Benkart-Kang-Kashiwara crystals for the general-linear Lie superalgebra

class sage.combinat.crystals.bkk_crystals.CrystalOfBKKTableaux(ct, shape)

Bases: CrystalOfWords

Crystal of tableaux for type \(A(m|n)\).

This is an implementation of the tableaux model of the Benkart-Kang-Kashiwara crystal [BKK2000] for the Lie superalgebra \(\mathfrak{gl}(m+1,n+1)\).

INPUT:

• ct – a super Lie Cartan type of type \(A(m|n)\)

• shape – shape specifying the highest weight; this should be a partition contained in a hook of height \(n+1\) and width \(m+1\)

EXAMPLES:

Sage

sage: T = crystals.Tableaux(['A', [1,1]], shape = [2,1])
sage: T.cardinality()
20

Python

>>> from sage.all import *
>>> T = crystals.Tableaux(['A', [Integer(1),Integer(1)]], shape = [Integer(2),Integer(1)])
>>> T.cardinality()
20

Sage Live

T = crystals.Tableaux(['A', [1,1]], shape = [2,1])
T.cardinality()

class Element

Bases: CrystalOfBKKTableauxElement

genuine_highest_weight_vectors(index_set=None)

Return a tuple of genuine highest weight elements.

A fake highest weight vector is one which is annihilated by \(e_i\) for all \(i\) in the index set, but whose weight is not bigger in dominance order than all other elements in the crystal. A genuine highest weight vector is a highest weight element that is not fake.

EXAMPLES:

Sage

sage: B = crystals.Tableaux(['A', [1,1]], shape=[3,2,1])
sage: B.genuine_highest_weight_vectors()
([[-2, -2, -2], [-1, -1], [1]],)
sage: B.highest_weight_vectors()
([[-2, -2, -2], [-1, -1], [1]],
 [[-2, -2, -2], [-1, 2], [1]],
 [[-2, -2, 2], [-1, -1], [1]])

Python

>>> from sage.all import *
>>> B = crystals.Tableaux(['A', [Integer(1),Integer(1)]], shape=[Integer(3),Integer(2),Integer(1)])
>>> B.genuine_highest_weight_vectors()
([[-2, -2, -2], [-1, -1], [1]],)
>>> B.highest_weight_vectors()
([[-2, -2, -2], [-1, -1], [1]],
 [[-2, -2, -2], [-1, 2], [1]],
 [[-2, -2, 2], [-1, -1], [1]])

Sage Live

B = crystals.Tableaux(['A', [1,1]], shape=[3,2,1])
B.genuine_highest_weight_vectors()
B.highest_weight_vectors()

shape()

Return the shape of self.

EXAMPLES:

Sage

sage: T = crystals.Tableaux(['A', [1, 2]], shape=[2,1])
sage: T.shape()
[2, 1]

Python

>>> from sage.all import *
>>> T = crystals.Tableaux(['A', [Integer(1), Integer(2)]], shape=[Integer(2),Integer(1)])
>>> T.shape()
[2, 1]

Sage Live

T = crystals.Tableaux(['A', [1, 2]], shape=[2,1])
T.shape()

Make live