Symmetric Group Algebra

class sage.combinat.symmetric_group_algebra.HeckeAlgebraSymmetricGroupElement_t(A, x)
sage.combinat.symmetric_group_algebra.HeckeAlgebraSymmetricGroupT(R, n, q=None)

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

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
sage: HeckeAlgebraSymmetricGroupT(QQ, 3, 2)
Hecke algebra of the symmetric group of order 3 with q=2 on the T basis over Rational Field
class sage.combinat.symmetric_group_algebra.HeckeAlgebraSymmetricGroup_generic(R, n, q=None)
__init__(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
_coerce_start(x)

EXAMPLES:

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

EXAMPLES:

sage: HeckeAlgebraSymmetricGroupT(QQ, 3).q()
q
sage: HeckeAlgebraSymmetricGroupT(QQ, 3, 2).q()
2
class sage.combinat.symmetric_group_algebra.HeckeAlgebraSymmetricGroup_t(R, n, q=None)
__init__(R, n, q=None)

TESTS:

sage: H3 = HeckeAlgebraSymmetricGroupT(QQ, 3)
sage: H3 == loads(dumps(H3))
True
_multiply_basis(perm1, perm2)

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]
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]
algebra_generators()

Return the generators of the algebra.

EXAMPLES:

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

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: H3.jucys_murphy(1)
...
ValueError: k must be between 2 and n (= 3)
t(i)

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)
...
ValueError: i must be between 1 and n-1 (= 2)
t_action(a, i)

Return the action of T_i on 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]
t_action_on_basis(perm, i)

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]
sage.combinat.symmetric_group_algebra.SymmetricGroupAlgebra(R, n)

Returns the symmetric group algebra of order n over R.

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
class sage.combinat.symmetric_group_algebra.SymmetricGroupAlgebraElement_n(A, x)
class sage.combinat.symmetric_group_algebra.SymmetricGroupAlgebra_n(R, n)
__init__(R, n)

TESTS:

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

Coerce things into the symmetric group algebra.

EXAMPLES:

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()

Returns the seminormal form of the discrete Fourier for self.

EXAMPLES:

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(left, right)

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

EXAMPLES:

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]
cpi(p)

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

EXAMPLES:

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()

Returns a list of the centrally primitive idempotents.

EXAMPLES:

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(form='seminormal')

Returns the discrete Fourier transform for self.

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]
epsilon_ik(itab, ktab, star=0)

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

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)
jucys_murphy(k)

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

EXAMPLES:

sage: QS3 = SymmetricGroupAlgebra(QQ, 3)
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]
seminormal_basis()

Returns a list of the seminormal basis elements of self.

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

The unnormalized Young projection operator.

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]

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

The Young projection operator, an idempotent in the rational group algebra.

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]

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

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

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

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)
...
ValueError: the two tableaux must be of the same shape
sage.combinat.symmetric_group_algebra.kappa(alpha)

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

EXAMPLES:

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

EXAMPLES:

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

EXAMPLES:

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

Previous topic

Combinatorial Algebras

Next topic

Schubert Polynomials

This Page