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 $gl (m + 1, n + 1)$ .

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

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: 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]])

shape()#

Return the shape of self.

EXAMPLES:

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