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

Skip to content
Back to top
View this page
Edit this page

# 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

  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]])

  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]