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

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

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

class Element
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]