### Navigation

- index
- modules |
- next |
- previous |
- Combinatorics »
- Rigged Configurations »

# Tensor Product of Kirillov-Reshetikhin Tableaux

A tensor product of `KirillovReshetikhinTableaux` which are tableaux of \(r\) rows and \(s\) columns which naturally arise in the bijection between rigged configurations and tableaux and which are in bijection with the elements of the Kirillov-Reshetikhin crystal \(B^{r,s}\), see `KirillovReshetikhinCrystal()`.

AUTHORS:

- Travis Scrimshaw (2010-09-26): Initial version

EXAMPLES:

Type \(A_n^{(1)}\) examples:

    sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['A',3,1], [[3,1], [2,1]])
    sage: KRT
    Tensor product of Kirillov-Reshetikhin tableaux of type ['A', 3, 1] and factor(s) ((3, 1), (2, 1))
    sage: KRT.cardinality()
    24
    sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['A',3,1], [[1,1], [2,1], [3,1]])
    sage: KRT
    Tensor product of Kirillov-Reshetikhin tableaux of type ['A', 3, 1] and factor(s) ((1, 1), (2, 1), (3, 1))
    sage: len(KRT.module_generators)
    5
    sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['A',3,1], [[1,1], [2,1], [3,1]])
    sage: KRT.cardinality()
    96

Type \(D_n^{(1)}\) examples:

    sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['D', 4, 1], [[1, 1], [2, 1], [1, 1]])
    sage: KRT
    Tensor product of Kirillov-Reshetikhin tableaux of type ['D', 4, 1] and factor(s) ((1, 1), (2, 1), (1, 1))
    sage: T = KRT(pathlist=[[1], [-2, 2], [1]])
    sage: T
    [[1]] (X) [[2], [-2]] (X) [[1]]
    sage: T2 = KRT(pathlist=[[1], [2, -2], [1]])
    sage: T2
    [[1]] (X) [[-2], [2]] (X) [[1]]
    sage: T == T2
    False

class `sage.combinat.rigged_configurations.tensor_product_kr_tableaux.``HighestWeightTensorKRT`(tp_krt)

  Bases: `sage.structure.unique_representation.UniqueRepresentation`

  Class so we do not have to build the module generators for `TensorProductOfKirillovReshetikhinTableaux` at initialization.

  Warning

  This class is for internal use only!

  `cardinality`()

    Return the cardinality of `self` which is the number of highest weight elements.

    EXAMPLES:

        sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['D',4,1], [[2,2]])
        sage: from sage.combinat.rigged_configurations.tensor_product_kr_tableaux import HighestWeightTensorKRT
        sage: HW = HighestWeightTensorKRT(KRT)
        sage: HW.cardinality()
        3
        sage: len(HW)
        3
        sage: len(KRT.module_generators)
        3

`sage.combinat.rigged_configurations.tensor_product_kr_tableaux.``HighestWeightTensorProductOfKirillovReshetikhinTableaux`(cartan_type, B)

  Deprecated in trac ticket #13872. Use instead the attribute `module_generators` of `TensorProductOfKirillovReshetikhinTableaux`.

  EXAMPLES:

      sage: HighestWeightTensorProductOfKirillovReshetikhinTableaux(['A',2,1], [[1,1]])
      doctest:...: DeprecationWarning: this class is deprecated.
       Use TensorProductOfKirillovReshetikhinTableaux(cartan_type, B).module_generators instead
      See http://trac.sagemath.org/13872 for details.
      Highest weight elements of Tensor product of Kirillov-Reshetikhin tableaux of type ['A', 2, 1] and factor(s) ((1, 1),)

class `sage.combinat.rigged_configurations.tensor_product_kr_tableaux.``TensorProductOfKirillovReshetikhinTableaux`(cartan_type, B)

  Bases: `sage.combinat.crystals.tensor_product.FullTensorProductOfRegularCrystals`

  A tensor product of `KirillovReshetikhinTableaux`.

  Through the bijection with rigged configurations, the tableaux that are produced in all nonexceptional types are all of rectangular shapes and do not necessarily obey the usual strict increase in columns and weak increase in rows. The relation between the elements of the Kirillov-Reshetikhin crystal, given by the Kashiwara-Nakashima tableaux, and the Kirillov-Reshetikhin tableaux is given by a filling map.

  Note

  The tableaux for all non-simply-laced types are provably correct if the bijection with `rigged configurations` holds. Therefore this is currently only proven for \(B^{r,1}\) or \(B^{1,s}\) and in general for types \(A_n^{(1)}\) and \(D_n^{(1)}\).

  For more information see [OSS2011] and `KirillovReshetikhinTableaux`.

  For more information on KR crystals, see `sage.combinat.crystals.kirillov_reshetikhin`.

  INPUT:

  - `cartan_type` – a Cartan type
  - `B` – an (ordered) list of pairs \((r,s)\) which give the dimension of a rectangle with \(r\) rows and \(s\) columns and corresponds to a Kirillov-Reshetikhin tableaux factor of \(B^{r,s}\).

  REFERENCES:

  [OSS2011] | Masato Okado, Reiho Sakamoto, Anne Schilling, Affine crystal structure on rigged configurations of type \(D_n^{(1)}\), J. Algebraic Combinatorics 37(3) (2013) 571-599 (Arxiv 1109.3523 [math.QA])

  EXAMPLES:

  We can go between tensor products of KR crystals and rigged configurations:

      sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['A',3,1], [[3,1],[2,2]])
      sage: tp_krt = KRT(pathlist=[[3,2,1],[3,2,3,2]]); tp_krt
      [[1], [2], [3]] (X) [[2, 2], [3, 3]]
      sage: RC = RiggedConfigurations(['A',3,1], [[3,1],[2,2]])
      sage: rc_elt = tp_krt.to_rigged_configuration(); rc_elt
      <BLANKLINE>
      -2[ ][ ]-2
      <BLANKLINE>
      0[ ][ ]0
      <BLANKLINE>
      (/)
      <BLANKLINE>
      sage: tp_krc = tp_krt.to_tensor_product_of_kirillov_reshetikhin_crystals(); tp_krc
      [[[1], [2], [3]], [[2, 2], [3, 3]]]
      sage: KRT(tp_krc) == tp_krt
      True
      sage: rc_elt == tp_krt.to_rigged_configuration()
      True
      sage: KR1 = crystals.KirillovReshetikhin(['A',3,1], 3,1)
      sage: KR2 = crystals.KirillovReshetikhin(['A',3,1], 2,2)
      sage: T = crystals.TensorProduct(KR1, KR2)
      sage: t = T(KR1(3,2,1), KR2(3,2,3,2))
      sage: KRT(t) == tp_krt
      True
      sage: t == tp_krc
      True

  We can get the highest weight elements by using the attribute `module_generators`:

      sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['A',3,1], [[3,1], [2,1]])
      sage: list(KRT.module_generators)
      [[[1], [2], [3]] (X) [[1], [2]], [[1], [3], [4]] (X) [[1], [2]]]

  To create elements directly (i.e. not passing in KR tableaux elements), there is the pathlist option will receive a list of lists which contain the reversed far-eastern reading word of the tableau. That is to say, in English notation, the word obtain from reading bottom-to-top, left-to-right.

      sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['A',3,1], [[3,2], [1,2], [2,1]])
      sage: elt = KRT(pathlist=[[3, 2, 1, 4, 2, 1], [1, 3], [3, 1]])
      sage: elt.pp()
        1  1 (X)   1  3 (X)   1
        2  2                  3
        3  4

  One can still create elements in the same way as tensor product of crystals:

      sage: K1 = crystals.KirillovReshetikhin(['A',3,1], 3, 2, model='KR')
      sage: K2 = crystals.KirillovReshetikhin(['A',3,1], 1, 2, model='KR')
      sage: K3 = crystals.KirillovReshetikhin(['A',3,1], 2, 1, model='KR')
      sage: eltlong = KRT(K1(3, 2, 1, 4, 2, 1), K2(1, 3), K3(3, 1))
      sage: eltlong == elt
      True

  `Element`

    alias of `TensorProductOfKirillovReshetikhinTableauxElement`

  `rigged_configurations`()

    Return the corresponding set of rigged configurations.

    EXAMPLES:

        sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['A',3,1], [[1,3], [2,1]])
        sage: KRT.rigged_configurations()
        Rigged configurations of type ['A', 3, 1] and factor(s) ((1, 3), (2, 1))

  `tensor`(*crystals, **options)

    Return the tensor product of `self` with `crystals`.

    If `crystals` is a list of (a tensor product of) KR tableaux, this returns a `TensorProductOfKirillovReshetikhinTableaux`.

    EXAMPLES:

        sage: TP = crystals.TensorProductOfKirillovReshetikhinTableaux(['A', 3, 1], [[1,3],[3,1]])
        sage: K = crystals.KirillovReshetikhin(['A', 3, 1], 2, 2, model='KR')
        sage: TP.tensor(K, TP)
        Tensor product of Kirillov-Reshetikhin tableaux of type ['A', 3, 1]
         and factor(s) ((1, 3), (3, 1), (2, 2), (1, 3), (3, 1))

        sage: C = crystals.KirillovReshetikhin(['A',3,1], 3, 1, model='KN')
        sage: TP.tensor(K, C)
        Full tensor product of the crystals
         [Kirillov-Reshetikhin tableaux of type ['A', 3, 1] and shape (1, 3),
          Kirillov-Reshetikhin tableaux of type ['A', 3, 1] and shape (3, 1),
          Kirillov-Reshetikhin tableaux of type ['A', 3, 1] and shape (2, 2),
          Kirillov-Reshetikhin crystal of type ['A', 3, 1] with (r,s)=(3,1)]

  `tensor_product_of_kirillov_reshetikhin_crystals`()

    Return the corresponding tensor product of Kirillov-Reshetikhin crystals.

    EXAMPLES:

        sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['A',3,1], [[3,1],[2,2]])
        sage: KRT.tensor_product_of_kirillov_reshetikhin_crystals()
        Full tensor product of the crystals [Kirillov-Reshetikhin crystal of type ['A', 3, 1] with (r,s)=(3,1),
        Kirillov-Reshetikhin crystal of type ['A', 3, 1] with (r,s)=(2,2)]

    TESTS:

        sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['D', 4, 1], [[4,1], [3,3]])
        sage: KR1 = crystals.KirillovReshetikhin(['D', 4, 1], 4, 1)
        sage: KR2 = crystals.KirillovReshetikhin(['D', 4, 1], 3, 3)
        sage: T = crystals.TensorProduct(KR1, KR2)
        sage: T == KRT.tensor_product_of_kirillov_reshetikhin_crystals()
        True
        sage: T is KRT.tensor_product_of_kirillov_reshetikhin_crystals()
        True

### Navigation

- index
- modules |
- next |
- previous |
- Combinatorics »
- Rigged Configurations »