okounkov_vershik(8*SGA5([1,4,2,5,3]) - 6*SGA5([1,3,2,5,4]) + 11*SGA5([3,2,1,4,5]), 2)
13*[1, 2]
sage: SGA5.retract_okounkov_vershik(8*SGA5([1,4,2,5,3]) - 6*SGA5([1,3,2,5,4]) + 11*SGA5([3,2,1,4,5]), 1)
13*[1]

sage: SGA5.retract_okounkov_vershik(8*SGA5([1,2,3,4,5]) - 6*SGA5([1,3,2,4,5]), 3)
8*[1, 2, 3] - 6*[1, 3, 2]
sage: SGA5.retract_okounkov_vershik(8*SGA5([1,2,3,4,5]) - 6*SGA5([1,3,2,4,5]), 1)
2*[1]
sage: SGA5.retract_okounkov_vershik(8*SGA5([1,2,3,4,5]) - 6*SGA5([1,3,2,4,5]), 0)
2*[]

>>> from sage.all import *
>>> SGA3 = SymmetricGroupAlgebra(QQ, Integer(3))
>>> SGA3.retract_okounkov_vershik(Integer(2)*SGA3([Integer(1),Integer(2),Integer(3)]) - Integer(4)*SGA3([Integer(2),Integer(1),Integer(3)]) + Integer(7)*SGA3([Integer(1),Integer(3),Integer(2)]), Integer(2))
9*[1, 2] - 4*[2, 1]
>>> SGA3.retract_okounkov_vershik(Integer(2)*SGA3([Integer(1),Integer(3),Integer(2)]) - Integer(5)*SGA3([Integer(2),Integer(3),Integer(1)]), Integer(2))
2*[1, 2] - 5*[2, 1]

>>> SGA5 = SymmetricGroupAlgebra(QQ, Integer(5))
>>> SGA5.retract_okounkov_vershik(Integer(8)*SGA5([Integer(1),Integer(4),Integer(2),Integer(5),Integer(3)]) - Integer(6)*SGA5([Integer(1),Integer(3),Integer(2),Integer(5),Integer(4)]) + Integer(11)*SGA5([Integer(3),Integer(2),Integer(1),Integer(4),Integer(5)]), Integer(4))
-6*[1, 3, 2, 4] + 8*[1, 4, 2, 3] + 11*[3, 2, 1, 4]
>>> SGA5.retract_okounkov_vershik(Integer(8)*SGA5([Integer(1),Integer(4),Integer(2),Integer(5),Integer(3)]) - Integer(6)*SGA5([Integer(1),Integer(3),Integer(2),Integer(5),Integer(4)]) + Integer(11)*SGA5([Integer(3),Integer(2),Integer(1),Integer(4),Integer(5)]), Integer(3))
2*[1, 3, 2] + 11*[3, 2, 1]
>>> SGA5.retract_okounkov_vershik(Integer(8)*SGA5([Integer(1),Integer(4),Integer(2),Integer(5),Integer(3)]) - Integer(6)*SGA5([Integer(1),Integer(3),Integer(2),Integer(5),Integer(4)]) + Integer(11)*SGA5([Integer(3),Integer(2),Integer(1),Integer(4),Integer(5)]), Integer(2))
13*[1, 2]
>>> SGA5.retract_okounkov_vershik(Integer(8)*SGA5([Integer(1),Integer(4),Integer(2),Integer(5),Integer(3)]) - Integer(6)*SGA5([Integer(1),Integer(3),Integer(2),Integer(5),Integer(4)]) + Integer(11)*SGA5([Integer(3),Integer(2),Integer(1),Integer(4),Integer(5)]), Integer(1))
13*[1]

>>> SGA5.retract_okounkov_vershik(Integer(8)*SGA5([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]) - Integer(6)*SGA5([Integer(1),Integer(3),Integer(2),Integer(4),Integer(5)]), Integer(3))
8*[1, 2, 3] - 6*[1, 3, 2]
>>> SGA5.retract_okounkov_vershik(Integer(8)*SGA5([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]) - Integer(6)*SGA5([Integer(1),Integer(3),Integer(2),Integer(4),Integer(5)]), Integer(1))
2*[1]
>>> SGA5.retract_okounkov_vershik(Integer(8)*SGA5([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]) - Integer(6)*SGA5([Integer(1),Integer(3),Integer(2),Integer(4),Integer(5)]), Integer(0))
2*[]

retract_plain(f, m)[source]#

Return the plain retract of the element $$f \in R S_n$$ to $$R S_m$$, where $$m \leq n$$ (and where $$R S_n$$ is self).

If $$m$$ is a nonnegative integer less or equal to $$n$$, then the plain retract from $$S_n$$ to $$S_m$$ is defined as an $$R$$-linear map $$S_n \to S_m$$ which sends every permutation $$p \in S_n$$ to

$\begin{split}\begin{cases} \mbox{pret}(p) &\mbox{if } \mbox{pret}(p)\mbox{ is defined;} \\ 0 & \mbox{otherwise} \end{cases}.\end{split}$

Here $$\mbox{pret}(p)$$ denotes the plain retract of the permutation $$p$$ to $$S_m$$, which is defined in retract_plain().

EXAMPLES:

sage: SGA3 = SymmetricGroupAlgebra(QQ, 3)
sage: SGA3.retract_plain(2*SGA3([1,2,3]) - 4*SGA3([2,1,3]) + 7*SGA3([1,3,2]), 2)
2*[1, 2] - 4*[2, 1]
sage: SGA3.retract_plain(2*SGA3([1,3,2]) - 5*SGA3([2,3,1]), 2)
0

sage: SGA5 = SymmetricGroupAlgebra(QQ, 5)
sage: SGA5.retract_plain(8*SGA5([1,4,2,5,3]) - 6*SGA5([1,3,2,5,4]) + 11*SGA5([3,2,1,4,5]), 4)
11*[3, 2, 1, 4]
sage: SGA5.retract_plain(8*SGA5([1,4,2,5,3]) - 6*SGA5([1,3,2,5,4]) + 11*SGA5([3,2,1,4,5]), 3)
11*[3, 2, 1]
sage: SGA5.retract_plain(8*SGA5([1,4,2,5,3]) - 6*SGA5([1,3,2,5,4]) + 11*SGA5([3,2,1,4,5]), 2)
0
sage: SGA5.retract_plain(8*SGA5([1,4,2,5,3]) - 6*SGA5([1,3,2,5,4]) + 11*SGA5([3,2,1,4,5]), 1)
0

sage: SGA5.retract_plain(8*SGA5([1,2,3,4,5]) - 6*SGA5([1,3,2,4,5]), 3)
8*[1, 2, 3] - 6*[1, 3, 2]
sage: SGA5.retract_plain(8*SGA5([1,2,3,4,5]) - 6*SGA5([1,3,2,4,5]), 1)
8*[1]
sage: SGA5.retract_plain(8*SGA5([1,2,3,4,5]) - 6*SGA5([1,3,2,4,5]), 0)
8*[]

>>> from sage.all import *
>>> SGA3 = SymmetricGroupAlgebra(QQ, Integer(3))
>>> SGA3.retract_plain(Integer(2)*SGA3([Integer(1),Integer(2),Integer(3)]) - Integer(4)*SGA3([Integer(2),Integer(1),Integer(3)]) + Integer(7)*SGA3([Integer(1),Integer(3),Integer(2)]), Integer(2))
2*[1, 2] - 4*[2, 1]
>>> SGA3.retract_plain(Integer(2)*SGA3([Integer(1),Integer(3),Integer(2)]) - Integer(5)*SGA3([Integer(2),Integer(3),Integer(1)]), Integer(2))
0

>>> SGA5 = SymmetricGroupAlgebra(QQ, Integer(5))
>>> SGA5.retract_plain(Integer(8)*SGA5([Integer(1),Integer(4),Integer(2),Integer(5),Integer(3)]) - Integer(6)*SGA5([Integer(1),Integer(3),Integer(2),Integer(5),Integer(4)]) + Integer(11)*SGA5([Integer(3),Integer(2),Integer(1),Integer(4),Integer(5)]), Integer(4))
11*[3, 2, 1, 4]
>>> SGA5.retract_plain(Integer(8)*SGA5([Integer(1),Integer(4),Integer(2),Integer(5),Integer(3)]) - Integer(6)*SGA5([Integer(1),Integer(3),Integer(2),Integer(5),Integer(4)]) + Integer(11)*SGA5([Integer(3),Integer(2),Integer(1),Integer(4),Integer(5)]), Integer(3))
11*[3, 2, 1]
>>> SGA5.retract_plain(Integer(8)*SGA5([Integer(1),Integer(4),Integer(2),Integer(5),Integer(3)]) - Integer(6)*SGA5([Integer(1),Integer(3),Integer(2),Integer(5),Integer(4)]) + Integer(11)*SGA5([Integer(3),Integer(2),Integer(1),Integer(4),Integer(5)]), Integer(2))
0
>>> SGA5.retract_plain(Integer(8)*SGA5([Integer(1),Integer(4),Integer(2),Integer(5),Integer(3)]) - Integer(6)*SGA5([Integer(1),Integer(3),Integer(2),Integer(5),Integer(4)]) + Integer(11)*SGA5([Integer(3),Integer(2),Integer(1),Integer(4),Integer(5)]), Integer(1))
0

>>> SGA5.retract_plain(Integer(8)*SGA5([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]) - Integer(6)*SGA5([Integer(1),Integer(3),Integer(2),Integer(4),Integer(5)]), Integer(3))
8*[1, 2, 3] - 6*[1, 3, 2]
>>> SGA5.retract_plain(Integer(8)*SGA5([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]) - Integer(6)*SGA5([Integer(1),Integer(3),Integer(2),Integer(4),Integer(5)]), Integer(1))
8*[1]
>>> SGA5.retract_plain(Integer(8)*SGA5([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]) - Integer(6)*SGA5([Integer(1),Integer(3),Integer(2),Integer(4),Integer(5)]), Integer(0))
8*[]

right_action_product(left, right)[source]#

Return the product of two elements left and right of self, where multiplication is defined in such a way that for two permutations $$p$$ and $$q$$, the product $$pq$$ is the permutation obtained by first applying $$p$$ and then applying $$q$$. This definition of multiplication is tailored to make multiplication of permutations associative with their action on numbers if permutations are to act on numbers from the right.

EXAMPLES:

sage: QS3 = SymmetricGroupAlgebra(QQ, 3)
sage: p1 = Permutation([2, 1, 3])
sage: p2 = Permutation([3, 1, 2])
sage: QS3.right_action_product(QS3(p1), QS3(p2))
[1, 3, 2]
sage: x = QS3([1, 2, 3]) - 2*QS3([1, 3, 2])
sage: y = 1/2 * QS3([3, 1, 2]) + 3*QS3([1, 2, 3])
sage: QS3.right_action_product(x, y)
3*[1, 2, 3] - 6*[1, 3, 2] + 1/2*[3, 1, 2] - [3, 2, 1]
sage: QS3.right_action_product(0, x)
0

>>> from sage.all import *
>>> QS3 = SymmetricGroupAlgebra(QQ, Integer(3))
>>> p1 = Permutation([Integer(2), Integer(1), Integer(3)])
>>> p2 = Permutation([Integer(3), Integer(1), Integer(2)])
>>> QS3.right_action_product(QS3(p1), QS3(p2))
[1, 3, 2]
>>> x = QS3([Integer(1), Integer(2), Integer(3)]) - Integer(2)*QS3([Integer(1), Integer(3), Integer(2)])
>>> y = Integer(1)/Integer(2) * QS3([Integer(3), Integer(1), Integer(2)]) + Integer(3)*QS3([Integer(1), Integer(2), Integer(3)])
>>> QS3.right_action_product(x, y)
3*[1, 2, 3] - 6*[1, 3, 2] + 1/2*[3, 1, 2] - [3, 2, 1]
>>> QS3.right_action_product(Integer(0), x)
0


The method coerces its input into the algebra self:

sage: QS4 = SymmetricGroupAlgebra(QQ, 4)
sage: QS4.right_action_product(QS3([1, 2, 3]), QS3([2, 1, 3]))
[2, 1, 3, 4]
sage: QS4.right_action_product(1, Permutation([4, 1, 2, 3]))
[4, 1, 2, 3]

>>> from sage.all import *
>>> QS4 = SymmetricGroupAlgebra(QQ, Integer(4))
>>> QS4.right_action_product(QS3([Integer(1), Integer(2), Integer(3)]), QS3([Integer(2), Integer(1), Integer(3)]))
[2, 1, 3, 4]
>>> QS4.right_action_product(Integer(1), Permutation([Integer(4), Integer(1), Integer(2), Integer(3)]))
[4, 1, 2, 3]


Warning

Note that coercion presently works from permutations of n into the n-th symmetric group algebra, and also from all smaller symmetric group algebras into the n-th symmetric group algebra, but not from permutations of integers smaller than n into the n-th symmetric group algebra.

rsw_shuffling_element(k)[source]#

Return the $$k$$-th Reiner-Saliola-Welker shuffling element in the group algebra self.

The $$k$$-th Reiner-Saliola-Welker shuffling element in the symmetric group algebra $$R S_n$$ over a ring $$R$$ is defined as the sum $$\sum_{\sigma \in S_n} \mathrm{noninv}_k(\sigma) \cdot \sigma$$, where for every permutation $$\sigma$$, the number $$\mathrm{noninv}_k(\sigma)$$ is the number of all $$k$$-noninversions of $$\sigma$$ (that is, the number of all $$k$$-element subsets of $$\{ 1, 2, \ldots, n \}$$ on which $$\sigma$$ restricts to a strictly increasing map). See sage.combinat.permutation.number_of_noninversions() for the $$\mathrm{noninv}$$ map.

This element is more or less the operator $$\nu_{k, 1^{n-k}}$$ introduced in [RSW2011]; more precisely, $$\nu_{k, 1^{n-k}}$$ is the left multiplication by this element.

It is a nontrivial theorem (Theorem 1.1 in [RSW2011]) that the operators $$\nu_{k, 1^{n-k}}$$ (for fixed $$n$$ and varying $$k$$) pairwise commute. It is a conjecture (Conjecture 1.2 in [RSW2011]) that all their eigenvalues are integers (which, in light of their commutativity and easily established symmetry, yields that they can be simultaneously diagonalized over $$\QQ$$ with only integer eigenvalues).

EXAMPLES:

The Reiner-Saliola-Welker shuffling elements on $$\QQ S_3$$:

sage: QS3 = SymmetricGroupAlgebra(QQ, 3)
sage: QS3.rsw_shuffling_element(0)
[1, 2, 3] + [1, 3, 2] + [2, 1, 3] + [2, 3, 1] + [3, 1, 2] + [3, 2, 1]
sage: QS3.rsw_shuffling_element(1)
3*[1, 2, 3] + 3*[1, 3, 2] + 3*[2, 1, 3] + 3*[2, 3, 1] + 3*[3, 1, 2] + 3*[3, 2, 1]
sage: QS3.rsw_shuffling_element(2)
3*[1, 2, 3] + 2*[1, 3, 2] + 2*[2, 1, 3] + [2, 3, 1] + [3, 1, 2]
sage: QS3.rsw_shuffling_element(3)
[1, 2, 3]
sage: QS3.rsw_shuffling_element(4)
0

>>> from sage.all import *
>>> QS3 = SymmetricGroupAlgebra(QQ, Integer(3))
>>> QS3.rsw_shuffling_element(Integer(0))
[1, 2, 3] + [1, 3, 2] + [2, 1, 3] + [2, 3, 1] + [3, 1, 2] + [3, 2, 1]
>>> QS3.rsw_shuffling_element(Integer(1))
3*[1, 2, 3] + 3*[1, 3, 2] + 3*[2, 1, 3] + 3*[2, 3, 1] + 3*[3, 1, 2] + 3*[3, 2, 1]
>>> QS3.rsw_shuffling_element(Integer(2))
3*[1, 2, 3] + 2*[1, 3, 2] + 2*[2, 1, 3] + [2, 3, 1] + [3, 1, 2]
>>> QS3.rsw_shuffling_element(Integer(3))
[1, 2, 3]
>>> QS3.rsw_shuffling_element(Integer(4))
0


Checking the commutativity of Reiner-Saliola-Welker shuffling elements (we leave out the ones for which it is trivial):

sage: def test_rsw_comm(n):
....:     QSn = SymmetricGroupAlgebra(QQ, n)
....:     rsws = [QSn.rsw_shuffling_element(k) for k in range(2, n)]
....:     return all(ri * rsws[j] == rsws[j] * ri
....:                for i, ri in enumerate(rsws) for j in range(i))
sage: test_rsw_comm(3)
True
sage: test_rsw_comm(4)   # long time
True
sage: test_rsw_comm(5)   # not tested
True

>>> from sage.all import *
>>> def test_rsw_comm(n):
...     QSn = SymmetricGroupAlgebra(QQ, n)
...     rsws = [QSn.rsw_shuffling_element(k) for k in range(Integer(2), n)]
...     return all(ri * rsws[j] == rsws[j] * ri
...                for i, ri in enumerate(rsws) for j in range(i))
>>> test_rsw_comm(Integer(3))
True
>>> test_rsw_comm(Integer(4))   # long time
True
>>> test_rsw_comm(Integer(5))   # not tested
True


Note

For large k (relative to n), it might be faster to call QSn.left_action_product(QSn.semi_rsw_element(k), QSn.antipode(binary_unshuffle_sum(k))) than QSn.rsw_shuffling_element(n).

semi_rsw_element(k)[source]#

Return the $$k$$-th semi-RSW element in the group algebra self.

The $$k$$-th semi-RSW element in the symmetric group algebra $$R S_n$$ over a ring $$R$$ is defined as the sum of all permutations $$\sigma \in S_n$$ satisfying $$\sigma(1) < \sigma(2) < \cdots < \sigma(k)$$.

This element has the property that, if it is denoted by $$s_k$$, then $$s_k S(s_k)$$ is $$(n-k)!$$ times the $$k$$-th Reiner-Saliola-Welker shuffling element of $$R S_n$$ (see rsw_shuffling_element()). Here, $$S$$ denotes the antipode of the group algebra $$R S_n$$.

The $$k$$-th semi-RSW element is the image of the complete non-commutative symmetric function $$S^{(k, 1^{n-k})}$$ in the ring of non-commutative symmetric functions under the canonical projection on the symmetric group algebra (through the descent algebra).

EXAMPLES:

The semi-RSW elements on $$\QQ S_3$$:

sage: QS3 = SymmetricGroupAlgebra(QQ, 3)
sage: QS3.semi_rsw_element(0)
[1, 2, 3] + [1, 3, 2] + [2, 1, 3] + [2, 3, 1] + [3, 1, 2] + [3, 2, 1]
sage: QS3.semi_rsw_element(1)
[1, 2, 3] + [1, 3, 2] + [2, 1, 3] + [2, 3, 1] + [3, 1, 2] + [3, 2, 1]
sage: QS3.semi_rsw_element(2)
[1, 2, 3] + [1, 3, 2] + [2, 3, 1]
sage: QS3.semi_rsw_element(3)
[1, 2, 3]
sage: QS3.semi_rsw_element(4)
0

>>> from sage.all import *
>>> QS3 = SymmetricGroupAlgebra(QQ, Integer(3))
>>> QS3.semi_rsw_element(Integer(0))
[1, 2, 3] + [1, 3, 2] + [2, 1, 3] + [2, 3, 1] + [3, 1, 2] + [3, 2, 1]
>>> QS3.semi_rsw_element(Integer(1))
[1, 2, 3] + [1, 3, 2] + [2, 1, 3] + [2, 3, 1] + [3, 1, 2] + [3, 2, 1]
>>> QS3.semi_rsw_element(Integer(2))
[1, 2, 3] + [1, 3, 2] + [2, 3, 1]
>>> QS3.semi_rsw_element(Integer(3))
[1, 2, 3]
>>> QS3.semi_rsw_element(Integer(4))
0


Let us check the relation with the $$k$$-th Reiner-Saliola-Welker shuffling element stated in the docstring:

sage: def test_rsw(n):
....:     ZSn = SymmetricGroupAlgebra(ZZ, n)
....:     for k in range(1, n):
....:         a = ZSn.semi_rsw_element(k)
....:         b = ZSn.left_action_product(a, ZSn.antipode(a))
....:         if factorial(n-k) * ZSn.rsw_shuffling_element(k) != b:
....:             return False
....:     return True
sage: test_rsw(3)
True
sage: test_rsw(4)
True
sage: test_rsw(5)  # long time
True

>>> from sage.all import *
>>> def test_rsw(n):
...     ZSn = SymmetricGroupAlgebra(ZZ, n)
...     for k in range(Integer(1), n):
...         a = ZSn.semi_rsw_element(k)
...         b = ZSn.left_action_product(a, ZSn.antipode(a))
...         if factorial(n-k) * ZSn.rsw_shuffling_element(k) != b:
...             return False
...     return True
>>> test_rsw(Integer(3))
True
>>> test_rsw(Integer(4))
True
>>> test_rsw(Integer(5))  # long time
True


Let us also check the statement about the complete non-commutative symmetric function:

sage: def test_rsw_ncsf(n):
....:     ZSn = SymmetricGroupAlgebra(ZZ, n)
....:     NSym = NonCommutativeSymmetricFunctions(ZZ)
....:     S = NSym.S()
....:     for k in range(1, n):
....:         a = S(Composition([k] + [1]*(n-k))).to_symmetric_group_algebra()
....:         if a != ZSn.semi_rsw_element(k):
....:             return False
....:     return True
sage: test_rsw_ncsf(3)
True
sage: test_rsw_ncsf(4)
True
sage: test_rsw_ncsf(5)  # long time
True

>>> from sage.all import *
>>> def test_rsw_ncsf(n):
...     ZSn = SymmetricGroupAlgebra(ZZ, n)
...     NSym = NonCommutativeSymmetricFunctions(ZZ)
...     S = NSym.S()
...     for k in range(Integer(1), n):
...         a = S(Composition([k] + [Integer(1)]*(n-k))).to_symmetric_group_algebra()
...         if a != ZSn.semi_rsw_element(k):
...             return False
...     return True
>>> test_rsw_ncsf(Integer(3))
True
>>> test_rsw_ncsf(Integer(4))
True
>>> test_rsw_ncsf(Integer(5))  # long time
True

seminormal_basis(mult='l2r')[source]#

Return a list of the seminormal basis elements of self.

The seminormal basis of a symmetric group algebra is defined as follows:

Let $$n$$ be a nonnegative integer. Let $$R$$ be a $$\QQ$$-algebra. In the following, we will use the “left action” convention for multiplying permutations. This means that for all permutations $$p$$ and $$q$$ in $$S_n$$, the product $$pq$$ is defined in such a way that $$(pq)(i) = p(q(i))$$ for each $$i \in \{ 1, 2, \ldots, n \}$$ (this is the same convention as in left_action_product(), but not the default semantics of the $$*$$ operator on permutations in Sage). Thus, for instance, $$s_2 s_1$$ is the permutation obtained by first transposing $$1$$ with $$2$$ and then transposing $$2$$ with $$3$$ (where $$s_i = (i, i+1)$$).

For every partition $$\lambda$$ of $$n$$, let

$\kappa_{\lambda} = \frac{n!}{f^{\lambda}}$

where $$f^{\lambda}$$ is the number of standard Young tableaux of shape $$\lambda$$. Note that $$\kappa_{\lambda}$$ is an integer, namely the product of all hook lengths of $$\lambda$$ (by the hook length formula). In Sage, this integer can be computed by using sage.combinat.symmetric_group_algebra.kappa().

Let $$T$$ be a standard tableau of size $$n$$.

Let $$a(T)$$ denote the formal sum (in $$R S_n$$) of all permutations in $$S_n$$ which stabilize the rows of $$T$$ (as sets), i. e., which map each entry $$i$$ of $$T$$ to an entry in the same row as $$i$$. (See sage.combinat.symmetric_group_algebra.a() for an implementation of this.)

Let $$b(T)$$ denote the signed formal sum (in $$R S_n$$) of all permutations in $$S_n$$ which stabilize the columns of $$T$$ (as sets). Here, “signed” means that each permutation is multiplied with its sign. (This is implemented in sage.combinat.symmetric_group_algebra.b().)

Define an element $$e(T)$$ of $$R S_n$$ to be $$a(T) b(T)$$. (This is implemented in sage.combinat.symmetric_group_algebra.e() for $$R = \QQ$$.)

Let $$\mathrm{sh}(T)$$ denote the shape of $$T$$. (See shape().)

Let $$\overline{T}$$ denote the standard tableau of size $$n-1$$ obtained by removing the letter $$n$$ (along with its cell) from $$T$$ (if $$n \geq 1$$).

Now, we define an element $$\epsilon(T)$$ of $$R S_n$$. We define it by induction on the size $$n$$ of $$T$$, so we set $$\epsilon(\emptyset) = 1$$ and only need to define $$\epsilon(T)$$ for $$n \geq 1$$, assuming that $$\epsilon(\overline{T})$$ is already defined. We do this by setting

$\epsilon(T) = \frac{1}{\kappa_{\mathrm{sh}(T)}} \epsilon(\overline{T}) e(T) \epsilon(\overline{T}).$

This element $$\epsilon(T)$$ is implemented as sage.combinat.symmetric_group_algebra.epsilon() for $$R = \QQ$$, but it is also a particular case of the elements $$\epsilon(T, S)$$ defined below.

Now let $$S$$ be a further tableau of the same shape as $$T$$ (possibly equal to $$T$$). Let $$\pi_{T, S}$$ denote the permutation in $$S_n$$ such that applying this permutation to the entries of $$T$$ yields the tableau $$S$$. Define an element $$\epsilon(T, S)$$ of $$R S_n$$ by

$\epsilon(T, S) = \frac{1}{\kappa_{\mathrm{sh}(T)}} \epsilon(\overline S) \pi_{T, S} e(T) \epsilon(\overline T) = \frac{1}{\kappa_{\mathrm{sh}(T)}} \epsilon(\overline S) a(S) \pi_{T, S} b(T) \epsilon(\overline T).$

This element $$\epsilon(T, S)$$ is called Young’s seminormal unit corresponding to the bitableau (T, S), and is the return value of epsilon_ik() applied to T and S. Note that $$\epsilon(T, T) = \epsilon(T)$$.

If we let $$\lambda$$ run through all partitions of $$n$$, and $$(T, S)$$ run through all pairs of tableaux of shape $$\lambda$$, then the elements $$\epsilon(T, S)$$ form a basis of $$R S_n$$. This basis is called Young’s seminormal basis and has the properties that

$\epsilon(T, S) \epsilon(U, V) = \delta_{T, V} \epsilon(U, S)$

(where $$\delta$$ stands for the Kronecker delta).

Warning

Because of our convention, we are multiplying our elements in reverse of those given in some papers, for example [Ram1997]. Using the other convention of multiplying permutations, we would instead have $$\epsilon(U, V) \epsilon(T, S) = \delta_{T, V} \epsilon(U, S)$$.

In other words, Young’s seminormal basis consists of the matrix units in a (particular) Artin-Wedderburn decomposition of $$R S_n$$ into a direct product of matrix algebras over $$\QQ$$.

The output of seminormal_basis() is a list of all elements of the seminormal basis of self.

INPUT:

• mult – string (default: 'l2r'). If set to 'r2l', this causes the method to return the list of the antipodes (antipode()) of all $$\epsilon(T, S)$$ instead of the $$\epsilon(T, S)$$ themselves.

EXAMPLES:

sage: QS3 = SymmetricGroupAlgebra(QQ,3)
sage: QS3.seminormal_basis()
[1/6*[1, 2, 3] + 1/6*[1, 3, 2] + 1/6*[2, 1, 3] + 1/6*[2, 3, 1] + 1/6*[3, 1, 2] + 1/6*[3, 2, 1],
1/3*[1, 2, 3] + 1/6*[1, 3, 2] - 1/3*[2, 1, 3] - 1/6*[2, 3, 1] - 1/6*[3, 1, 2] + 1/6*[3, 2, 1],
1/3*[1, 3, 2] + 1/3*[2, 3, 1] - 1/3*[3, 1, 2] - 1/3*[3, 2, 1],
1/4*[1, 3, 2] - 1/4*[2, 3, 1] + 1/4*[3, 1, 2] - 1/4*[3, 2, 1],
1/3*[1, 2, 3] - 1/6*[1, 3, 2] + 1/3*[2, 1, 3] - 1/6*[2, 3, 1] - 1/6*[3, 1, 2] - 1/6*[3, 2, 1],
1/6*[1, 2, 3] - 1/6*[1, 3, 2] - 1/6*[2, 1, 3] + 1/6*[2, 3, 1] + 1/6*[3, 1, 2] - 1/6*[3, 2, 1]]

>>> from sage.all import *
>>> QS3 = SymmetricGroupAlgebra(QQ,Integer(3))
>>> QS3.seminormal_basis()
[1/6*[1, 2, 3] + 1/6*[1, 3, 2] + 1/6*[2, 1, 3] + 1/6*[2, 3, 1] + 1/6*[3, 1, 2] + 1/6*[3, 2, 1],
1/3*[1, 2, 3] + 1/6*[1, 3, 2] - 1/3*[2, 1, 3] - 1/6*[2, 3, 1] - 1/6*[3, 1, 2] + 1/6*[3, 2, 1],
1/3*[1, 3, 2] + 1/3*[2, 3, 1] - 1/3*[3, 1, 2] - 1/3*[3, 2, 1],
1/4*[1, 3, 2] - 1/4*[2, 3, 1] + 1/4*[3, 1, 2] - 1/4*[3, 2, 1],
1/3*[1, 2, 3] - 1/6*[1, 3, 2] + 1/3*[2, 1, 3] - 1/6*[2, 3, 1] - 1/6*[3, 1, 2] - 1/6*[3, 2, 1],
1/6*[1, 2, 3] - 1/6*[1, 3, 2] - 1/6*[2, 1, 3] + 1/6*[2, 3, 1] + 1/6*[3, 1, 2] - 1/6*[3, 2, 1]]

simple_module(la)[source]#

Return the simple module of self indexed by the partition la.

Over a field of characteristic $$0$$, this simply returns the Specht module.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(GF(3), 5)
sage: D = SGA.simple_module(Partition([3,1,1]))
sage: D
Simple module of [3, 1, 1] over Finite Field of size 3
sage: D.brauer_character()
(6, 0, -2, 0, 1)

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(GF(Integer(3)), Integer(5))
>>> D = SGA.simple_module(Partition([Integer(3),Integer(1),Integer(1)]))
>>> D
Simple module of [3, 1, 1] over Finite Field of size 3
>>> D.brauer_character()
(6, 0, -2, 0, 1)

simple_module_dimension(la)[source]#

Return the dimension of the simple module of self indexed by the partition la.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(GF(5), 6)
sage: SGA.simple_module_dimension(Partition([4,1,1]))
10

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(GF(Integer(5)), Integer(6))
>>> SGA.simple_module_dimension(Partition([Integer(4),Integer(1),Integer(1)]))
10

simple_module_parameterization()[source]#

Return a parameterization of the simple modules of self.

The symmetric group algebra of $$S_n$$ over a field of characteristic $$p$$ has its simple modules indexed by all $$p$$-regular partitions of $$n$$.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(QQ, 6)
sage: SGA.simple_module_parameterization()
Partitions of the integer 6

sage: SGA = SymmetricGroupAlgebra(GF(2), 6)
sage: SGA.simple_module_parameterization()
2-Regular Partitions of the integer 6

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(QQ, Integer(6))
>>> SGA.simple_module_parameterization()
Partitions of the integer 6

>>> SGA = SymmetricGroupAlgebra(GF(Integer(2)), Integer(6))
>>> SGA.simple_module_parameterization()
2-Regular Partitions of the integer 6

specht_module(D)[source]#

Return the Specht module of self indexed by the diagram D.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(QQ, 5)
sage: SM = SGA.specht_module(Partition([3,1,1]))
sage: SM
Specht module of [3, 1, 1] over Rational Field
sage: SM.frobenius_image()
s[3, 1, 1]

sage: SM = SGA.specht_module([(1,1),(1,3),(2,2),(3,1),(3,2)])
sage: SM
Specht module of [(1, 1), (1, 3), (2, 2), (3, 1), (3, 2)] over Rational Field
sage: SM.frobenius_image()
s[2, 2, 1] + s[3, 1, 1] + s[3, 2]

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(QQ, Integer(5))
>>> SM = SGA.specht_module(Partition([Integer(3),Integer(1),Integer(1)]))
>>> SM
Specht module of [3, 1, 1] over Rational Field
>>> SM.frobenius_image()
s[3, 1, 1]

>>> SM = SGA.specht_module([(Integer(1),Integer(1)),(Integer(1),Integer(3)),(Integer(2),Integer(2)),(Integer(3),Integer(1)),(Integer(3),Integer(2))])
>>> SM
Specht module of [(1, 1), (1, 3), (2, 2), (3, 1), (3, 2)] over Rational Field
>>> SM.frobenius_image()
s[2, 2, 1] + s[3, 1, 1] + s[3, 2]

specht_module_dimension(D)[source]#

Return the dimension of the Specht module of self indexed by D.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(QQ, 5)
sage: SGA.specht_module_dimension(Partition([3,1,1]))
6
sage: SGA.specht_module_dimension([(1,1),(1,3),(2,2),(3,1),(3,2)])
16

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(QQ, Integer(5))
>>> SGA.specht_module_dimension(Partition([Integer(3),Integer(1),Integer(1)]))
6
>>> SGA.specht_module_dimension([(Integer(1),Integer(1)),(Integer(1),Integer(3)),(Integer(2),Integer(2)),(Integer(3),Integer(1)),(Integer(3),Integer(2))])
16

tabloid_module(D)[source]#

Return the module of tabloids with the natural action of self.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(QQ, 5)
sage: TM = SGA.tabloid_module(Partition([3,1,1]))
sage: TM
Tabloid module of [3, 1, 1] over Rational Field
sage: s = SymmetricFunctions(QQ).s()
sage: s(TM.frobenius_image())
s[3, 1, 1] + s[3, 2] + 2*s[4, 1] + s[5]

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(QQ, Integer(5))
>>> TM = SGA.tabloid_module(Partition([Integer(3),Integer(1),Integer(1)]))
>>> TM
Tabloid module of [3, 1, 1] over Rational Field
>>> s = SymmetricFunctions(QQ).s()
>>> s(TM.frobenius_image())
s[3, 1, 1] + s[3, 2] + 2*s[4, 1] + s[5]

young_symmetrizer(la)[source]#

Return the Young symmetrizer of shape la of self.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(QQ, SymmetricGroup(3))
sage: SGA.young_symmetrizer([2,1])
() + (1,2) - (1,3,2) - (1,3)
sage: SGA = SymmetricGroupAlgebra(QQ, 4)
sage: SGA.young_symmetrizer([2,1,1])
[1, 2, 3, 4] - [1, 2, 4, 3] + [2, 1, 3, 4] - [2, 1, 4, 3]
- [3, 1, 2, 4] + [3, 1, 4, 2] - [3, 2, 1, 4] + [3, 2, 4, 1]
+ [4, 1, 2, 3] - [4, 1, 3, 2] + [4, 2, 1, 3] - [4, 2, 3, 1]
sage: SGA.young_symmetrizer([5,1,1])
Traceback (most recent call last):
...
ValueError: the partition [5, 1, 1] is not of size 4

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(QQ, SymmetricGroup(Integer(3)))
>>> SGA.young_symmetrizer([Integer(2),Integer(1)])
() + (1,2) - (1,3,2) - (1,3)
>>> SGA = SymmetricGroupAlgebra(QQ, Integer(4))
>>> SGA.young_symmetrizer([Integer(2),Integer(1),Integer(1)])
[1, 2, 3, 4] - [1, 2, 4, 3] + [2, 1, 3, 4] - [2, 1, 4, 3]
- [3, 1, 2, 4] + [3, 1, 4, 2] - [3, 2, 1, 4] + [3, 2, 4, 1]
+ [4, 1, 2, 3] - [4, 1, 3, 2] + [4, 2, 1, 3] - [4, 2, 3, 1]
>>> SGA.young_symmetrizer([Integer(5),Integer(1),Integer(1)])
Traceback (most recent call last):
...
ValueError: the partition [5, 1, 1] is not of size 4

sage.combinat.symmetric_group_algebra.a(tableau, star=0, base_ring=Rational Field)[source]#

The row projection operator corresponding to the Young tableau tableau (which is supposed to contain every integer from $$1$$ to its size precisely once, but may and may not be standard).

This is the sum (in the group algebra of the relevant symmetric group over $$\QQ$$) of all the permutations which preserve the rows of tableau. It is called $$a_{\text{tableau}}$$ in [EGHLSVY], Section 4.2.

INPUT:

• tableau – Young tableau which contains every integer from $$1$$ to its size precisely once.

• star – nonnegative integer (default: $$0$$). When this optional variable is set, the method computes not the row projection operator of tableau, but the row projection operator of the restriction of tableau to the entries 1, 2, ..., tableau.size() - star instead.

• base_ring – commutative ring (default: QQ). When this optional variable is set, the row projection operator is computed over a user-determined base ring instead of $$\QQ$$. (Note that symmetric group algebras currently don’t preserve coercion, so e. g. a symmetric group algebra over $$\ZZ$$ does not coerce into the corresponding one over $$\QQ$$; so convert manually or choose your base rings wisely!)

EXAMPLES:

sage: from sage.combinat.symmetric_group_algebra import a
sage: a([[1,2]])
[1, 2] + [2, 1]
sage: a([[1],[2]])
[1, 2]
sage: a([])
[]
sage: a([[1, 5], [2, 3], [4]])
[1, 2, 3, 4, 5] + [1, 3, 2, 4, 5] + [5, 2, 3, 4, 1] + [5, 3, 2, 4, 1]
sage: a([[1,4], [2,3]], base_ring=ZZ)
[1, 2, 3, 4] + [1, 3, 2, 4] + [4, 2, 3, 1] + [4, 3, 2, 1]

>>> from sage.all import *
>>> from sage.combinat.symmetric_group_algebra import a
>>> a([[Integer(1),Integer(2)]])
[1, 2] + [2, 1]
>>> a([[Integer(1)],[Integer(2)]])
[1, 2]
>>> a([])
[]
>>> a([[Integer(1), Integer(5)], [Integer(2), Integer(3)], [Integer(4)]])
[1, 2, 3, 4, 5] + [1, 3, 2, 4, 5] + [5, 2, 3, 4, 1] + [5, 3, 2, 4, 1]
>>> a([[Integer(1),Integer(4)], [Integer(2),Integer(3)]], base_ring=ZZ)
[1, 2, 3, 4] + [1, 3, 2, 4] + [4, 2, 3, 1] + [4, 3, 2, 1]


The same with a skew tableau:

sage: a([[None,1,4], [2,3]], base_ring=ZZ)
[1, 2, 3, 4] + [1, 3, 2, 4] + [4, 2, 3, 1] + [4, 3, 2, 1]

>>> from sage.all import *
>>> a([[None,Integer(1),Integer(4)], [Integer(2),Integer(3)]], base_ring=ZZ)
[1, 2, 3, 4] + [1, 3, 2, 4] + [4, 2, 3, 1] + [4, 3, 2, 1]

sage.combinat.symmetric_group_algebra.b(tableau, star=0, base_ring=Rational Field)[source]#

The column projection operator corresponding to the Young tableau tableau (which is supposed to contain every integer from $$1$$ to its size precisely once, but may and may not be standard).

This is the signed sum (in the group algebra of the relevant symmetric group over $$\QQ$$) of all the permutations which preserve the column of tableau (where the signs are the usual signs of the permutations). It is called $$b_{\text{tableau}}$$ in [EGHLSVY], Section 4.2.

INPUT:

• tableau – Young tableau which contains every integer from $$1$$ to its size precisely once.

• star – nonnegative integer (default: $$0$$). When this optional variable is set, the method computes not the column projection operator of tableau, but the column projection operator of the restriction of tableau to the entries 1, 2, ..., tableau.size() - star instead.

• base_ring – commutative ring (default: QQ). When this optional variable is set, the column projection operator is computed over a user-determined base ring instead of $$\QQ$$. (Note that symmetric group algebras currently don’t preserve coercion, so e. g. a symmetric group algebra over $$\ZZ$$ does not coerce into the corresponding one over $$\QQ$$; so convert manually or choose your base rings wisely!)

EXAMPLES:

sage: from sage.combinat.symmetric_group_algebra import b
sage: b([[1,2]])
[1, 2]
sage: b([[1],[2]])
[1, 2] - [2, 1]
sage: b([])
[]
sage: b([[1, 2, 4], [5, 3]])
[1, 2, 3, 4, 5] - [1, 3, 2, 4, 5] - [5, 2, 3, 4, 1] + [5, 3, 2, 4, 1]
sage: b([[1, 4], [2, 3]], base_ring=ZZ)
[1, 2, 3, 4] - [1, 2, 4, 3] - [2, 1, 3, 4] + [2, 1, 4, 3]
sage: b([[1, 4], [2, 3]], base_ring=Integers(5))
[1, 2, 3, 4] + 4*[1, 2, 4, 3] + 4*[2, 1, 3, 4] + [2, 1, 4, 3]

>>> from sage.all import *
>>> from sage.combinat.symmetric_group_algebra import b
>>> b([[Integer(1),Integer(2)]])
[1, 2]
>>> b([[Integer(1)],[Integer(2)]])
[1, 2] - [2, 1]
>>> b([])
[]
>>> b([[Integer(1), Integer(2), Integer(4)], [Integer(5), Integer(3)]])
[1, 2, 3, 4, 5] - [1, 3, 2, 4, 5] - [5, 2, 3, 4, 1] + [5, 3, 2, 4, 1]
>>> b([[Integer(1), Integer(4)], [Integer(2), Integer(3)]], base_ring=ZZ)
[1, 2, 3, 4] - [1, 2, 4, 3] - [2, 1, 3, 4] + [2, 1, 4, 3]
>>> b([[Integer(1), Integer(4)], [Integer(2), Integer(3)]], base_ring=Integers(Integer(5)))
[1, 2, 3, 4] + 4*[1, 2, 4, 3] + 4*[2, 1, 3, 4] + [2, 1, 4, 3]


The same with a skew tableau:

sage: b([[None, 2, 4], [1, 3], [5]])
[1, 2, 3, 4, 5] - [1, 3, 2, 4, 5] - [5, 2, 3, 4, 1] + [5, 3, 2, 4, 1]

>>> from sage.all import *
>>> b([[None, Integer(2), Integer(4)], [Integer(1), Integer(3)], [Integer(5)]])
[1, 2, 3, 4, 5] - [1, 3, 2, 4, 5] - [5, 2, 3, 4, 1] + [5, 3, 2, 4, 1]


With the l2r setting for multiplication, the unnormalized Young symmetrizer e(tableau) should be the product b(tableau) * a(tableau) for every tableau. Let us check this on the standard tableaux of size 5:

sage: from sage.combinat.symmetric_group_algebra import a, b, e
sage: all( e(t) == b(t) * a(t) for t in StandardTableaux(5) )
True

>>> from sage.all import *
>>> from sage.combinat.symmetric_group_algebra import a, b, e
>>> all( e(t) == b(t) * a(t) for t in StandardTableaux(Integer(5)) )
True

sage.combinat.symmetric_group_algebra.e(tableau, star=0)[source]#

The unnormalized Young projection operator corresponding to the Young tableau tableau (which is supposed to contain every integer from $$1$$ to its size precisely once, but may and may not be standard).

If $$n$$ is a nonnegative integer, and $$T$$ is a Young tableau containing every integer from $$1$$ to $$n$$ exactly once, then the unnormalized Young projection operator $$e(T)$$ is defined by

$e(T) = a(T) b(T) \in \QQ S_n,$

where $$a(T) \in \QQ S_n$$ is the sum of all permutations in $$S_n$$ which fix the rows of $$T$$ (as sets), and $$b(T) \in \QQ S_n$$ is the signed sum of all permutations in $$S_n$$ which fix the columns of $$T$$ (as sets). Here, “signed” means that each permutation is multiplied with its sign; and the product on the group $$S_n$$ is defined in such a way that $$(pq)(i) = p(q(i))$$ for any permutations $$p$$ and $$q$$ and any $$1 \leq i \leq n$$.

Note that the definition of $$e(T)$$ is not uniform across literature. Others define it as $$b(T) a(T)$$ instead, or include certain scalar factors (we do not, whence “unnormalized”).

EXAMPLES:

sage: from sage.combinat.symmetric_group_algebra import e
sage: e([[1,2]])
[1, 2] + [2, 1]
sage: e([[1],[2]])
[1, 2] - [2, 1]
sage: e([])
[]

>>> from sage.all import *
>>> from sage.combinat.symmetric_group_algebra import e
>>> e([[Integer(1),Integer(2)]])
[1, 2] + [2, 1]
>>> e([[Integer(1)],[Integer(2)]])
[1, 2] - [2, 1]
>>> e([])
[]


There are differing conventions for the order of the symmetrizers and antisymmetrizers. This example illustrates our conventions:

sage: e([[1,2],[3]])
[1, 2, 3] + [2, 1, 3] - [3, 1, 2] - [3, 2, 1]

>>> from sage.all import *
>>> e([[Integer(1),Integer(2)],[Integer(3)]])
[1, 2, 3] + [2, 1, 3] - [3, 1, 2] - [3, 2, 1]


To obtain the product $$b(T) a(T)$$, one has to take the antipode of this:

sage: QS3 = parent(e([[1,2],[3]]))
sage: QS3.antipode(e([[1,2],[3]]))
[1, 2, 3] + [2, 1, 3] - [2, 3, 1] - [3, 2, 1]

>>> from sage.all import *
>>> QS3 = parent(e([[Integer(1),Integer(2)],[Integer(3)]]))
>>> QS3.antipode(e([[Integer(1),Integer(2)],[Integer(3)]]))
[1, 2, 3] + [2, 1, 3] - [2, 3, 1] - [3, 2, 1]


And here is an example for a skew tableau:

sage: e([[None, 2, 1], [4, 3]])
[1, 2, 3, 4] + [1, 2, 4, 3] - [1, 3, 2, 4] - [1, 4, 2, 3]
+ [2, 1, 3, 4] + [2, 1, 4, 3] - [2, 3, 1, 4] - [2, 4, 1, 3]

>>> from sage.all import *
>>> e([[None, Integer(2), Integer(1)], [Integer(4), Integer(3)]])
[1, 2, 3, 4] + [1, 2, 4, 3] - [1, 3, 2, 4] - [1, 4, 2, 3]
+ [2, 1, 3, 4] + [2, 1, 4, 3] - [2, 3, 1, 4] - [2, 4, 1, 3]

sage.combinat.symmetric_group_algebra.e_hat(tab, star=0)[source]#

The Young projection operator corresponding to the Young tableau tab (which is supposed to contain every integer from $$1$$ to its size precisely once, but may and may not be standard). This is an idempotent in the rational group algebra.

If $$n$$ is a nonnegative integer, and $$T$$ is a Young tableau containing every integer from $$1$$ to $$n$$ exactly once, then the Young projection operator $$\widehat{e}(T)$$ is defined by

$\widehat{e}(T) = \frac{1}{\kappa_\lambda} a(T) b(T) \in \QQ S_n,$

where $$\lambda$$ is the shape of $$T$$, where $$\kappa_\lambda$$ is $$n!$$ divided by the number of standard tableaux of shape $$\lambda$$, where $$a(T) \in \QQ S_n$$ is the sum of all permutations in $$S_n$$ which fix the rows of $$T$$ (as sets), and where $$b(T) \in \QQ S_n$$ is the signed sum of all permutations in $$S_n$$ which fix the columns of $$T$$ (as sets). Here, “signed” means that each permutation is multiplied with its sign; and the product on the group $$S_n$$ is defined in such a way that $$(pq)(i) = p(q(i))$$ for any permutations $$p$$ and $$q$$ and any $$1 \leq i \leq n$$.

Note that the definition of $$\widehat{e}(T)$$ is not uniform across literature. Others define it as $$\frac{1}{\kappa_\lambda} b(T) a(T)$$ instead.

EXAMPLES:

sage: from sage.combinat.symmetric_group_algebra import e_hat
sage: e_hat([[1,2,3]])
1/6*[1, 2, 3] + 1/6*[1, 3, 2] + 1/6*[2, 1, 3] + 1/6*[2, 3, 1] + 1/6*[3, 1, 2] + 1/6*[3, 2, 1]
sage: e_hat([[1],[2]])
1/2*[1, 2] - 1/2*[2, 1]

>>> from sage.all import *
>>> from sage.combinat.symmetric_group_algebra import e_hat
>>> e_hat([[Integer(1),Integer(2),Integer(3)]])
1/6*[1, 2, 3] + 1/6*[1, 3, 2] + 1/6*[2, 1, 3] + 1/6*[2, 3, 1] + 1/6*[3, 1, 2] + 1/6*[3, 2, 1]
>>> e_hat([[Integer(1)],[Integer(2)]])
1/2*[1, 2] - 1/2*[2, 1]


There are differing conventions for the order of the symmetrizers and antisymmetrizers. This example illustrates our conventions:

sage: e_hat([[1,2],[3]])
1/3*[1, 2, 3] + 1/3*[2, 1, 3] - 1/3*[3, 1, 2] - 1/3*[3, 2, 1]

>>> from sage.all import *
>>> e_hat([[Integer(1),Integer(2)],[Integer(3)]])
1/3*[1, 2, 3] + 1/3*[2, 1, 3] - 1/3*[3, 1, 2] - 1/3*[3, 2, 1]

sage.combinat.symmetric_group_algebra.e_ik(itab, ktab, star=0)[source]#

EXAMPLES:

sage: from sage.combinat.symmetric_group_algebra import e_ik
sage: e_ik([[1,2,3]], [[1,2,3]])
[1, 2, 3] + [1, 3, 2] + [2, 1, 3] + [2, 3, 1] + [3, 1, 2] + [3, 2, 1]
sage: e_ik([[1,2,3]], [[1,2,3]], star=1)
[1, 2] + [2, 1]

>>> from sage.all import *
>>> from sage.combinat.symmetric_group_algebra import e_ik
>>> e_ik([[Integer(1),Integer(2),Integer(3)]], [[Integer(1),Integer(2),Integer(3)]])
[1, 2, 3] + [1, 3, 2] + [2, 1, 3] + [2, 3, 1] + [3, 1, 2] + [3, 2, 1]
>>> e_ik([[Integer(1),Integer(2),Integer(3)]], [[Integer(1),Integer(2),Integer(3)]], star=Integer(1))
[1, 2] + [2, 1]

sage.combinat.symmetric_group_algebra.epsilon(tab, star=0)[source]#

The $$(T, T)$$-th element of the seminormal basis of the group algebra $$\QQ[S_n]$$, where $$T$$ is the tableau tab (with its star highest entries removed if the optional variable star is set).

See the docstring of seminormal_basis() for the notation used herein.

EXAMPLES:

sage: from sage.combinat.symmetric_group_algebra import epsilon
sage: epsilon([[1,2]])
1/2*[1, 2] + 1/2*[2, 1]
sage: epsilon([[1],[2]])
1/2*[1, 2] - 1/2*[2, 1]

>>> from sage.all import *
>>> from sage.combinat.symmetric_group_algebra import epsilon
>>> epsilon([[Integer(1),Integer(2)]])
1/2*[1, 2] + 1/2*[2, 1]
>>> epsilon([[Integer(1)],[Integer(2)]])
1/2*[1, 2] - 1/2*[2, 1]

sage.combinat.symmetric_group_algebra.epsilon_ik(itab, ktab, star=0)[source]#

Return the seminormal basis element of the symmetric group algebra $$\QQ S_n$$ corresponding to the pair of tableaux itab and ktab (or restrictions of these tableaux, if the optional variable star is set).

INPUT:

• itab, ktab – two standard tableaux of same size.

• star – integer (default: $$0$$).

OUTPUT:

The element $$\epsilon(I, K) \in \QQ S_n$$, where $$I$$ and $$K$$ are the tableaux obtained by removing all entries higher than $$n - \mathrm{star}$$ from itab and ktab, respectively (where $$n$$ is the size of itab and ktab). Here, we are using the notations from seminormal_basis().

EXAMPLES:

sage: from sage.combinat.symmetric_group_algebra import epsilon_ik
sage: epsilon_ik([[1,2],[3]], [[1,3],[2]])
1/4*[1, 3, 2] - 1/4*[2, 3, 1] + 1/4*[3, 1, 2] - 1/4*[3, 2, 1]
sage: epsilon_ik([[1,2],[3]], [[1,3],[2]], star=1)
Traceback (most recent call last):
...
ValueError: the two tableaux must be of the same shape

>>> from sage.all import *
>>> from sage.combinat.symmetric_group_algebra import epsilon_ik
>>> epsilon_ik([[Integer(1),Integer(2)],[Integer(3)]], [[Integer(1),Integer(3)],[Integer(2)]])
1/4*[1, 3, 2] - 1/4*[2, 3, 1] + 1/4*[3, 1, 2] - 1/4*[3, 2, 1]
>>> epsilon_ik([[Integer(1),Integer(2)],[Integer(3)]], [[Integer(1),Integer(3)],[Integer(2)]], star=Integer(1))
Traceback (most recent call last):
...
ValueError: the two tableaux must be of the same shape

sage.combinat.symmetric_group_algebra.kappa(alpha)[source]#

Return $$\kappa_\alpha$$, which is $$n!$$ divided by the number of standard tableaux of shape $$\alpha$$ (where $$\alpha$$ is a partition of $$n$$).

INPUT:

• alpha – integer partition (can be encoded as a list).

OUTPUT:

The factorial of the size of alpha, divided by the number of standard tableaux of shape alpha. Equivalently, the product of all hook lengths of alpha.

EXAMPLES:

sage: from sage.combinat.symmetric_group_algebra import kappa
sage: kappa(Partition([2,1]))
3
sage: kappa([2,1])
3

>>> from sage.all import *
>>> from sage.combinat.symmetric_group_algebra import kappa
>>> kappa(Partition([Integer(2),Integer(1)]))
3
>>> kappa([Integer(2),Integer(1)])
3

sage.combinat.symmetric_group_algebra.pi_ik(itab, ktab)[source]#

Return the permutation $$p$$ which sends every entry of the tableau itab to the respective entry of the tableau ktab, as an element of the corresponding symmetric group algebra.

This assumes that itab and ktab are tableaux (possibly given just as lists of lists) of the same shape. Both tableaux are allowed to be skew.

EXAMPLES:

sage: from sage.combinat.symmetric_group_algebra import pi_ik
sage: pi_ik([[1,3],[2]], [[1,2],[3]])
[1, 3, 2]

>>> from sage.all import *
>>> from sage.combinat.symmetric_group_algebra import pi_ik
>>> pi_ik([[Integer(1),Integer(3)],[Integer(2)]], [[Integer(1),Integer(2)],[Integer(3)]])
[1, 3, 2]


The same with skew tableaux:

sage: from sage.combinat.symmetric_group_algebra import pi_ik
sage: pi_ik([[None,1,3],[2]], [[None,1,2],[3]])
[1, 3, 2]

>>> from sage.all import *
>>> from sage.combinat.symmetric_group_algebra import pi_ik
>>> pi_ik([[None,Integer(1),Integer(3)],[Integer(2)]], [[None,Integer(1),Integer(2)],[Integer(3)]])
[1, 3, 2]

sage.combinat.symmetric_group_algebra.seminormal_test(n)[source]#

Run a variety of tests to verify that the construction of the seminormal basis works as desired. The numbers appearing are results in James and Kerber’s ‘Representation Theory of the Symmetric Group’ [JK1981].

EXAMPLES:

sage: from sage.combinat.symmetric_group_algebra import seminormal_test
sage: seminormal_test(3)
True

>>> from sage.all import *
>>> from sage.combinat.symmetric_group_algebra import seminormal_test
>>> seminormal_test(Integer(3))
True