18.34 Symmetric Group Algebra

Module: sage.combinat.symmetric_group_algebra

Symmetric Group Algebra

Module-level Functions

HeckeAlgebraSymmetricGroupT( R, n, [q=None])

Returns the Hecke algebra of the symmetric group on the T basis.

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

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

SymmetricGroupAlgebra( R, n)

Returns the symmetric group algebra of order n over R.

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

e( tableau, [star=0])

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]

e_hat( tab, [star=0])

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]

e_ik( itab, ktab, [star=0])

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]

epsilon( tab, [star=0])

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]

epsilon_ik( itab, ktab, [star=0])

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

kappa( alpha)

Returns $ \kappa_\alpha$ which is n! divided by the number of standard tableaux of shape $ \alpha$ .

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

pi_ik( itab, ktab)

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

seminormal_test( n)

Runs a variety of tests to verify that the construction of the seminormal basis works as desired. The numbers appearing are Theorems in James and Kerber's 'Representation Theory of the Symmetric Group'.

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

Class: HeckeAlgebraSymmetricGroup_generic

class HeckeAlgebraSymmetricGroup_generic
HeckeAlgebraSymmetricGroup_generic( self, R, n, [q=None])

TESTS:

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

sage: HeckeAlgebraSymmetricGroupT(QQ, 3, q=1)
Hecke algebra of the symmetric group of order 3 with q=1 on the T basis
over Rational Field

Functions: q

q( self)

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

Special Functions: __init__,$ \,$ _coerce_start

_coerce_start( self, x)

sage: H3 = HeckeAlgebraSymmetricGroupT(QQ, 3)
sage: H3._coerce_start([2,1])
T[2, 1, 3]

Class: HeckeAlgebraSymmetricGroup_t

class HeckeAlgebraSymmetricGroup_t
HeckeAlgebraSymmetricGroup_t( self, R, n, [q=None])

TESTS:

sage: H3 = HeckeAlgebraSymmetricGroupT(QQ, 3)
sage: H3 == loads(dumps(H3))
True

Functions: algebra_generators,$ \,$ jucys_murphy,$ \,$ t,$ \,$ t_action,$ \,$ t_action_on_basis

algebra_generators( self)

Return the generators of the algebra.

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

jucys_murphy( self, k)

Returns the Jucys-Murphy element J_k of the Hecke algebra. The Jucys-Murphy elements generate the maximal commutative sub-algebra of the Hecke algebra.

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: H3.jucys_murphy(1)
Traceback (most recent call last):
...
ValueError: k must be between 2 and n (= 3)

t( self, i)

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 must be between 1 and n-1 (= 2)

t_action( self, a, i)

Return the action of T_i on a.

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]

t_action_on_basis( self, perm, i)

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]

Special Functions: __init__,$ \,$ _multiply_basis

_multiply_basis( self, perm1, perm2)

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]

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]

Class: HeckeAlgebraSymmetricGroupElement_t

class HeckeAlgebraSymmetricGroupElement_t

Class: SymmetricGroupAlgebra_n

class SymmetricGroupAlgebra_n
SymmetricGroupAlgebra_n( self, R, n)

TESTS:

sage: QS3 = SymmetricGroupAlgebra(QQ, 3)
sage: QS3 == loads(dumps(QS3))
True

Functions: cpi,$ \,$ cpis,$ \,$ dft,$ \,$ epsilon_ik,$ \,$ jucys_murphy,$ \,$ seminormal_basis

cpi( self, p)

Returns the centrally primitive idempotent for the symmetric group of order n for the irreducible corresponding indexed by the partition p.

sage: QS3 = SymmetricGroupAlgebra(QQ,3)
sage: QS3.cpi([2,1])
2/3*[1, 2, 3] - 1/3*[2, 3, 1] - 1/3*[3, 1, 2]
sage: QS3.cpi([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: QS3.cpi([1,1,1])
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]

cpis( self)

Returns a list of the centrally primitive idempotents.

sage: QS3 = SymmetricGroupAlgebra(QQ,3)
sage: a = QS3.cpis()
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]

dft( self, [form=seminormal])

Returns the discrete Fourier transform for self.

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]

epsilon_ik( self, itab, ktab, [star=0])

Returns the seminormal basis element of self corresponding to the pair of tableaux itab and ktab.

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)

jucys_murphy( self, k)

Returns the Jucys-Murphy element J_k for the symmetric group algebra.

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

seminormal_basis( self)

Returns a list of the seminormal basis elements of self.

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]]

Special Functions: __init__,$ \,$ _coerce_start,$ \,$ _dft_seminormal,$ \,$ _multiply_basis

_coerce_start( self, x)

Coerce things into the symmetric group algebra.

sage: QS3 = SymmetricGroupAlgebra(QQ, 3)
sage: QS3._coerce_start([])
[1, 2, 3]
sage: QS3._coerce_start([2,1])
[2, 1, 3]
sage: _.parent()
Symmetric group algebra of order 3 over Rational Field

_dft_seminormal( self)

Returns the seminormal form of the discrete Fourier for self.

sage: QS3 = SymmetricGroupAlgebra(QQ, 3)
sage: QS3._dft_seminormal()
[   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]

_multiply_basis( self, left, right)

Returns the product of the basis elements indexed by left and right.

sage: QS3 = SymmetricGroupAlgebra(QQ, 3)
sage: p1 = Permutation([1,2,3])
sage: p2 = Permutation([2,1,3])
sage: QS3._multiply_basis(p1,p2)
[2, 1, 3]

Class: SymmetricGroupAlgebraElement_n

class SymmetricGroupAlgebraElement_n
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