Representations of the Symmetric Group

Todo

  • construct the product of two irreducible representations.

  • implement Induction/Restriction of representations.

Warning

This code uses a different convention than in Sagan’s book “The Symmetric Group”

class sage.combinat.symmetric_group_representations.GarsiaProcesiModule(SGA, shape)[source]

Bases: UniqueRepresentation, QuotientRing_generic, SymmetricGroupRepresentation

A Garsia-Procesi module.

Let \(\lambda\) be a partition of \(n\) and \(R\) be a commutative ring. The Garsia-Procesi module is defined by \(R_{\lambda} := R[x_1, \ldots, x_n] / I_{\lambda}\), where

\[I_{\lambda} := \langle e_r(x_{i_1}, \ldots, x_{i_k}) \mid \{i_1, \ldots, i_k\} \subseteq [n] \text{ and } k \geq r > k - d_k(\lambda) \rangle,\]

with \(e_r\) being the \(r\)-the elementary symmetric function and \(d_k(\lambda) = \lambda'_n + \cdots + \lambda'_{n+1-k}\), is the Tanisaki ideal.

If we consider \(R = \QQ\), then the Garsia-Procesi module has the following interpretation. Let \(\mathcal{F}_n = GL_n / B\) denote the (complex type A) flag variety. Consider the Springer fiber \(F_{\lambda} \subseteq \mathcal{F}_n\) associated to a nilpotent matrix with Jordan blocks sizes \(\lambda\). Springer showed that the cohomology ring \(H^*(F_{\lambda})\) admits a graded \(S_n\)-action that agrees with the induced representation of the sign representation of the Young subgroup \(S_{\lambda}\). From work of De Concini and Procesi, this \(S_n\)-representation is isomorphic to \(R_{\lambda}\). Moreover, the graded Frobenius image is known to be a modified Hall-Littlewood polynomial.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(QQ, 7)
sage: GP421 = SGA.garsia_procesi_module([4, 2, 1])
sage: GP421.dimension()
105
sage: v = GP421.an_element(); v
-gp1 - gp2 - gp3 - gp4 - gp5 - gp6
sage: SGA.an_element() * v
-6*gp1 - 6*gp2 - 6*gp3 - 6*gp4 - 6*gp5 - 5*gp6

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(QQ, Integer(7))
>>> GP421 = SGA.garsia_procesi_module([Integer(4), Integer(2), Integer(1)])
>>> GP421.dimension()
105
>>> v = GP421.an_element(); v
-gp1 - gp2 - gp3 - gp4 - gp5 - gp6
>>> SGA.an_element() * v
-6*gp1 - 6*gp2 - 6*gp3 - 6*gp4 - 6*gp5 - 5*gp6

We verify the result is a modified Hall-Littlewood polynomial by using the \(Q'\) Hall-Littlewood polynomials, replacing \(q \mapsto q^{-1}\) and multiplying by the smallest power of \(q\) so the coefficients are again polynomials:

sage: GP421.graded_frobenius_image()
q^4*s[4, 2, 1] + q^3*s[4, 3] + q^3*s[5, 1, 1] + (q^3+q^2)*s[5, 2]
 + (q^2+q)*s[6, 1] + s[7]
sage: R.<q> = QQ[]
sage: Sym = SymmetricFunctions(R)
sage: s = Sym.s()
sage: Qp = Sym.hall_littlewood(q).Qp()
sage: mHL = s(Qp[4,2,1]); mHL
s[4, 2, 1] + q*s[4, 3] + q*s[5, 1, 1] + (q^2+q)*s[5, 2]
 + (q^3+q^2)*s[6, 1] + q^4*s[7]
sage: mHL.map_coefficients(lambda c: R(q^4*c(q^-1)))
q^4*s[4, 2, 1] + q^3*s[4, 3] + q^3*s[5, 1, 1] + (q^3+q^2)*s[5, 2]
 + (q^2+q)*s[6, 1] + s[7]

>>> from sage.all import *
>>> GP421.graded_frobenius_image()
q^4*s[4, 2, 1] + q^3*s[4, 3] + q^3*s[5, 1, 1] + (q^3+q^2)*s[5, 2]
 + (q^2+q)*s[6, 1] + s[7]
>>> R = QQ['q']; (q,) = R._first_ngens(1)
>>> Sym = SymmetricFunctions(R)
>>> s = Sym.s()
>>> Qp = Sym.hall_littlewood(q).Qp()
>>> mHL = s(Qp[Integer(4),Integer(2),Integer(1)]); mHL
s[4, 2, 1] + q*s[4, 3] + q*s[5, 1, 1] + (q^2+q)*s[5, 2]
 + (q^3+q^2)*s[6, 1] + q^4*s[7]
>>> mHL.map_coefficients(lambda c: R(q**Integer(4)*c(q**-Integer(1))))
q^4*s[4, 2, 1] + q^3*s[4, 3] + q^3*s[5, 1, 1] + (q^3+q^2)*s[5, 2]
 + (q^2+q)*s[6, 1] + s[7]

We show that the maximal degree component corresponds to the Yamanouchi words of content \(\lambda\):

sage: B = GP421.graded_decomposition(4).basis()
sage: top_deg = [Word([i+1 for i in b.lift().lift().exponents()[0]]) for b in B]
sage: yamanouchi = [P.to_packed_word() for P in OrderedSetPartitions(range(7), [4, 2, 1])
....:               if P.to_packed_word().reversal().is_yamanouchi()]
sage: set(top_deg) == set(yamanouchi)
True

>>> from sage.all import *
>>> B = GP421.graded_decomposition(Integer(4)).basis()
>>> top_deg = [Word([i+Integer(1) for i in b.lift().lift().exponents()[Integer(0)]]) for b in B]
>>> yamanouchi = [P.to_packed_word() for P in OrderedSetPartitions(range(Integer(7)), [Integer(4), Integer(2), Integer(1)])
...               if P.to_packed_word().reversal().is_yamanouchi()]
>>> set(top_deg) == set(yamanouchi)
True
class Element(parent, rep, reduce=True)[source]

Bases: QuotientRingElement

degree()[source]

Return the degree of self.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(GF(3), 4)
sage: GP22 = SGA.garsia_procesi_module([2, 2])
sage: for b in GP22.basis():
....:     print(b, b.degree())
gp2*gp3 2
gp1*gp3 2
gp3 1
gp2 1
gp1 1
1 0
sage: v = sum(GP22.basis())
sage: v.degree()
2

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(GF(Integer(3)), Integer(4))
>>> GP22 = SGA.garsia_procesi_module([Integer(2), Integer(2)])
>>> for b in GP22.basis():
...     print(b, b.degree())
gp2*gp3 2
gp1*gp3 2
gp3 1
gp2 1
gp1 1
1 0
>>> v = sum(GP22.basis())
>>> v.degree()
2
homogeneous_degree()[source]

Return the (homogeneous) degree of self if homogeneous otherwise raise an error.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(GF(2), 4)
sage: GP31 = SGA.garsia_procesi_module([3, 1])
sage: for b in GP31.basis():
....:     print(b, b.homogeneous_degree())
gp3 1
gp2 1
gp1 1
1 0
sage: v = sum(GP31.basis()); v
gp1 + gp2 + gp3 + 1
sage: v.homogeneous_degree()
Traceback (most recent call last):
...
ValueError: element is not homogeneous

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(GF(Integer(2)), Integer(4))
>>> GP31 = SGA.garsia_procesi_module([Integer(3), Integer(1)])
>>> for b in GP31.basis():
...     print(b, b.homogeneous_degree())
gp3 1
gp2 1
gp1 1
1 0
>>> v = sum(GP31.basis()); v
gp1 + gp2 + gp3 + 1
>>> v.homogeneous_degree()
Traceback (most recent call last):
...
ValueError: element is not homogeneous
monomial_coefficients(copy=None)[source]

Return the monomial coefficients of self.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(GF(3), 4)
sage: GP31 = SGA.garsia_procesi_module([3, 1])
sage: v = GP31.an_element(); v
-gp1 - gp2 - gp3
sage: v.monomial_coefficients()
{0: 2, 1: 2, 2: 2, 3: 0}

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(GF(Integer(3)), Integer(4))
>>> GP31 = SGA.garsia_procesi_module([Integer(3), Integer(1)])
>>> v = GP31.an_element(); v
-gp1 - gp2 - gp3
>>> v.monomial_coefficients()
{0: 2, 1: 2, 2: 2, 3: 0}
to_vector(order=None)[source]

Return self as a (dense) free module vector.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(GF(3), 4)
sage: GP22 = SGA.garsia_procesi_module([2, 2])
sage: v = GP22.an_element(); v
-gp1 - gp2 - gp3
sage: v.to_vector()
(0, 0, 2, 2, 2, 0)

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(GF(Integer(3)), Integer(4))
>>> GP22 = SGA.garsia_procesi_module([Integer(2), Integer(2)])
>>> v = GP22.an_element(); v
-gp1 - gp2 - gp3
>>> v.to_vector()
(0, 0, 2, 2, 2, 0)
basis()[source]

Return a basis of self.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(QQ, 4)
sage: GP = SGA.garsia_procesi_module([2, 2])
sage: GP.basis()
Family (gp2*gp3, gp1*gp3, gp3, gp2, gp1, 1)

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(QQ, Integer(4))
>>> GP = SGA.garsia_procesi_module([Integer(2), Integer(2)])
>>> GP.basis()
Family (gp2*gp3, gp1*gp3, gp3, gp2, gp1, 1)
dimension()[source]

Return the dimension of self.

The graded Frobenius character of the Garsia-Procesi module \(R_{\lambda}\) is given by the modified Hall-Littlewood polynomial \(\widetilde{H}_{\lambda'}(x; q)\).

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(QQ, 5)
sage: Sym = SymmetricFunctions(QQ)
sage: s = Sym.s()
sage: Qp = Sym.hall_littlewood(1).Qp()
sage: for la in Partitions(5):
....:     print(SGA.garsia_procesi_module(la).dimension(),
....:           sum(c * StandardTableaux(la).cardinality()
....:               for la, c in s(Qp[la])))
1 1
5 5
10 10
20 20
30 30
60 60
120 120

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(QQ, Integer(5))
>>> Sym = SymmetricFunctions(QQ)
>>> s = Sym.s()
>>> Qp = Sym.hall_littlewood(Integer(1)).Qp()
>>> for la in Partitions(Integer(5)):
...     print(SGA.garsia_procesi_module(la).dimension(),
...           sum(c * StandardTableaux(la).cardinality()
...               for la, c in s(Qp[la])))
1 1
5 5
10 10
20 20
30 30
60 60
120 120
get_order()[source]

Return the order of the elements in the basis.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(QQ, 4)
sage: GP = SGA.garsia_procesi_module([2, 2])
sage: GP.get_order()
(0, 1, 2, 3, 4, 5)

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(QQ, Integer(4))
>>> GP = SGA.garsia_procesi_module([Integer(2), Integer(2)])
>>> GP.get_order()
(0, 1, 2, 3, 4, 5)
graded_brauer_character()[source]

Return the graded Brauer character of self.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(GF(2), 5)
sage: GP311 = SGA.garsia_procesi_module([3, 1, 1])
sage: GP311.graded_brauer_character()
(6*q^3 + 9*q^2 + 4*q + 1, q + 1, q^3 - q^2 - q + 1)

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(GF(Integer(2)), Integer(5))
>>> GP311 = SGA.garsia_procesi_module([Integer(3), Integer(1), Integer(1)])
>>> GP311.graded_brauer_character()
(6*q^3 + 9*q^2 + 4*q + 1, q + 1, q^3 - q^2 - q + 1)
graded_character()[source]

Return the graded character of self.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(QQ, 5)
sage: GP = SGA.garsia_procesi_module([2, 2, 1])
sage: gchi = GP.graded_character(); gchi
(5*q^4 + 11*q^3 + 9*q^2 + 4*q + 1, -q^4 + q^3 + 3*q^2 + 2*q + 1,
 q^4 - q^3 + q^2 + 1, -q^4 - q^3 + q + 1, -q^4 + q^3 - q + 1,
 q^4 - q^3 - q^2 + 1, q^3 - q^2 - q + 1)
sage: R.<q> = QQ[]
sage: gchi == sum(q^d * D.character()
....:             for d, D in GP.graded_decomposition().items())
True

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(QQ, Integer(5))
>>> GP = SGA.garsia_procesi_module([Integer(2), Integer(2), Integer(1)])
>>> gchi = GP.graded_character(); gchi
(5*q^4 + 11*q^3 + 9*q^2 + 4*q + 1, -q^4 + q^3 + 3*q^2 + 2*q + 1,
 q^4 - q^3 + q^2 + 1, -q^4 - q^3 + q + 1, -q^4 + q^3 - q + 1,
 q^4 - q^3 - q^2 + 1, q^3 - q^2 - q + 1)
>>> R = QQ['q']; (q,) = R._first_ngens(1)
>>> gchi == sum(q**d * D.character()
...             for d, D in GP.graded_decomposition().items())
True
graded_decomposition(k=None)[source]

Return the decomposition of self as a direct sum of representations given by a fixed grading.

INPUT:

  • k – (optional) integer; if given, return the \(k\)-th graded part

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(GF(2), 5)
sage: GP32 = SGA.garsia_procesi_module([3, 2])
sage: decomp = GP32.graded_decomposition(); decomp
{0: Subrepresentation with basis {0} of Garsia-Procesi ...,
 1: Subrepresentation with basis {0, 1, 2, 3} of Garsia-Procesi ...,
 2: Subrepresentation with basis {0, 1, 2, 3, 4} of Garsia-Procesi ...}
sage: decomp[2] is GP32.graded_decomposition(2)
True
sage: GP32.graded_decomposition(10)
Subrepresentation with basis {} of Garsia-Procesi module
 of shape [3, 2] over Finite Field of size 2

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(GF(Integer(2)), Integer(5))
>>> GP32 = SGA.garsia_procesi_module([Integer(3), Integer(2)])
>>> decomp = GP32.graded_decomposition(); decomp
{0: Subrepresentation with basis {0} of Garsia-Procesi ...,
 1: Subrepresentation with basis {0, 1, 2, 3} of Garsia-Procesi ...,
 2: Subrepresentation with basis {0, 1, 2, 3, 4} of Garsia-Procesi ...}
>>> decomp[Integer(2)] is GP32.graded_decomposition(Integer(2))
True
>>> GP32.graded_decomposition(Integer(10))
Subrepresentation with basis {} of Garsia-Procesi module
 of shape [3, 2] over Finite Field of size 2
graded_frobenius_image()[source]

Return the graded Frobenius image of self.

The graded Frobenius image is the sum of the frobenius_image() of each graded component, which is known to result in the modified Hall-Littlewood polynomial \(\widetilde{H}_{\lambda}(x; q)\).

EXAMPLES:

We verify that the result is the modified Hall-Littlewood polynomial for \(n = 5\):

sage: R.<q> = QQ[]
sage: Sym = SymmetricFunctions(R)
sage: s = Sym.s()
sage: Qp = Sym.hall_littlewood(q).Qp()
sage: SGA = SymmetricGroupAlgebra(QQ, 5)
sage: for la in Partitions(5):
....:     f = SGA.garsia_procesi_module(la).graded_frobenius_image()
....:     d = f[la].degree()
....:     assert f.map_coefficients(lambda c: R(c(~q)*q^d)) == s(Qp[la])

>>> from sage.all import *
>>> R = QQ['q']; (q,) = R._first_ngens(1)
>>> Sym = SymmetricFunctions(R)
>>> s = Sym.s()
>>> Qp = Sym.hall_littlewood(q).Qp()
>>> SGA = SymmetricGroupAlgebra(QQ, Integer(5))
>>> for la in Partitions(Integer(5)):
...     f = SGA.garsia_procesi_module(la).graded_frobenius_image()
...     d = f[la].degree()
...     assert f.map_coefficients(lambda c: R(c(~q)*q**d)) == s(Qp[la])
graded_representation_matrix(elt, q=None)[source]

Return the matrix corresponding to the left action of the symmetric group (algebra) element elt on self.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(GF(3), 3)
sage: GP = SGA.garsia_procesi_module([1, 1, 1])
sage: elt = SGA.an_element(); elt
[1, 2, 3] + 2*[1, 3, 2] + [3, 1, 2]
sage: X = GP.graded_representation_matrix(elt); X
[  0   0   0   0   0   0]
[  0 q^2   0   0   0   0]
[  0 q^2   0   0   0   0]
[  0   0   0   q   0   0]
[  0   0   0   q   0   0]
[  0   0   0   0   0   1]
sage: X.parent()
Full MatrixSpace of 6 by 6 dense matrices over
 Univariate Polynomial Ring in q over Finite Field of size 3
sage: R.<q> = GF(3)[]
sage: t = R.quotient([q^2+2*q+1]).gen()
sage: GP.graded_representation_matrix(elt, t)
[       0        0        0        0        0        0]
[       0 qbar + 2        0        0        0        0]
[       0 qbar + 2        0        0        0        0]
[       0        0        0     qbar        0        0]
[       0        0        0     qbar        0        0]
[       0        0        0        0        0        1]

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(GF(Integer(3)), Integer(3))
>>> GP = SGA.garsia_procesi_module([Integer(1), Integer(1), Integer(1)])
>>> elt = SGA.an_element(); elt
[1, 2, 3] + 2*[1, 3, 2] + [3, 1, 2]
>>> X = GP.graded_representation_matrix(elt); X
[  0   0   0   0   0   0]
[  0 q^2   0   0   0   0]
[  0 q^2   0   0   0   0]
[  0   0   0   q   0   0]
[  0   0   0   q   0   0]
[  0   0   0   0   0   1]
>>> X.parent()
Full MatrixSpace of 6 by 6 dense matrices over
 Univariate Polynomial Ring in q over Finite Field of size 3
>>> R = GF(Integer(3))['q']; (q,) = R._first_ngens(1)
>>> t = R.quotient([q**Integer(2)+Integer(2)*q+Integer(1)]).gen()
>>> GP.graded_representation_matrix(elt, t)
[       0        0        0        0        0        0]
[       0 qbar + 2        0        0        0        0]
[       0 qbar + 2        0        0        0        0]
[       0        0        0     qbar        0        0]
[       0        0        0     qbar        0        0]
[       0        0        0        0        0        1]
one_basis()[source]

Return the index of the basis element \(1\).

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(QQ, 4)
sage: GP = SGA.garsia_procesi_module([2, 2])
sage: GP.one_basis()
5

>>> from sage.all import *
>>> SGA = SymmetricGroupAlgebra(QQ, Integer(4))
>>> GP = SGA.garsia_procesi_module([Integer(2), Integer(2)])
>>> GP.one_basis()
5
class sage.combinat.symmetric_group_representations.SpechtRepresentation(parent, partition)[source]

Bases: SymmetricGroupRepresentation_generic_class

representation_matrix(permutation)[source]

Return the matrix representing the permutation in this irreducible representation.

Note

This method caches the results.

EXAMPLES:

sage: spc = SymmetricGroupRepresentation([3,1], 'specht')
sage: spc.representation_matrix(Permutation([2,1,3,4]))
[ 0 -1  0]
[-1  0  0]
[ 0  0  1]
sage: spc.representation_matrix(Permutation([3,2,1,4]))
[0 0 1]
[0 1 0]
[1 0 0]

>>> from sage.all import *
>>> spc = SymmetricGroupRepresentation([Integer(3),Integer(1)], 'specht')
>>> spc.representation_matrix(Permutation([Integer(2),Integer(1),Integer(3),Integer(4)]))
[ 0 -1  0]
[-1  0  0]
[ 0  0  1]
>>> spc.representation_matrix(Permutation([Integer(3),Integer(2),Integer(1),Integer(4)]))
[0 0 1]
[0 1 0]
[1 0 0]
scalar_product(u, v)[source]

Return 0 if u+v is not a permutation, and the signature of the permutation otherwise.

This is the scalar product of a vertex u of the underlying Yang-Baxter graph with the vertex v in the ‘dual’ Yang-Baxter graph.

EXAMPLES:

sage: spc = SymmetricGroupRepresentation([3,2], 'specht')
sage: spc.scalar_product((1,0,2,1,0),(0,3,0,3,0))
-1
sage: spc.scalar_product((1,0,2,1,0),(3,0,0,3,0))
0

>>> from sage.all import *
>>> spc = SymmetricGroupRepresentation([Integer(3),Integer(2)], 'specht')
>>> spc.scalar_product((Integer(1),Integer(0),Integer(2),Integer(1),Integer(0)),(Integer(0),Integer(3),Integer(0),Integer(3),Integer(0)))
-1
>>> spc.scalar_product((Integer(1),Integer(0),Integer(2),Integer(1),Integer(0)),(Integer(3),Integer(0),Integer(0),Integer(3),Integer(0)))
0
scalar_product_matrix(permutation=None)[source]

Return the scalar product matrix corresponding to permutation.

The entries are given by the scalar products of u and permutation.action(v), where u is a vertex in the underlying Yang-Baxter graph and v is a vertex in the dual graph.

EXAMPLES:

sage: spc = SymmetricGroupRepresentation([3,1], 'specht')
sage: spc.scalar_product_matrix()
[ 1  0  0]
[ 0 -1  0]
[ 0  0  1]

>>> from sage.all import *
>>> spc = SymmetricGroupRepresentation([Integer(3),Integer(1)], 'specht')
>>> spc.scalar_product_matrix()
[ 1  0  0]
[ 0 -1  0]
[ 0  0  1]
class sage.combinat.symmetric_group_representations.SpechtRepresentations(n, ring=None, cache_matrices=True)[source]

Bases: SymmetricGroupRepresentations_class

Element[source]

alias of SpechtRepresentation

sage.combinat.symmetric_group_representations.SymmetricGroupRepresentation(partition, implementation='specht', ring=None, cache_matrices=True)[source]

The irreducible representation of the symmetric group corresponding to partition.

INPUT:

  • partition – a partition of a positive integer

  • implementation – string (default: 'specht'); one of:

    • 'seminormal' – for Young’s seminormal representation

    • 'orthogonal' – for Young’s orthogonal representation

    • 'specht' – for Specht’s representation

    • 'unitary' – for the unitary representation

  • ring – the ring over which the representation is defined

  • cache_matrices – boolean (default: True); if True, then any representation matrices that are computed are cached

EXAMPLES:

Young’s orthogonal representation: the matrices are orthogonal.

sage: orth = SymmetricGroupRepresentation([2,1], "orthogonal"); orth            # needs sage.symbolic
Orthogonal representation of the symmetric group corresponding to [2, 1]
sage: all(a*a.transpose() == a.parent().identity_matrix() for a in orth)        # needs sage.symbolic
True

>>> from sage.all import *
>>> orth = SymmetricGroupRepresentation([Integer(2),Integer(1)], "orthogonal"); orth            # needs sage.symbolic
Orthogonal representation of the symmetric group corresponding to [2, 1]
>>> all(a*a.transpose() == a.parent().identity_matrix() for a in orth)        # needs sage.symbolic
True


sage: # needs sage.symbolic
sage: orth = SymmetricGroupRepresentation([3,2], "orthogonal"); orth
Orthogonal representation of the symmetric group corresponding to [3, 2]
sage: orth([2,1,3,4,5])
[ 1  0  0  0  0]
[ 0  1  0  0  0]
[ 0  0 -1  0  0]
[ 0  0  0  1  0]
[ 0  0  0  0 -1]
sage: orth([1,3,2,4,5])
[          1           0           0           0           0]
[          0        -1/2 1/2*sqrt(3)           0           0]
[          0 1/2*sqrt(3)         1/2           0           0]
[          0           0           0        -1/2 1/2*sqrt(3)]
[          0           0           0 1/2*sqrt(3)         1/2]
sage: orth([1,2,4,3,5])
[       -1/3 2/3*sqrt(2)           0           0           0]
[2/3*sqrt(2)         1/3           0           0           0]
[          0           0           1           0           0]
[          0           0           0           1           0]
[          0           0           0           0          -1]

>>> from sage.all import *
>>> # needs sage.symbolic
>>> orth = SymmetricGroupRepresentation([Integer(3),Integer(2)], "orthogonal"); orth
Orthogonal representation of the symmetric group corresponding to [3, 2]
>>> orth([Integer(2),Integer(1),Integer(3),Integer(4),Integer(5)])
[ 1  0  0  0  0]
[ 0  1  0  0  0]
[ 0  0 -1  0  0]
[ 0  0  0  1  0]
[ 0  0  0  0 -1]
>>> orth([Integer(1),Integer(3),Integer(2),Integer(4),Integer(5)])
[          1           0           0           0           0]
[          0        -1/2 1/2*sqrt(3)           0           0]
[          0 1/2*sqrt(3)         1/2           0           0]
[          0           0           0        -1/2 1/2*sqrt(3)]
[          0           0           0 1/2*sqrt(3)         1/2]
>>> orth([Integer(1),Integer(2),Integer(4),Integer(3),Integer(5)])
[       -1/3 2/3*sqrt(2)           0           0           0]
[2/3*sqrt(2)         1/3           0           0           0]
[          0           0           1           0           0]
[          0           0           0           1           0]
[          0           0           0           0          -1]

The Specht representation:

sage: spc = SymmetricGroupRepresentation([3,2], "specht")
sage: spc.scalar_product_matrix(Permutation([1,2,3,4,5]))
[ 1  0  0  0  0]
[ 0 -1  0  0  0]
[ 0  0  1  0  0]
[ 0  0  0  1  0]
[-1  0  0  0 -1]
sage: spc.scalar_product_matrix(Permutation([5,4,3,2,1]))
[ 1 -1  0  1  0]
[ 0  0  1  0 -1]
[ 0  0  0 -1  1]
[ 0  1 -1 -1  1]
[-1  0  0  0 -1]
sage: spc([5,4,3,2,1])
[ 1 -1  0  1  0]
[ 0  0 -1  0  1]
[ 0  0  0 -1  1]
[ 0  1 -1 -1  1]
[ 0  1  0 -1  1]
sage: spc.verify_representation()
True

>>> from sage.all import *
>>> spc = SymmetricGroupRepresentation([Integer(3),Integer(2)], "specht")
>>> spc.scalar_product_matrix(Permutation([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]))
[ 1  0  0  0  0]
[ 0 -1  0  0  0]
[ 0  0  1  0  0]
[ 0  0  0  1  0]
[-1  0  0  0 -1]
>>> spc.scalar_product_matrix(Permutation([Integer(5),Integer(4),Integer(3),Integer(2),Integer(1)]))
[ 1 -1  0  1  0]
[ 0  0  1  0 -1]
[ 0  0  0 -1  1]
[ 0  1 -1 -1  1]
[-1  0  0  0 -1]
>>> spc([Integer(5),Integer(4),Integer(3),Integer(2),Integer(1)])
[ 1 -1  0  1  0]
[ 0  0 -1  0  1]
[ 0  0  0 -1  1]
[ 0  1 -1 -1  1]
[ 0  1  0 -1  1]
>>> spc.verify_representation()
True

The unitary representation:

sage: unitary = SymmetricGroupRepresentation([3,1], "unitary"); unitary
Unitary representation of the symmetric group corresponding to [3, 1]
sage: unitary_GF49 = SymmetricGroupRepresentation([3,1], "unitary", ring=GF(7**2)); unitary_GF49
Unitary representation of the symmetric group corresponding to [3, 1]
sage: unitary_GF49([2,1,3,4])
[6 0 0]
[0 1 0]
[0 0 1]
sage: unitary_GF49([3,2,1,4])
[       4 2*z2 + 3        0]
[5*z2 + 5        3        0]
[       0        0        1]
sage: unitary_GF49.verify_representation()
True

>>> from sage.all import *
>>> unitary = SymmetricGroupRepresentation([Integer(3),Integer(1)], "unitary"); unitary
Unitary representation of the symmetric group corresponding to [3, 1]
>>> unitary_GF49 = SymmetricGroupRepresentation([Integer(3),Integer(1)], "unitary", ring=GF(Integer(7)**Integer(2))); unitary_GF49
Unitary representation of the symmetric group corresponding to [3, 1]
>>> unitary_GF49([Integer(2),Integer(1),Integer(3),Integer(4)])
[6 0 0]
[0 1 0]
[0 0 1]
>>> unitary_GF49([Integer(3),Integer(2),Integer(1),Integer(4)])
[       4 2*z2 + 3        0]
[5*z2 + 5        3        0]
[       0        0        1]
>>> unitary_GF49.verify_representation()
True

By default, any representation matrices that are computed are cached:

sage: spc = SymmetricGroupRepresentation([3,2], "specht")
sage: spc([5,4,3,2,1])
[ 1 -1  0  1  0]
[ 0  0 -1  0  1]
[ 0  0  0 -1  1]
[ 0  1 -1 -1  1]
[ 0  1  0 -1  1]
sage: spc._cache__representation_matrix
{(([5, 4, 3, 2, 1],), ()): [ 1 -1  0  1  0]
[ 0  0 -1  0  1]
[ 0  0  0 -1  1]
[ 0  1 -1 -1  1]
[ 0  1  0 -1  1]}

>>> from sage.all import *
>>> spc = SymmetricGroupRepresentation([Integer(3),Integer(2)], "specht")
>>> spc([Integer(5),Integer(4),Integer(3),Integer(2),Integer(1)])
[ 1 -1  0  1  0]
[ 0  0 -1  0  1]
[ 0  0  0 -1  1]
[ 0  1 -1 -1  1]
[ 0  1  0 -1  1]
>>> spc._cache__representation_matrix
{(([5, 4, 3, 2, 1],), ()): [ 1 -1  0  1  0]
[ 0  0 -1  0  1]
[ 0  0  0 -1  1]
[ 0  1 -1 -1  1]
[ 0  1  0 -1  1]}

This can be turned off with the keyword cache_matrices:

sage: spc = SymmetricGroupRepresentation([3,2], "specht", cache_matrices=False)
sage: spc([5,4,3,2,1])
[ 1 -1  0  1  0]
[ 0  0 -1  0  1]
[ 0  0  0 -1  1]
[ 0  1 -1 -1  1]
[ 0  1  0 -1  1]
sage: hasattr(spc, '_cache__representation_matrix')
False

>>> from sage.all import *
>>> spc = SymmetricGroupRepresentation([Integer(3),Integer(2)], "specht", cache_matrices=False)
>>> spc([Integer(5),Integer(4),Integer(3),Integer(2),Integer(1)])
[ 1 -1  0  1  0]
[ 0  0 -1  0  1]
[ 0  0  0 -1  1]
[ 0  1 -1 -1  1]
[ 0  1  0 -1  1]
>>> hasattr(spc, '_cache__representation_matrix')
False

Note

The implementation is based on the paper [Las].

REFERENCES:

[Las] (1,2)

Alain Lascoux, ‘Young representations of the symmetric group.’ http://phalanstere.univ-mlv.fr/~al/ARTICLES/ProcCrac.ps.gz

AUTHORS:

  • Franco Saliola (2009-04-23)

class sage.combinat.symmetric_group_representations.SymmetricGroupRepresentation_generic_class(parent, partition)[source]

Bases: Element

Generic methods for a representation of the symmetric group.

to_character()[source]

Return the character of the representation.

EXAMPLES:

The trivial character:

sage: rho = SymmetricGroupRepresentation([3])
sage: chi = rho.to_character(); chi
Character of Symmetric group of order 3! as a permutation group
sage: chi.values()
[1, 1, 1]
sage: all(chi(g) == 1 for g in SymmetricGroup(3))
True

>>> from sage.all import *
>>> rho = SymmetricGroupRepresentation([Integer(3)])
>>> chi = rho.to_character(); chi
Character of Symmetric group of order 3! as a permutation group
>>> chi.values()
[1, 1, 1]
>>> all(chi(g) == Integer(1) for g in SymmetricGroup(Integer(3)))
True

The sign character:

sage: rho = SymmetricGroupRepresentation([1,1,1])
sage: chi = rho.to_character(); chi
Character of Symmetric group of order 3! as a permutation group
sage: chi.values()
[1, -1, 1]
sage: all(chi(g) == g.sign() for g in SymmetricGroup(3))
True

>>> from sage.all import *
>>> rho = SymmetricGroupRepresentation([Integer(1),Integer(1),Integer(1)])
>>> chi = rho.to_character(); chi
Character of Symmetric group of order 3! as a permutation group
>>> chi.values()
[1, -1, 1]
>>> all(chi(g) == g.sign() for g in SymmetricGroup(Integer(3)))
True

The defining representation:

sage: triv = SymmetricGroupRepresentation([4])
sage: hook = SymmetricGroupRepresentation([3,1])
sage: def_rep = lambda p : triv(p).block_sum(hook(p)).trace()
sage: list(map(def_rep, Permutations(4)))
[4, 2, 2, 1, 1, 2, 2, 0, 1, 0, 0, 1, 1, 0, 2, 1, 0, 0, 0, 1, 1, 2, 0, 0]
sage: [p.to_matrix().trace() for p in Permutations(4)]
[4, 2, 2, 1, 1, 2, 2, 0, 1, 0, 0, 1, 1, 0, 2, 1, 0, 0, 0, 1, 1, 2, 0, 0]

>>> from sage.all import *
>>> triv = SymmetricGroupRepresentation([Integer(4)])
>>> hook = SymmetricGroupRepresentation([Integer(3),Integer(1)])
>>> def_rep = lambda p : triv(p).block_sum(hook(p)).trace()
>>> list(map(def_rep, Permutations(Integer(4))))
[4, 2, 2, 1, 1, 2, 2, 0, 1, 0, 0, 1, 1, 0, 2, 1, 0, 0, 0, 1, 1, 2, 0, 0]
>>> [p.to_matrix().trace() for p in Permutations(Integer(4))]
[4, 2, 2, 1, 1, 2, 2, 0, 1, 0, 0, 1, 1, 0, 2, 1, 0, 0, 0, 1, 1, 2, 0, 0]
verify_representation()[source]

Verify the representation.

This tests that the images of the simple transpositions are involutions and tests that the braid relations hold.

EXAMPLES:

sage: spc = SymmetricGroupRepresentation([1,1,1])
sage: spc.verify_representation()
True
sage: spc = SymmetricGroupRepresentation([4,2,1])
sage: spc.verify_representation()
True

>>> from sage.all import *
>>> spc = SymmetricGroupRepresentation([Integer(1),Integer(1),Integer(1)])
>>> spc.verify_representation()
True
>>> spc = SymmetricGroupRepresentation([Integer(4),Integer(2),Integer(1)])
>>> spc.verify_representation()
True
sage.combinat.symmetric_group_representations.SymmetricGroupRepresentations(n, implementation='specht', ring=None, cache_matrices=True)[source]

Irreducible representations of the symmetric group.

INPUT:

  • n – positive integer

  • implementation – string (default: 'specht'); one of:

    • 'seminormal' – for Young’s seminormal representation

    • 'orthogonal' – for Young’s orthogonal representation

    • 'specht' – for Specht’s representation

    • 'unitary' – for the unitary representation

  • ring – the ring over which the representation is defined

  • cache_matrices – boolean (default: True); if True, then any representation matrices that are computed are cached

EXAMPLES:

Young’s orthogonal representation: the matrices are orthogonal.

sage: orth = SymmetricGroupRepresentations(3, "orthogonal"); orth               # needs sage.symbolic
Orthogonal representations of the symmetric group of order 3! over Symbolic Ring
sage: orth.list()                                                               # needs sage.symbolic
[Orthogonal representation of the symmetric group corresponding to [3],
 Orthogonal representation of the symmetric group corresponding to [2, 1],
 Orthogonal representation of the symmetric group corresponding to [1, 1, 1]]
sage: orth([2,1])([1,2,3])                                                      # needs sage.symbolic
[1 0]
[0 1]

>>> from sage.all import *
>>> orth = SymmetricGroupRepresentations(Integer(3), "orthogonal"); orth               # needs sage.symbolic
Orthogonal representations of the symmetric group of order 3! over Symbolic Ring
>>> orth.list()                                                               # needs sage.symbolic
[Orthogonal representation of the symmetric group corresponding to [3],
 Orthogonal representation of the symmetric group corresponding to [2, 1],
 Orthogonal representation of the symmetric group corresponding to [1, 1, 1]]
>>> orth([Integer(2),Integer(1)])([Integer(1),Integer(2),Integer(3)])                                                      # needs sage.symbolic
[1 0]
[0 1]

Young’s seminormal representation.

sage: snorm = SymmetricGroupRepresentations(3, "seminormal"); snorm
Seminormal representations of the symmetric group of order 3! over Rational Field
sage: sgn = snorm([1,1,1]); sgn
Seminormal representation of the symmetric group corresponding to [1, 1, 1]
sage: list(map(sgn, Permutations(3)))
[[1], [-1], [-1], [1], [1], [-1]]

>>> from sage.all import *
>>> snorm = SymmetricGroupRepresentations(Integer(3), "seminormal"); snorm
Seminormal representations of the symmetric group of order 3! over Rational Field
>>> sgn = snorm([Integer(1),Integer(1),Integer(1)]); sgn
Seminormal representation of the symmetric group corresponding to [1, 1, 1]
>>> list(map(sgn, Permutations(Integer(3))))
[[1], [-1], [-1], [1], [1], [-1]]

The Specht Representation.

sage: spc = SymmetricGroupRepresentations(5, "specht"); spc
Specht representations of the symmetric group of order 5! over Integer Ring
sage: spc([3,2])([5,4,3,2,1])
[ 1 -1  0  1  0]
[ 0  0 -1  0  1]
[ 0  0  0 -1  1]
[ 0  1 -1 -1  1]
[ 0  1  0 -1  1]

>>> from sage.all import *
>>> spc = SymmetricGroupRepresentations(Integer(5), "specht"); spc
Specht representations of the symmetric group of order 5! over Integer Ring
>>> spc([Integer(3),Integer(2)])([Integer(5),Integer(4),Integer(3),Integer(2),Integer(1)])
[ 1 -1  0  1  0]
[ 0  0 -1  0  1]
[ 0  0  0 -1  1]
[ 0  1 -1 -1  1]
[ 0  1  0 -1  1]

The unitary representation.

sage: unitary = SymmetricGroupRepresentations(3, "unitary"); unitary
Unitary representations of the symmetric group of order 3! over Symbolic Ring
sage: unitary_GF49 = SymmetricGroupRepresentations(4, "unitary", ring=GF(7**2)); unitary_GF49
Unitary representations of the symmetric group of order 4! over Finite Field in z2 of size 7^2
sage: unitary_GF49([3,1])([2,1,3,4])
[6 0 0]
[0 1 0]
[0 0 1]
sage: unitary_GF49([3,1])([3,2,1,4])
[       4 2*z2 + 3        0]
[5*z2 + 5        3        0]
[       0        0        1]

>>> from sage.all import *
>>> unitary = SymmetricGroupRepresentations(Integer(3), "unitary"); unitary
Unitary representations of the symmetric group of order 3! over Symbolic Ring
>>> unitary_GF49 = SymmetricGroupRepresentations(Integer(4), "unitary", ring=GF(Integer(7)**Integer(2))); unitary_GF49
Unitary representations of the symmetric group of order 4! over Finite Field in z2 of size 7^2
>>> unitary_GF49([Integer(3),Integer(1)])([Integer(2),Integer(1),Integer(3),Integer(4)])
[6 0 0]
[0 1 0]
[0 0 1]
>>> unitary_GF49([Integer(3),Integer(1)])([Integer(3),Integer(2),Integer(1),Integer(4)])
[       4 2*z2 + 3        0]
[5*z2 + 5        3        0]
[       0        0        1]

Note

The implementation is based on the paper [Las].

AUTHORS:

  • Franco Saliola (2009-04-23)

class sage.combinat.symmetric_group_representations.SymmetricGroupRepresentations_class(n, ring=None, cache_matrices=True)[source]

Bases: UniqueRepresentation, Parent

Generic methods for the CombinatorialClass of irreducible representations of the symmetric group.

cardinality()[source]

Return the cardinality of self.

EXAMPLES:

sage: sp = SymmetricGroupRepresentations(4, "specht")
sage: sp.cardinality()
5

>>> from sage.all import *
>>> sp = SymmetricGroupRepresentations(Integer(4), "specht")
>>> sp.cardinality()
5
class sage.combinat.symmetric_group_representations.UnitaryRepresentation(parent, partition)[source]

Bases: SymmetricGroupRepresentation_generic_class

A unitary representation of the symmetric group.

In characteristic zero, this is the same as the orthogonal representation since they are defined over the reals. In positive characteristic, we are able to construct representations when the field is finite, has square order, and the characteristic does not divide the order of the group. In this case, we construct the representation by first constructing the orthogonal representation \(\rho\) and then taking the extended Cholesky decomposition of the unique solution \(U\) to the equation \(\rho(g)^T U \rho(g) = U\) for all \(g\) in \(G\).

representation_matrix(permutation)[source]

Return the matrix representing the permutation in this irreducible representation.

Note

This method caches the results.

EXAMPLES:

sage: unitary_specht = SymmetricGroupRepresentation([3,1], 'unitary', ring=GF(7**2))
sage: unitary_specht.representation_matrix(Permutation([2,1,3,4]))
[6 0 0]
[0 1 0]
[0 0 1]
sage: unitary_specht.representation_matrix(Permutation([3,2,1,4]))
[       4 2*z2 + 3        0]
[5*z2 + 5        3        0]
[       0        0        1]
sage: all(A * A.H == 1 for A in [unitary_specht.representation_matrix(g)
....:                            for g in Permutations(4)])
True

>>> from sage.all import *
>>> unitary_specht = SymmetricGroupRepresentation([Integer(3),Integer(1)], 'unitary', ring=GF(Integer(7)**Integer(2)))
>>> unitary_specht.representation_matrix(Permutation([Integer(2),Integer(1),Integer(3),Integer(4)]))
[6 0 0]
[0 1 0]
[0 0 1]
>>> unitary_specht.representation_matrix(Permutation([Integer(3),Integer(2),Integer(1),Integer(4)]))
[       4 2*z2 + 3        0]
[5*z2 + 5        3        0]
[       0        0        1]
>>> all(A * A.H == Integer(1) for A in [unitary_specht.representation_matrix(g)
...                            for g in Permutations(Integer(4))])
True
class sage.combinat.symmetric_group_representations.UnitaryRepresentations(n, ring=None, cache_matrices=True)[source]

Bases: SymmetricGroupRepresentations_class

Element[source]

alias of UnitaryRepresentation

class sage.combinat.symmetric_group_representations.YoungRepresentation_Orthogonal(parent, partition)[source]

Bases: YoungRepresentation_generic

class sage.combinat.symmetric_group_representations.YoungRepresentation_Seminormal(parent, partition)[source]

Bases: YoungRepresentation_generic

class sage.combinat.symmetric_group_representations.YoungRepresentation_generic(parent, partition)[source]

Bases: SymmetricGroupRepresentation_generic_class

Generic methods for Young’s representations of the symmetric group.

representation_matrix(permutation)[source]

Return the matrix representing permutation.

EXAMPLES:

sage: orth = SymmetricGroupRepresentation([2,1], "orthogonal")              # needs sage.symbolic
sage: orth.representation_matrix(Permutation([2,1,3]))                      # needs sage.symbolic
[ 1  0]
[ 0 -1]
sage: orth.representation_matrix(Permutation([1,3,2]))                      # needs sage.symbolic
[       -1/2 1/2*sqrt(3)]
[1/2*sqrt(3)         1/2]

>>> from sage.all import *
>>> orth = SymmetricGroupRepresentation([Integer(2),Integer(1)], "orthogonal")              # needs sage.symbolic
>>> orth.representation_matrix(Permutation([Integer(2),Integer(1),Integer(3)]))                      # needs sage.symbolic
[ 1  0]
[ 0 -1]
>>> orth.representation_matrix(Permutation([Integer(1),Integer(3),Integer(2)]))                      # needs sage.symbolic
[       -1/2 1/2*sqrt(3)]
[1/2*sqrt(3)         1/2]


sage: norm = SymmetricGroupRepresentation([2,1], "seminormal")
sage: p = PermutationGroupElement([2,1,3])
sage: norm.representation_matrix(p)
[ 1  0]
[ 0 -1]
sage: p = PermutationGroupElement([1,3,2])
sage: norm.representation_matrix(p)
[-1/2  3/2]
[ 1/2  1/2]

>>> from sage.all import *
>>> norm = SymmetricGroupRepresentation([Integer(2),Integer(1)], "seminormal")
>>> p = PermutationGroupElement([Integer(2),Integer(1),Integer(3)])
>>> norm.representation_matrix(p)
[ 1  0]
[ 0 -1]
>>> p = PermutationGroupElement([Integer(1),Integer(3),Integer(2)])
>>> norm.representation_matrix(p)
[-1/2  3/2]
[ 1/2  1/2]
representation_matrix_for_simple_transposition(i)[source]

Return the matrix representing the transposition that swaps i and i+1.

EXAMPLES:

sage: orth = SymmetricGroupRepresentation([2,1], "orthogonal")              # needs sage.symbolic
sage: orth.representation_matrix_for_simple_transposition(1)                # needs sage.symbolic
[ 1  0]
[ 0 -1]
sage: orth.representation_matrix_for_simple_transposition(2)                # needs sage.symbolic
[       -1/2 1/2*sqrt(3)]
[1/2*sqrt(3)         1/2]

sage: norm = SymmetricGroupRepresentation([2,1], "seminormal")
sage: norm.representation_matrix_for_simple_transposition(1)
[ 1  0]
[ 0 -1]
sage: norm.representation_matrix_for_simple_transposition(2)
[-1/2  3/2]
[ 1/2  1/2]

>>> from sage.all import *
>>> orth = SymmetricGroupRepresentation([Integer(2),Integer(1)], "orthogonal")              # needs sage.symbolic
>>> orth.representation_matrix_for_simple_transposition(Integer(1))                # needs sage.symbolic
[ 1  0]
[ 0 -1]
>>> orth.representation_matrix_for_simple_transposition(Integer(2))                # needs sage.symbolic
[       -1/2 1/2*sqrt(3)]
[1/2*sqrt(3)         1/2]

>>> norm = SymmetricGroupRepresentation([Integer(2),Integer(1)], "seminormal")
>>> norm.representation_matrix_for_simple_transposition(Integer(1))
[ 1  0]
[ 0 -1]
>>> norm.representation_matrix_for_simple_transposition(Integer(2))
[-1/2  3/2]
[ 1/2  1/2]
class sage.combinat.symmetric_group_representations.YoungRepresentations_Orthogonal(n, ring=None, cache_matrices=True)[source]

Bases: SymmetricGroupRepresentations_class

Element[source]

alias of YoungRepresentation_Orthogonal

class sage.combinat.symmetric_group_representations.YoungRepresentations_Seminormal(n, ring=None, cache_matrices=True)[source]

Bases: SymmetricGroupRepresentations_class

Element[source]

alias of YoungRepresentation_Seminormal

sage.combinat.symmetric_group_representations.partition_to_vector_of_contents(partition, reverse=False)[source]

Return the “vector of contents” associated to partition.

EXAMPLES:

sage: from sage.combinat.symmetric_group_representations import partition_to_vector_of_contents
sage: partition_to_vector_of_contents([3,2])
(0, 1, 2, -1, 0)

>>> from sage.all import *
>>> from sage.combinat.symmetric_group_representations import partition_to_vector_of_contents
>>> partition_to_vector_of_contents([Integer(3),Integer(2)])
(0, 1, 2, -1, 0)