Specht Modules

AUTHORS:

  • Travis Scrimshaw (2023-1-22): initial version

  • Travis Scrimshaw (2023-11-23): added simple modules based on code from Sacha Goldman

Todo

Integrate this with the implementations in sage.modules.with_basis.representation.

class sage.combinat.specht_module.MaximalSpechtSubmodule(specht_module)[source]

Bases: SymmetricGroupRepresentation, SubmoduleWithBasis

The maximal submodule \(U^{\lambda}\) of the Specht module \(S^{\lambda}\).

ALGORITHM:

We construct \(U^{\lambda}\) as the intersection \(S \cap S^{\perp}\), where \(S^{\perp}\) is the orthogonal complement of the Specht module \(S\) inside of the tabloid module \(T\) (with respect to the natural bilinear form on \(T\)).

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(GF(3), 5)
sage: SM = SGA.specht_module([3,2])
sage: U = SM.maximal_submodule()
sage: u = U.an_element(); u
2*U[0] + 2*U[1]
sage: [p * u for p in list(SGA.basis())[:4]]
[2*U[0] + 2*U[1], 2*U[2] + 2*U[3], 2*U[0] + 2*U[1], U[0] + 2*U[2]]
sage: sum(SGA.basis()) * u
0

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(GF(Integer(3)), Integer(5))
>>> SM = SGA.specht_module([Integer(3),Integer(2)])
>>> U = SM.maximal_submodule()
>>> u = U.an_element(); u
2*U[0] + 2*U[1]
>>> [p * u for p in list(SGA.basis())[:Integer(4)]]
[2*U[0] + 2*U[1], 2*U[2] + 2*U[3], 2*U[0] + 2*U[1], U[0] + 2*U[2]]
>>> sum(SGA.basis()) * u
0
Element[source]

alias of Element

class sage.combinat.specht_module.SimpleModule(specht_module)[source]

Bases: SymmetricGroupRepresentation, QuotientModuleWithBasis

The simple \(S_n\)-module associated with a partition \(\lambda\).

The simple module \(D^{\lambda}\) is the quotient of the Specht module \(S^{\lambda}\) by its maximal submodule \(U^{\lambda}\).

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(GF(3), 5)
sage: SM = SGA.specht_module([3,1,1])
sage: D = SM.simple_module()
sage: v = D.an_element(); v
2*D[[[1, 3, 5], [2], [4]]] + 2*D[[[1, 4, 5], [2], [3]]]
sage: SGA.an_element() * v
2*D[[[1, 2, 4], [3], [5]]] + 2*D[[[1, 3, 5], [2], [4]]]

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(GF(Integer(3)), Integer(5))
>>> SM = SGA.specht_module([Integer(3),Integer(1),Integer(1)])
>>> D = SM.simple_module()
>>> v = D.an_element(); v
2*D[[[1, 3, 5], [2], [4]]] + 2*D[[[1, 4, 5], [2], [3]]]
>>> SGA.an_element() * v
2*D[[[1, 2, 4], [3], [5]]] + 2*D[[[1, 3, 5], [2], [4]]]

We give an example on how to construct the decomposition matrix (the Specht modules are a complete set of irreducible projective modules) and the Cartan matrix of a symmetric group algebra:

sage: SGA = SymmetricGroupAlgebra(GF(3), 4)
sage: BM = matrix(SGA.simple_module(la).brauer_character()
....:             for la in Partitions(4, regular=3))
sage: SBT = matrix(SGA.specht_module(la).brauer_character()
....:              for la in Partitions(4))
sage: D = SBT * ~BM; D
[1 0 0 0]
[0 1 0 0]
[1 0 1 0]
[0 0 0 1]
[0 0 1 0]
sage: D.transpose() * D
[2 0 1 0]
[0 1 0 0]
[1 0 2 0]
[0 0 0 1]

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(GF(Integer(3)), Integer(4))
>>> BM = matrix(SGA.simple_module(la).brauer_character()
...             for la in Partitions(Integer(4), regular=Integer(3)))
>>> SBT = matrix(SGA.specht_module(la).brauer_character()
...              for la in Partitions(Integer(4)))
>>> D = SBT * ~BM; D
[1 0 0 0]
[0 1 0 0]
[1 0 1 0]
[0 0 0 1]
[0 0 1 0]
>>> D.transpose() * D
[2 0 1 0]
[0 1 0 0]
[1 0 2 0]
[0 0 0 1]

We verify this against the direct computation (up to reindexing the rows and columns):

sage: SGA.cartan_invariants_matrix()  # long time
[1 0 0 0]
[0 1 0 0]
[0 0 2 1]
[0 0 1 2]

>>> from sage.all import *
>>> SGA.cartan_invariants_matrix()  # long time
[1 0 0 0]
[0 1 0 0]
[0 0 2 1]
[0 0 1 2]
Element[source]

alias of Element

class sage.combinat.specht_module.SpechtModule(SGA, D)[source]

Bases: SymmetricGroupRepresentation, SubmoduleWithBasis

A Specht module.

Let \(S_n\) be the symmetric group on \(n\) letters and \(R\) be a commutative ring. The Specht module \(S^D\) for a diagram \(D\) is an \(S_n\)-module defined as follows. Let

\[R(D) := \sum_{w \in R_D} w, \qquad\qquad C(D) := \sum_{w \in C_D} (-1)^w w,\]

where \(R_D\) (resp. \(C_D\)) is the row (resp. column) stabilizer of \(D\). Then, we construct the Specht module \(S^D\) as the left ideal

\[S^D = R[S_n] C(D) R(D),\]

where \(R[S_n]\) is the group algebra of \(S_n\) over \(R\).

INPUT:

  • SGA – a symmetric group algebra

  • D – a diagram

EXAMPLES:

We begin by constructing all irreducible Specht modules for the symmetric group \(S_4\) and show that they give a full set of irreducible representations both by having distinct Frobenius characters and the sum of the square of their dimensions is equal to \(4!\):

sage: SP = [la.specht_module(QQ) for la in Partitions(4)]
sage: s = SymmetricFunctions(QQ).s()
sage: [s(S.frobenius_image()) for S in SP]
[s[4], s[3, 1], s[2, 2], s[2, 1, 1], s[1, 1, 1, 1]]
sage: sum(S.dimension()^2 for S in SP)
24

>>> from sage.all import *
>>> SP = [la.specht_module(QQ) for la in Partitions(Integer(4))]
>>> s = SymmetricFunctions(QQ).s()
>>> [s(S.frobenius_image()) for S in SP]
[s[4], s[3, 1], s[2, 2], s[2, 1, 1], s[1, 1, 1, 1]]
>>> sum(S.dimension()**Integer(2) for S in SP)
24

Next, we compute the Specht module for a more general diagram for \(S_5\) and compute its irreducible decomposition by using its Frobenius character:

sage: D = [(0,0), (0,1), (1,1), (1,2), (0,3)]
sage: SGA = SymmetricGroupAlgebra(QQ, 5)
sage: SM = SGA.specht_module(D)
sage: SM.dimension()
9
sage: s(SM.frobenius_image())
s[3, 2] + s[4, 1]

>>> from sage.all import *
>>> D = [(Integer(0),Integer(0)), (Integer(0),Integer(1)), (Integer(1),Integer(1)), (Integer(1),Integer(2)), (Integer(0),Integer(3))]
>>> SGA = SymmetricGroupAlgebra(QQ, Integer(5))
>>> SM = SGA.specht_module(D)
>>> SM.dimension()
9
>>> s(SM.frobenius_image())
s[3, 2] + s[4, 1]

This carries a natural (left) action of the symmetric group (algebra):

sage: S5 = SGA.group()
sage: v = SM.an_element(); v
2*S[0] + 2*S[1] + 3*S[2]
sage: S5([2,1,5,3,4]) * v
3*S[0] + 2*S[1] + 2*S[2]
sage: x = SGA.an_element(); x
[1, 2, 3, 4, 5] + 2*[1, 2, 3, 5, 4] + 3*[1, 2, 4, 3, 5] + [5, 1, 2, 3, 4]
sage: x * v
15*S[0] + 14*S[1] + 16*S[2] - 7*S[5] + 2*S[6] + 2*S[7]

>>> from sage.all import *
>>> S5 = SGA.group()
>>> v = SM.an_element(); v
2*S[0] + 2*S[1] + 3*S[2]
>>> S5([Integer(2),Integer(1),Integer(5),Integer(3),Integer(4)]) * v
3*S[0] + 2*S[1] + 2*S[2]
>>> x = SGA.an_element(); x
[1, 2, 3, 4, 5] + 2*[1, 2, 3, 5, 4] + 3*[1, 2, 4, 3, 5] + [5, 1, 2, 3, 4]
>>> x * v
15*S[0] + 14*S[1] + 16*S[2] - 7*S[5] + 2*S[6] + 2*S[7]

See also

SpechtRepresentation for an implementation of the representation by matrices.

class Element[source]

Bases: IndexedFreeModuleElement

class sage.combinat.specht_module.SpechtModuleTableauxBasis(ambient)[source]

Bases: SpechtModule

A Specht module of a partition in the classical standard tableau basis.

This is constructed as a \(S_n\)-submodule of the TabloidModule (also referred to as the standard module).

See also

  • SpechtModule for the generic diagram implementation constructed as a left ideal of the group algebra

  • SpechtRepresentation for an implementation of the representation by matrices.

bilinear_form(u, v)[source]

Return the natural bilinear form of self applied to u and v.

The natural bilinear form is given by the pullback of the natural bilinear form on the tabloid module (where the tabloid basis is an orthonormal basis).

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(QQ, 5)
sage: SM = SGA.specht_module([2,2,1])
sage: u = SM.an_element(); u
3*S[[1, 2], [3, 5], [4]] + 2*S[[1, 3], [2, 5], [4]] + 2*S[[1, 4], [2, 5], [3]]
sage: v = sum(SM.basis())
sage: SM.bilinear_form(u, v)
140

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(QQ, Integer(5))
>>> SM = SGA.specht_module([Integer(2),Integer(2),Integer(1)])
>>> u = SM.an_element(); u
3*S[[1, 2], [3, 5], [4]] + 2*S[[1, 3], [2, 5], [4]] + 2*S[[1, 4], [2, 5], [3]]
>>> v = sum(SM.basis())
>>> SM.bilinear_form(u, v)
140
gram_matrix()[source]

Return the Gram matrix of the natural bilinear form of self.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(QQ, 5)
sage: SM = SGA.specht_module([2,2,1])
sage: M = SM.gram_matrix(); M
[12  4 -4 -4  4]
[ 4 12  4  4  4]
[-4  4 12  4  4]
[-4  4  4 12  4]
[ 4  4  4  4 12]
sage: M.det() != 0
True

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(QQ, Integer(5))
>>> SM = SGA.specht_module([Integer(2),Integer(2),Integer(1)])
>>> M = SM.gram_matrix(); M
[12  4 -4 -4  4]
[ 4 12  4  4  4]
[-4  4 12  4  4]
[-4  4  4 12  4]
[ 4  4  4  4 12]
>>> M.det() != Integer(0)
True
intrinsic_arrangement(base_ring=None)[source]

Return the intrinsic arrangement of self.

Consider the Specht module \(S^{\lambda}\) with \(\lambda\) a (integer) partition of \(n\) (i.e., \(S^{\lambda}\) is an \(S_n\)-module). The intrinsic arrangement of \(S^{\lambda}\) is the central hyperplane arrangement in \(S^{\lambda}\) given by the hyperplanes \(H_{\alpha}\), indexed by a set partition \(\alpha\) of \(\{1, \ldots, n\}\) of size \(\lambda\), defined by

\[H_{\alpha} := \bigoplus_{\tau \in T_{\alpha}} (S^{\lambda})^{\tau},\]

where \(T_{\alpha}\) is some set of generating transpositions of the Young subgroup \(S_{\alpha}\) and \(V^{\tau}\) denotes the \(\tau\)-invariant subspace of \(V\). (These hyperplanes do not depend on the choice of \(T_{\alpha}\).)

This was introduced in [TVY2020] as a generalization of the braid arrangement, which is the case when \(\lambda = (n-1, 1)\) (equivalently, for the irreducible representation of \(S_n\) given by the type \(A_{n-1}\) root system).

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(QQ, 4)
sage: SM = SGA.specht_module([2, 1, 1])
sage: A = SM.intrinsic_arrangement()
sage: A.hyperplanes()
(Hyperplane T0 - T1 - 3*T2 + 0,
 Hyperplane T0 - T1 + T2 + 0,
 Hyperplane T0 + 3*T1 + T2 + 0,
 Hyperplane 3*T0 + T1 - T2 + 0)
sage: A.is_free()
False

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(QQ, Integer(4))
>>> SM = SGA.specht_module([Integer(2), Integer(1), Integer(1)])
>>> A = SM.intrinsic_arrangement()
>>> A.hyperplanes()
(Hyperplane T0 - T1 - 3*T2 + 0,
 Hyperplane T0 - T1 + T2 + 0,
 Hyperplane T0 + 3*T1 + T2 + 0,
 Hyperplane 3*T0 + T1 - T2 + 0)
>>> A.is_free()
False

We reproduce Example 3 of [TVY2020]:

sage: SGA = SymmetricGroupAlgebra(QQ, 5)
sage: for la in Partitions(5):
....:     SM = SGA.specht_module(la)
....:     A = SM.intrinsic_arrangement()
....:     print(la, A.characteristic_polynomial())
[5] 1
[4, 1] x^4 - 10*x^3 + 35*x^2 - 50*x + 24
[3, 2] x^5 - 15*x^4 + 90*x^3 - 260*x^2 + 350*x - 166
[3, 1, 1] x^6 - 10*x^5 + 45*x^4 - 115*x^3 + 175*x^2 - 147*x + 51
[2, 2, 1] x^5 - 10*x^4 + 45*x^3 - 105*x^2 + 120*x - 51
[2, 1, 1, 1] x^4 - 5*x^3 + 10*x^2 - 10*x + 4
[1, 1, 1, 1, 1] 1

sage: A = SGA.specht_module([4, 1]).intrinsic_arrangement()
sage: A.characteristic_polynomial().factor()
(x - 4) * (x - 3) * (x - 2) * (x - 1)

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(QQ, Integer(5))
>>> for la in Partitions(Integer(5)):
...     SM = SGA.specht_module(la)
...     A = SM.intrinsic_arrangement()
...     print(la, A.characteristic_polynomial())
[5] 1
[4, 1] x^4 - 10*x^3 + 35*x^2 - 50*x + 24
[3, 2] x^5 - 15*x^4 + 90*x^3 - 260*x^2 + 350*x - 166
[3, 1, 1] x^6 - 10*x^5 + 45*x^4 - 115*x^3 + 175*x^2 - 147*x + 51
[2, 2, 1] x^5 - 10*x^4 + 45*x^3 - 105*x^2 + 120*x - 51
[2, 1, 1, 1] x^4 - 5*x^3 + 10*x^2 - 10*x + 4
[1, 1, 1, 1, 1] 1

>>> A = SGA.specht_module([Integer(4), Integer(1)]).intrinsic_arrangement()
>>> A.characteristic_polynomial().factor()
(x - 4) * (x - 3) * (x - 2) * (x - 1)
lift()[source]

The lift (embedding) map from self to the ambient space.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(QQ, 5)
sage: SM = SGA.specht_module([3, 1, 1])
sage: SM.lift
Generic morphism:
  From: Specht module of [3, 1, 1] over Rational Field
  To:   Tabloid module of [3, 1, 1] over Rational Field

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(QQ, Integer(5))
>>> SM = SGA.specht_module([Integer(3), Integer(1), Integer(1)])
>>> SM.lift
Generic morphism:
  From: Specht module of [3, 1, 1] over Rational Field
  To:   Tabloid module of [3, 1, 1] over Rational Field
maximal_submodule()[source]

Return the maximal submodule of self.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(GF(3), 5)
sage: SM = SGA.specht_module([3,2])
sage: U = SM.maximal_submodule()
sage: U.dimension()
4

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(GF(Integer(3)), Integer(5))
>>> SM = SGA.specht_module([Integer(3),Integer(2)])
>>> U = SM.maximal_submodule()
>>> U.dimension()
4
retract()[source]

The retract map from the ambient space.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(QQ, 5)
sage: X = SGA.tabloid_module([2,2,1])
sage: Y = X.specht_module()
sage: Y.retract
Generic morphism:
  From: Tabloid module of [2, 2, 1] over Rational Field
  To:   Specht module of [2, 2, 1] over Rational Field
sage: all(Y.retract(u.lift()) == u for u in Y.basis())
True

sage: Y.retract(X.zero())
0
sage: Y.retract(sum(X.basis()))
Traceback (most recent call last):
...
ValueError: ... is not in the image

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(QQ, Integer(5))
>>> X = SGA.tabloid_module([Integer(2),Integer(2),Integer(1)])
>>> Y = X.specht_module()
>>> Y.retract
Generic morphism:
  From: Tabloid module of [2, 2, 1] over Rational Field
  To:   Specht module of [2, 2, 1] over Rational Field
>>> all(Y.retract(u.lift()) == u for u in Y.basis())
True

>>> Y.retract(X.zero())
0
>>> Y.retract(sum(X.basis()))
Traceback (most recent call last):
...
ValueError: ... is not in the image
simple_module()[source]

Return the simple (or irreducible) \(S_n\)-submodule of self.

See also

SimpleModule

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(GF(3), 5)
sage: SM = SGA.specht_module([3,2])
sage: L = SM.simple_module()
sage: L.dimension()
1

sage: SGA = SymmetricGroupAlgebra(QQ, 5)
sage: SM = SGA.specht_module([3,2])
sage: SM.simple_module() is SM
True

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(GF(Integer(3)), Integer(5))
>>> SM = SGA.specht_module([Integer(3),Integer(2)])
>>> L = SM.simple_module()
>>> L.dimension()
1

>>> SGA = SymmetricGroupAlgebra(QQ, Integer(5))
>>> SM = SGA.specht_module([Integer(3),Integer(2)])
>>> SM.simple_module() is SM
True
class sage.combinat.specht_module.SymmetricGroupRepresentation(SGA)[source]

Bases: Representation_abstract

Mixin class for symmetric group (algebra) representations.

frobenius_image()[source]

Return the Frobenius image of self.

The Frobenius map is defined as the map to symmetric functions

\[F(\chi) = \frac{1}{n!} \sum_{w \in S_n} \chi(w) p_{\rho(w)},\]

where \(\chi\) is the character of the \(S_n\)-module self, \(p_{\lambda}\) is the powersum symmetric function basis element indexed by \(\lambda\), and \(\rho(w)\) is the cycle type of \(w\) as a partition. Specifically, this map takes irreducible representations indexed by \(\lambda\) to the Schur function \(s_{\lambda}\).

EXAMPLES:

sage: SM = Partition([2,2,1]).specht_module(QQ)
sage: SM.frobenius_image()
s[2, 2, 1]
sage: SM = Partition([4,1]).specht_module(CyclotomicField(5))
sage: SM.frobenius_image()
s[4, 1]

>>> from sage.all import *
>>> SM = Partition([Integer(2),Integer(2),Integer(1)]).specht_module(QQ)
>>> SM.frobenius_image()
s[2, 2, 1]
>>> SM = Partition([Integer(4),Integer(1)]).specht_module(CyclotomicField(Integer(5)))
>>> SM.frobenius_image()
s[4, 1]

We verify the regular representation:

sage: from sage.combinat.diagram import Diagram
sage: D = Diagram([(0,0), (1,1), (2,2), (3,3), (4,4)])
sage: F = D.specht_module(QQ).frobenius_image(); F
s[1, 1, 1, 1, 1] + 4*s[2, 1, 1, 1] + 5*s[2, 2, 1]
 + 6*s[3, 1, 1] + 5*s[3, 2] + 4*s[4, 1] + s[5]
sage: s = SymmetricFunctions(QQ).s()
sage: F == sum(StandardTableaux(la).cardinality() * s[la]
....:          for la in Partitions(5))
True
sage: all(s[la] == la.specht_module(QQ).frobenius_image()
....:     for n in range(1, 5) for la in Partitions(n))
True

sage: D = Diagram([(0,0), (1,1), (1,2), (2,3), (2,4)])
sage: SM = D.specht_module(QQ)
sage: SM.frobenius_image()
s[2, 2, 1] + s[3, 1, 1] + 2*s[3, 2] + 2*s[4, 1] + s[5]

>>> from sage.all import *
>>> from sage.combinat.diagram import Diagram
>>> D = Diagram([(Integer(0),Integer(0)), (Integer(1),Integer(1)), (Integer(2),Integer(2)), (Integer(3),Integer(3)), (Integer(4),Integer(4))])
>>> F = D.specht_module(QQ).frobenius_image(); F
s[1, 1, 1, 1, 1] + 4*s[2, 1, 1, 1] + 5*s[2, 2, 1]
 + 6*s[3, 1, 1] + 5*s[3, 2] + 4*s[4, 1] + s[5]
>>> s = SymmetricFunctions(QQ).s()
>>> F == sum(StandardTableaux(la).cardinality() * s[la]
...          for la in Partitions(Integer(5)))
True
>>> all(s[la] == la.specht_module(QQ).frobenius_image()
...     for n in range(Integer(1), Integer(5)) for la in Partitions(n))
True

>>> D = Diagram([(Integer(0),Integer(0)), (Integer(1),Integer(1)), (Integer(1),Integer(2)), (Integer(2),Integer(3)), (Integer(2),Integer(4))])
>>> SM = D.specht_module(QQ)
>>> SM.frobenius_image()
s[2, 2, 1] + s[3, 1, 1] + 2*s[3, 2] + 2*s[4, 1] + s[5]

An example using the tabloid module:

sage: SGA = SymmetricGroupAlgebra(QQ, 5)
sage: TM = SGA.tabloid_module([2, 2, 1])
sage: TM.frobenius_image()
s[2, 2, 1] + s[3, 1, 1] + 2*s[3, 2] + 2*s[4, 1] + s[5]

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(QQ, Integer(5))
>>> TM = SGA.tabloid_module([Integer(2), Integer(2), Integer(1)])
>>> TM.frobenius_image()
s[2, 2, 1] + s[3, 1, 1] + 2*s[3, 2] + 2*s[4, 1] + s[5]
class sage.combinat.specht_module.TabloidModule(SGA, shape)[source]

Bases: SymmetricGroupRepresentation, CombinatorialFreeModule

The vector space of all tabloids of a fixed shape with the natural symmetric group action.

A tabloid is an OrderedSetPartition whose underlying set is \(\{1, \ldots, n\}\). The symmetric group acts by permuting the entries of the set. Hence, this is a representation of the symmetric group defined over any field.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(GF(3), 5)
sage: TM = SGA.tabloid_module([2, 2, 1])
sage: TM.dimension()
30
sage: TM.brauer_character()
(30, 6, 2, 0, 0)
sage: IM = TM.invariant_module()
sage: IM.dimension()
1
sage: IM.basis()[0].lift() == sum(TM.basis())
True

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(GF(Integer(3)), Integer(5))
>>> TM = SGA.tabloid_module([Integer(2), Integer(2), Integer(1)])
>>> TM.dimension()
30
>>> TM.brauer_character()
(30, 6, 2, 0, 0)
>>> IM = TM.invariant_module()
>>> IM.dimension()
1
>>> IM.basis()[Integer(0)].lift() == sum(TM.basis())
True
class Element[source]

Bases: IndexedFreeModuleElement

bilinear_form(u, v)[source]

Return the natural bilinear form of self applied to u and v.

The natural bilinear form is given by defining the tabloid basis to be orthonormal.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(QQ, 5)
sage: TM = SGA.tabloid_module([2,2,1])
sage: u = TM.an_element(); u
2*T[{1, 2}, {3, 4}, {5}] + 2*T[{1, 2}, {3, 5}, {4}] + 3*T[{1, 2}, {4, 5}, {3}]
sage: v = sum(TM.basis())
sage: TM.bilinear_form(u, v)
7
sage: TM.bilinear_form(u, TM.zero())
0

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(QQ, Integer(5))
>>> TM = SGA.tabloid_module([Integer(2),Integer(2),Integer(1)])
>>> u = TM.an_element(); u
2*T[{1, 2}, {3, 4}, {5}] + 2*T[{1, 2}, {3, 5}, {4}] + 3*T[{1, 2}, {4, 5}, {3}]
>>> v = sum(TM.basis())
>>> TM.bilinear_form(u, v)
7
>>> TM.bilinear_form(u, TM.zero())
0
specht_module()[source]

Return the Specht submodule of self.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(QQ, 5)
sage: TM = SGA.tabloid_module([2,2,1])
sage: TM.specht_module() is SGA.specht_module([2,2,1])
True

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(QQ, Integer(5))
>>> TM = SGA.tabloid_module([Integer(2),Integer(2),Integer(1)])
>>> TM.specht_module() is SGA.specht_module([Integer(2),Integer(2),Integer(1)])
True
sage.combinat.specht_module.polytabloid(T)[source]

Compute the polytabloid element associated to a tableau T.

For a tableau \(T\), the polytabloid associated to \(T\) is

\[e_T = \sum_{\sigma \in C_T} (-1)^{\sigma} \{\sigma T\},\]

where \(\{\}\) is the row-equivalence class, i.e. a tabloid, and \(C_T\) is the column stabilizer of \(T\). The sum takes place in the module spanned by tabloids \(\{T\}\).

OUTPUT:

A dict whose keys are tabloids represented by tuples of frozensets and whose values are the coefficient.

EXAMPLES:

sage: from sage.combinat.specht_module import polytabloid
sage: T = StandardTableau([[1,3,4],[2,5]])
sage: polytabloid(T)
{(frozenset({1, 3, 4}), frozenset({2, 5})): 1,
 (frozenset({1, 4, 5}), frozenset({2, 3})): -1,
 (frozenset({2, 3, 4}), frozenset({1, 5})): -1,
 (frozenset({2, 4, 5}), frozenset({1, 3})): 1}

>>> from sage.all import *
>>> from sage.combinat.specht_module import polytabloid
>>> T = StandardTableau([[Integer(1),Integer(3),Integer(4)],[Integer(2),Integer(5)]])
>>> polytabloid(T)
{(frozenset({1, 3, 4}), frozenset({2, 5})): 1,
 (frozenset({1, 4, 5}), frozenset({2, 3})): -1,
 (frozenset({2, 3, 4}), frozenset({1, 5})): -1,
 (frozenset({2, 4, 5}), frozenset({1, 3})): 1}
sage.combinat.specht_module.simple_module_rank(la, base_ring)[source]

Return the rank of the simple \(S_n\)-module corresponding to the partition la of size \(n\) over base_ring.

EXAMPLES:

sage: from sage.combinat.specht_module import simple_module_rank
sage: simple_module_rank([3,2,1,1], GF(3))
13

>>> from sage.all import *
>>> from sage.combinat.specht_module import simple_module_rank
>>> simple_module_rank([Integer(3),Integer(2),Integer(1),Integer(1)], GF(Integer(3)))
13
sage.combinat.specht_module.specht_module_rank(D, base_ring=None)[source]

Return the rank of the Specht module of diagram D.

EXAMPLES:

sage: from sage.combinat.specht_module import specht_module_rank
sage: specht_module_rank([(0,0), (1,1), (2,2)])
6

>>> from sage.all import *
>>> from sage.combinat.specht_module import specht_module_rank
>>> specht_module_rank([(Integer(0),Integer(0)), (Integer(1),Integer(1)), (Integer(2),Integer(2))])
6
sage.combinat.specht_module.specht_module_spanning_set(D, SGA=None)[source]

Return a spanning set of the Specht module of diagram D.

INPUT:

  • D – list of cells (r,c) for row r and column c

  • SGA – (optional) a symmetric group algebra

EXAMPLES:

sage: from sage.combinat.specht_module import specht_module_spanning_set
sage: specht_module_spanning_set([(0,0), (1,1), (2,2)])
([1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1])
sage: specht_module_spanning_set([(0,0), (1,1), (2,1)])
([1, 2, 3] - [1, 3, 2], -[1, 2, 3] + [1, 3, 2], [2, 1, 3] - [3, 1, 2],
 [2, 3, 1] - [3, 2, 1], -[2, 1, 3] + [3, 1, 2], -[2, 3, 1] + [3, 2, 1])

sage: SGA = SymmetricGroup(3).algebra(QQ)
sage: specht_module_spanning_set([(0,0), (1,1), (2,1)], SGA)
(() - (2,3), -(1,2) + (1,3,2), (1,2,3) - (1,3),
 -() + (2,3), -(1,2,3) + (1,3), (1,2) - (1,3,2))

>>> from sage.all import *
>>> from sage.combinat.specht_module import specht_module_spanning_set
>>> specht_module_spanning_set([(Integer(0),Integer(0)), (Integer(1),Integer(1)), (Integer(2),Integer(2))])
([1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1])
>>> specht_module_spanning_set([(Integer(0),Integer(0)), (Integer(1),Integer(1)), (Integer(2),Integer(1))])
([1, 2, 3] - [1, 3, 2], -[1, 2, 3] + [1, 3, 2], [2, 1, 3] - [3, 1, 2],
 [2, 3, 1] - [3, 2, 1], -[2, 1, 3] + [3, 1, 2], -[2, 3, 1] + [3, 2, 1])

>>> SGA = SymmetricGroup(Integer(3)).algebra(QQ)
>>> specht_module_spanning_set([(Integer(0),Integer(0)), (Integer(1),Integer(1)), (Integer(2),Integer(1))], SGA)
(() - (2,3), -(1,2) + (1,3,2), (1,2,3) - (1,3),
 -() + (2,3), -(1,2,3) + (1,3), (1,2) - (1,3,2))
sage.combinat.specht_module.tabloid_gram_matrix(la, base_ring)[source]

Compute the Gram matrix of the bilinear form of a Specht module pulled back from the tabloid module.

For the module spanned by all tabloids, we define an bilinear form by having the tabloids be an orthonormal basis. We then pull this bilinear form back across the natural injection of the Specht module into the tabloid module.

EXAMPLES:

sage: from sage.combinat.specht_module import tabloid_gram_matrix
sage: tabloid_gram_matrix([3,2], GF(5))
[4 2 2 1 4]
[2 4 1 2 1]
[2 1 4 2 1]
[1 2 2 4 2]
[4 1 1 2 4]

>>> from sage.all import *
>>> from sage.combinat.specht_module import tabloid_gram_matrix
>>> tabloid_gram_matrix([Integer(3),Integer(2)], GF(Integer(5)))
[4 2 2 1 4]
[2 4 1 2 1]
[2 1 4 2 1]
[1 2 2 4 2]
[4 1 1 2 4]