Benkart-Kang-Kashiwara crystals for the general-linear Lie superalgebra¶
- class sage.combinat.crystals.bkk_crystals.CrystalOfBKKTableaux(ct, shape)[source]¶
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: T = crystals.Tableaux(['A', [1,1]], shape = [2,1]) sage: T.cardinality() 20
>>> from sage.all import * >>> T = crystals.Tableaux(['A', [Integer(1),Integer(1)]], shape = [Integer(2),Integer(1)]) >>> T.cardinality() 20
- class Element[source]¶
Bases:
CrystalOfBKKTableauxElement
- genuine_highest_weight_vectors(index_set=None)[source]¶
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]])
>>> 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]])