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# Kirillov-Reshetikhin Tableaux

Kirillov-Reshetikhin tableaux are rectangular tableaux with \(r\) rows and \(s\) columns that naturally arise under the bijection between rigged configurations and tableaux [RigConBijection]. They are in bijection with the elements of the Kirillov-Reshetikhin crystal \(B^{r,s}\) under the (inverse) filling map [OSS13] [SchScr]. They do not have to satisfy the semistandard row or column restrictions. These tensor products are the result from the bijection from rigged configurations [RigConBijection].

For more information, see `KirillovReshetikhinTableaux` and `TensorProductOfKirillovReshetikhinTableaux`.

AUTHORS:

- Travis Scrimshaw (2012-01-03): Initial version
- Travis Scrimshaw (2012-11-14): Added bijection to KR crystals

REFERENCES:

[OSS13] | Masato Okado, Reiho Sakamoto, and Anne Schilling. Affine crystal structure on rigged configurations of type \(D_n^{(1)}\). J. Algebraic Combinatorics, 37 (2013). 571-599. Arxiv 1109.3523.

class `sage.combinat.rigged_configurations.kr_tableaux.``KRTableauxBn`(cartan_type, r, s)

  Bases: `sage.combinat.rigged_configurations.kr_tableaux.KRTableauxTypeHorizonal`

  Kirillov-Reshetkhin tableaux \(B^{n,s}\) of type \(B_n^{(1)}\).

  TESTS:

      sage: KRT = crystals.KirillovReshetikhin(['B', 2, 1], 2, 3, model='KR')
      sage: TestSuite(KRT).run()

  `Element`

    alias of `KRTableauxSpinElement`

  `from_kirillov_reshetikhin_crystal`(krc)

    Construct an element of `self` from the Kirillov-Reshetikhin crystal element `krc`.

    EXAMPLES:

        sage: KR = crystals.KirillovReshetikhin(['B',3,1], 3, 3, model='KR')
        sage: C = crystals.KirillovReshetikhin(['B',3,1], 3, 3, model='KN')
        sage: krc = C.module_generators[1].f_string([3,2,3,1,3,3]); krc
        [++-, [[2], [0], [-3]]]
        sage: KR.from_kirillov_reshetikhin_crystal(krc)
        [[1, 1, 2], [2, 2, -3], [-3, -3, -1]]

class `sage.combinat.rigged_configurations.kr_tableaux.``KRTableauxDTwistedSpin`(cartan_type, r, s)

  Bases: `sage.combinat.rigged_configurations.kr_tableaux.KRTableauxRectangle`

  Kirillov-Reshetikhin tableaux \(B^{r,s}\) of type \(D_n^{(2)}\) with \(r = n\).

  EXAMPLES:

      sage: KRT = crystals.KirillovReshetikhin(['D', 4, 2], 1, 1, model='KR')
      sage: KRT.cardinality()
      8
      sage: KRC = crystals.KirillovReshetikhin(['D', 4, 2], 1, 1, model='KN')
      sage: KRT.cardinality() == KRC.cardinality()
      True

  `Element`

    alias of `KRTableauxSpinElement`

class `sage.combinat.rigged_configurations.kr_tableaux.``KRTableauxRectangle`(cartan_type, r, s)

  Bases: `sage.combinat.rigged_configurations.kr_tableaux.KirillovReshetikhinTableaux`

  Kirillov-Reshetkhin tableaux \(B^{r,s}\) whose module generator is a single \(r \times s\) rectangle.

  These are Kirillov-Reshetkhin tableaux \(B^{r,s}\) of type:

  - \(A_n^{(1)}\) for all \(1 \leq r \leq n\),
  - \(C_n^{(1)}\) when \(r = n\).

  TESTS:

      sage: KRT = crystals.KirillovReshetikhin(['A', 3, 1], 2, 2, model='KR')
      sage: TestSuite(KRT).run()
      sage: KRT = crystals.KirillovReshetikhin(['C', 3, 1], 3, 2, model='KR')
      sage: TestSuite(KRT).run() # long time

  `from_kirillov_reshetikhin_crystal`(krc)

    Construct a `KirillovReshetikhinTableauxElement`.

    EXAMPLES:

        sage: KRT = crystals.KirillovReshetikhin(['A', 4, 1], 2, 1, model='KR')
        sage: C = crystals.KirillovReshetikhin(['A',4,1], 2, 1, model='KN')
        sage: krc = C(4,3); krc
        [[3], [4]]
        sage: KRT.from_kirillov_reshetikhin_crystal(krc)
        [[3], [4]]

class `sage.combinat.rigged_configurations.kr_tableaux.``KRTableauxSpin`(cartan_type, r, s)

  Bases: `sage.combinat.rigged_configurations.kr_tableaux.KRTableauxRectangle`

  Kirillov-Reshetikhin tableaux \(B^{r,s}\) of type \(D_n^{(1)}\) with \(r = n, n-1\).

  TESTS:

      sage: KRT = crystals.KirillovReshetikhin(['D', 4, 1], 3, 2, model='KR')
      sage: TestSuite(KRT).run()
      sage: KRT = crystals.KirillovReshetikhin(['D', 4, 1], 4, 2, model='KR')
      sage: TestSuite(KRT).run()

  `Element`

    alias of `KRTableauxSpinElement`

class `sage.combinat.rigged_configurations.kr_tableaux.``KRTableauxSpinElement`(parent, list, **options)

  Bases: `sage.combinat.rigged_configurations.kr_tableaux.KirillovReshetikhinTableauxElement`

  Kirillov-Reshetikhin tableau for spinors.

  Here we are in the embedding \(B(\Lambda_n) \hookrightarrow B(2 \Lambda_n)\), so \(e_i\) and \(f_i\) act by \(e_i^2\) and \(f_i^2\) respectively for all \(i \neq 0\). We do this so our columns are full width (as opposed to half width and/or uses a \(\pm\) representation).

  `e`(i)

    Calculate the action of \(e_i\) on `self`.

    EXAMPLES:

        sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 4, 1, model='KR')
        sage: KRT(-1, -4, 3, 2).e(1)
        [[1], [3], [-4], [-2]]
        sage: KRT(-1, -4, 3, 2).e(3)

  `epsilon`(i)

    Compute \(\varepsilon_i\) of `self`.

    EXAMPLES:

        sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 4, 1, model='KR')
        sage: KRT(-1, -4, 3, 2).epsilon(1)
        1
        sage: KRT(-1, -4, 3, 2).epsilon(3)
        0

  `f`(i)

    Calculate the action of \(f_i\) on `self`.

    EXAMPLES:

        sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 4, 1, model='KR')
        sage: KRT(-1, -4, 3, 2).f(1)
        sage: KRT(-1, -4, 3, 2).f(3)
        [[2], [4], [-3], [-1]]

  `left_split`()

    Return the image of `self` under the left column splitting map.

    EXAMPLES:

        sage: KRT = crystals.KirillovReshetikhin(['D', 4, 1], 4, 3, model='KR')
        sage: elt = KRT(-3,-4,2,1,-3,-4,2,1,-2,-4,3,1); elt.pp()
          1  1  1
          2  2  3
         -4 -4 -4
         -3 -3 -2
        sage: elt.left_split().pp()
          1 (X)   1  1
          2       2  3
         -4      -4 -4
         -3      -3 -2

  `phi`(i)

    Compute \(\varphi_i\) of `self`.

    EXAMPLES:

        sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 4, 1, model='KR')
        sage: KRT(-1, -4, 3, 2).phi(1)
        0
        sage: KRT(-1, -4, 3, 2).phi(3)
        1

  `to_array`(rows=True)

    Return a 2-dimensional array representation of this Kirillov-Reshetikhin element.

    If the output is in rows, then it outputs the top row first (in the English convention) from left to right.

    For example: if the reading word is \([2, 1, 4, 3]\), so as a \(2 \times 2\) tableau:

        1 3
        2 4

    we output `[[1, 3], [2, 4]]`.

    If the output is in columns, then it outputs the leftmost column first with the bottom element first. In other words this parses the reading word into its columns.

    Continuing with the previous example, the output would be `[[2, 1], [4, 3]]`.

    INPUT:

    - `rows` – (Default: `True`) Set to `True` if the resulting array is by row, otherwise it is by column.

    EXAMPLES:

        sage: KRT = crystals.KirillovReshetikhin(['D', 4, 1], 4, 3, model='KR')
        sage: elt = KRT(-3,-4,2,1,-3,-4,2,1,-2,-4,3,1)
        sage: elt.to_array()
        [[1, 1, 1], [2, 2, 3], [-4, -4, -4], [-3, -3, -2]]
        sage: elt.to_array(False)
        [[-3, -4, 2, 1], [-3, -4, 2, 1], [-2, -4, 3, 1]]

class `sage.combinat.rigged_configurations.kr_tableaux.``KRTableauxTypeBox`(cartan_type, r, s)

  Bases: `sage.combinat.rigged_configurations.kr_tableaux.KRTableauxTypeVertical`

  Kirillov-Reshetikhin tableaux \(B^{r,s}\) of type:

  - \(A_{2n}^{(2)}\) for all \(r \leq n\),
  - \(D_{n+1}^{(2)}\) for all \(r < n\).

  TESTS:

      sage: KRT = crystals.KirillovReshetikhin(['A', 4, 2], 2, 2, model='KR')
      sage: TestSuite(KRT).run()
      sage: KRT = crystals.KirillovReshetikhin(['D', 4, 2], 2, 2, model='KR')
      sage: TestSuite(KRT).run() # long time

class `sage.combinat.rigged_configurations.kr_tableaux.``KRTableauxTypeHorizonal`(cartan_type, r, s)

  Bases: `sage.combinat.rigged_configurations.kr_tableaux.KirillovReshetikhinTableaux`

  Kirillov-Reshetikhin tableaux \(B^{r,s}\) of type:

  - \(C_n^{(1)}\) for \(1 \leq r < n\),
  - \(A_{2n}^{(2)\dagger}\) for \(1 \leq r \leq n\).

  TESTS:

      sage: KRT = crystals.KirillovReshetikhin(['C', 3, 1], 2, 2, model='KR')
      sage: TestSuite(KRT).run() # long time
      sage: KRT = crystals.KirillovReshetikhin(CartanType(['A', 4, 2]).dual(), 2, 2, model='KR')
      sage: TestSuite(KRT).run()

  `from_kirillov_reshetikhin_crystal`(krc)

    Construct an element of `self` from the Kirillov-Reshetikhin crystal element `krc`.

    EXAMPLES:

        sage: KRT = crystals.KirillovReshetikhin(['C',4,1], 2, 3, model='KR')
        sage: C = crystals.KirillovReshetikhin(['C',4,1], 2, 3, model='KN')
        sage: krc = C(4,3); krc
        [[3], [4]]
        sage: KRT.from_kirillov_reshetikhin_crystal(krc)
        [[3, -2, 1], [4, -1, 2]]

class `sage.combinat.rigged_configurations.kr_tableaux.``KRTableauxTypeVertical`(cartan_type, r, s)

  Bases: `sage.combinat.rigged_configurations.kr_tableaux.KirillovReshetikhinTableaux`

  Kirillov-Reshetkihn tableaux \(B^{r,s}\) of type:

  - \(D_n^{(1)}\) for all \(1 \leq r < n-1\),
  - \(B_n^{(1)}\) for all \(1 \leq r < n\),
  - \(A_{2n-1}^{(2)}\) for all \(1 \leq r \leq n\).

  TESTS:

      sage: KRT = crystals.KirillovReshetikhin(['D', 4, 1], 1, 1, model='KR')
      sage: TestSuite(KRT).run()
      sage: KRT = crystals.KirillovReshetikhin(['B', 3, 1], 2, 2, model='KR')
      sage: TestSuite(KRT).run() # long time
      sage: KRT = crystals.KirillovReshetikhin(['A', 5, 2], 2, 2, model='KR')
      sage: TestSuite(KRT).run() # long time

  `from_kirillov_reshetikhin_crystal`(krc)

    Construct an element of `self` from the Kirillov-Reshetikhin crystal element `krc`.

    EXAMPLES:

        sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 2, 3, model='KR')
        sage: C = crystals.KirillovReshetikhin(['D',4,1], 2, 3, model='KN')
        sage: krc = C(4,3); krc
        [[3], [4]]
        sage: KRT.from_kirillov_reshetikhin_crystal(krc)
        [[3, -2, 1], [4, -1, 2]]

class `sage.combinat.rigged_configurations.kr_tableaux.``KirillovReshetikhinTableaux`(cartan_type, r, s)

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

  Kirillov-Reshetikhin tableaux.

  Kirillov-Reshetikhin tableaux are rectangular tableaux with \(r\) rows and \(s\) columns that naturally arise under the bijection between rigged configurations and tableaux [RigConBijection]. They are in bijection with the elements of the Kirillov-Reshetikhin crystal \(B^{r,s}\) under the (inverse) filling map.

  Whenever \(B^{r,s} \cong B(s\Lambda_r)\) as a classical crystal (which is the case for \(B^{r,s}\) in type \(A_n^{(1)}\), \(B^{n,s}\) in type \(C_n^{(1)}\) and \(D_{n+1}^{(2)}\), \(B^{n,s}\) and \(B^{n-1,s}\) in type \(D_n^{(1)}\)) then the filling map is trivial.

  For \(B^{r,s}\) in:

  - type \(D_n^{(1)}\) when \(r \leq n-2\),
  - type \(B_n^{(1)}\) when \(r < n\),
  - type \(A_{2n-1}^{(2)}\) for all \(r\),

  the filling map is defined in [OSS2011].

  For the spinor cases in type \(D_n^{(1)}\), the crystal \(B^{k,s}\) where \(k = n-1, n\), is isomorphic as a classical crystal to \(B(s\Lambda_k)\), and here we consider the Kirillov-Reshetikhin tableaux as living in \(B(2s \Lambda_k)\) under the natural doubling map. In this case, the crystal operators \(e_i\) and \(f_i\) act as \(e_i^2\) and \(f_i^2\) respectively. See [BijectionDn].

  For the spinor case in type \(B_n^{(1)}\), the crystal \(B^{n,s}\), we consider the images under the natural doubling map into \(B^{n,2s}\). The classical components of this crystal are now given by removing \(2 \times 2\) boxes. The filling map is the same as below (see the non-spin type \(C_n^{(1)}\)).

  For \(B^{r,s}\) in:

  - type \(C_n^{(1)}\) when \(r < n\),
  - type \(A_{2n}^{(2)\dagger}\) for all \(r\),

  the filling map is given as follows. Suppose we are considering the (classically) highest weight element in the classical component \(B(\lambda)\). Then we fill it in with the horizontal dominoes \([\bar{\imath}, i]\) in the \(i\)-th row from the top (in English notation) and reordering the columns so that they are increasing. Recall from above that \(B^{n,s} \cong B(s\Lambda_n)\) in type \(C^{(1)}_n\).

  For \(B^{r,s}\) in:

  - type \(A_{2n}^{(2)}\) for all \(r\),
  - type \(D_{n+1}^{(2)}\) when \(r < n\),

  the filling map is the same as given in [OSS2011] except for the rightmost column which is given by the column \([1, 2, \ldots, k, \emptyset, \ldots \emptyset]\) where \(k = (r+x-1)/2\) in Step 3 of [OSS2011].

  For the spinor case in type \(D_{n+1}^{(2)}\), the crystal \(B^{n,s}\), we define the filling map in the same way as in type \(D_n^{(1)}\).

  Note

  The filling map and classical decompositions in non-spinor cases can be classified by how the special node \(0\) connects with the corresponding classical diagram.

  The classical crystal stucture is given by the usual Kashiwara-Nakashima tableaux rules. That is to embed this into \(B(\Lambda_1)^{\otimes n s}\) by using the reading word and then applying the classical crystal operator. The affine crystal stucture is given by converting to the corresponding KR crystal element, performing the affine crystal operator, and pulling back to a KR tableau.

  For more information about the bijection between rigged configurations and tensor products of Kirillov-Reshetikhin tableaux, see `TensorProductOfKirillovReshetikhinTableaux`.

  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 \(D_n^{(1)}\).

  INPUT:

  - `cartan_type` – the Cartan type
  - `r` – the Dynkin diagram index (typically the number of rows)
  - `s` – the number of columns

  EXAMPLES:

      sage: KRT = crystals.KirillovReshetikhin(['A', 4, 1], 2, 1, model='KR')
      sage: elt = KRT(4, 3); elt
      [[3], [4]]

      sage: KRT = crystals.KirillovReshetikhin(['D', 4, 1], 2, 1, model='KR')
      sage: elt = KRT(-1, 1); elt
      [[1], [-1]]

  We can create highest weight crystals from a given shape or weight:

      sage: KRT = crystals.KirillovReshetikhin(['D', 4, 1], 2, 2, model='KR')
      sage: KRT.module_generator(shape=[1,1])
      [[1, 1], [2, -1]]
      sage: KRT.module_generator(column_shape=[2])
      [[1, 1], [2, -1]]
      sage: WS = RootSystem(['D',4,1]).weight_space()
      sage: KRT.module_generator(weight=WS.sum_of_terms([[0,-2],[2,1]]))
      [[1, 1], [2, -1]]
      sage: WSC = RootSystem(['D',4]).weight_space()
      sage: KRT.module_generator(classical_weight=WSC.fundamental_weight(2))
      [[1, 1], [2, -1]]

  We can go between `KashiwaraNakashimaTableaux()` and `KirillovReshetikhinTableaux` elements:

      sage: KRCrys = crystals.KirillovReshetikhin(['D', 4, 1], 2, 2, model='KN')
      sage: KRTab = crystals.KirillovReshetikhin(['D', 4, 1], 2, 2, model='KR')
      sage: elt = KRCrys(3, 2); elt
      [[2], [3]]
      sage: k = KRTab(elt); k
      [[2, 1], [3, -1]]
      sage: KRCrys(k)
      [[2], [3]]

  We check that the classical weights in the classical decompositions agree in a few different type:

      sage: KRCrys = crystals.KirillovReshetikhin(['D', 4, 1], 2, 2, model='KN')
      sage: KRTab = crystals.KirillovReshetikhin(['D', 4, 1], 2, 2, model='KR')
      sage: all(t.classical_weight() == KRCrys(t).classical_weight() for t in KRTab)
      True
      sage: KRCrys = crystals.KirillovReshetikhin(['B', 3, 1], 2, 2, model='KN')
      sage: KRTab = crystals.KirillovReshetikhin(['B', 3, 1], 2, 2, model='KR')
      sage: all(t.classical_weight() == KRCrys(t).classical_weight() for t in KRTab)
      True
      sage: KRCrys = crystals.KirillovReshetikhin(['C', 3, 1], 2, 2, model='KN')
      sage: KRTab = crystals.KirillovReshetikhin(['C', 3, 1], 2, 2, model='KR')
      sage: all(t.classical_weight() == KRCrys(t).classical_weight() for t in KRTab)
      True
      sage: KRCrys = crystals.KirillovReshetikhin(['D', 4, 2], 2, 2, model='KN')
      sage: KRTab = crystals.KirillovReshetikhin(['D', 4, 2], 2, 2, model='KR')
      sage: all(t.classical_weight() == KRCrys(t).classical_weight() for t in KRTab)
      True
      sage: KRCrys = crystals.KirillovReshetikhin(['A', 4, 2], 2, 2, model='KN')
      sage: KRTab = crystals.KirillovReshetikhin(['A', 4, 2], 2, 2, model='KR')
      sage: all(t.classical_weight() == KRCrys(t).classical_weight() for t in KRTab)
      True

  `Element`

    alias of `KirillovReshetikhinTableauxElement`

  `classical_decomposition`()

    Return the classical crystal decomposition of `self`.

    EXAMPLES:

        sage: crystals.KirillovReshetikhin(['D', 4, 1], 2, 2, model='KR').classical_decomposition()
        The crystal of tableaux of type ['D', 4] and shape(s) [[], [1, 1], [2, 2]]

  `from_kirillov_reshetikhin_crystal`(krc)

    Construct an element of `self` from the Kirillov-Reshetikhin crystal element `krc`.

    EXAMPLES:

        sage: KRT = crystals.KirillovReshetikhin(['A', 4, 1], 2, 1, model='KR')
        sage: C = crystals.KirillovReshetikhin(['A',4,1], 2, 1, model='KN')
        sage: krc = C(4,3); krc
        [[3], [4]]
        sage: KRT.from_kirillov_reshetikhin_crystal(krc)
        [[3], [4]]

  `kirillov_reshetikhin_crystal`()

    Return the corresponding KR crystal in the `Kashiwara-Nakashima model`.

    EXAMPLES:

        sage: crystals.KirillovReshetikhin(['A', 4, 1], 2, 1, model='KR').kirillov_reshetikhin_crystal()
        Kirillov-Reshetikhin crystal of type ['A', 4, 1] with (r,s)=(2,1)

  `module_generator`(i=None, **options)

    Return the specified module generator.

    INPUT:

    - `i` – The index of the module generator

    We can also get a module generator by using one of the following optional arguments:

    - `shape` – The associated shape
    - `column_shape` – The shape given as columns (a column of length \(k\) correspond to a classical weight \(\omega_k\))
    - `weight` – The weight
    - `classical_weight` – The classical weight

    EXAMPLES:

        sage: KRT = crystals.KirillovReshetikhin(['D', 4, 1], 2, 2, model='KR')
        sage: KRT.module_generator(1)
        [[1, 1], [2, -1]]
        sage: KRT.module_generator(shape=[1,1])
        [[1, 1], [2, -1]]
        sage: KRT.module_generator(column_shape=[2])
        [[1, 1], [2, -1]]
        sage: WS = RootSystem(['D',4,1]).weight_space()
        sage: KRT.module_generator(weight=WS.sum_of_terms([[0,-2],[2,1]]))
        [[1, 1], [2, -1]]
        sage: WSC = RootSystem(['D',4]).weight_space()
        sage: KRT.module_generator(classical_weight=WSC.fundamental_weight(2))
        [[1, 1], [2, -1]]

        sage: KRT = crystals.KirillovReshetikhin(['A', 3, 1], 2, 2, model='KR')
        sage: KRT.module_generator()
        [[1, 1], [2, 2]]

  `r`()

    Return the value \(r\) for this tableaux class which corresponds to the number of rows.

    EXAMPLES:

        sage: KRT = crystals.KirillovReshetikhin(['A', 4, 1], 2, 1, model='KR')
        sage: KRT.r()
        2

  `s`()

    Return the value \(s\) for this tableaux class which corresponds to the number of columns.

    EXAMPLES:

        sage: KRT = crystals.KirillovReshetikhin(['A', 4, 1], 2, 1, model='KR')
        sage: KRT.s()
        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: K = crystals.KirillovReshetikhin(['A', 3, 1], 2, 2, model='KR')
        sage: TP = crystals.TensorProductOfKirillovReshetikhinTableaux(['A', 3, 1], [[1,3],[3,1]])
        sage: K.tensor(TP, K)
        Tensor product of Kirillov-Reshetikhin tableaux of type ['A', 3, 1]
         and factor(s) ((2, 2), (1, 3), (3, 1), (2, 2))

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

class `sage.combinat.rigged_configurations.kr_tableaux.``KirillovReshetikhinTableauxElement`(parent, list, **options)

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

  A Kirillov-Reshetikhin tableau.

  For more information, see `KirillovReshetikhinTableaux` and `TensorProductOfKirillovReshetikhinTableaux`.

  `classical_weight`()

    Return the classical weight of `self`.

    EXAMPLES:

        sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 2, 2, model='KR')
        sage: elt = KRT(3,2,-1,1); elt
        [[2, 1], [3, -1]]
        sage: elt.classical_weight()
        (0, 1, 1, 0)

  `e`(i)

    Perform the action of \(e_i\) on `self`.

    Todo

    Implement a direct action of \(e_0\) without moving to KR crystals.

    EXAMPLES:

        sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 2, 2, model='KR')
        sage: KRT.module_generators[0].e(0)
        [[-2, 1], [-1, -1]]

  `epsilon`(i)

    Compute \(\epsilon_i\) of `self`.

    Todo

    Implement a direct action of \(\epsilon_0\) without moving to KR crystals.

    EXAMPLES:

        sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 2, 2, model='KR')
        sage: KRT.module_generators[0].epsilon(0)
        2

  `f`(i)

    Perform the action of \(f_i\) on `self`.

    Todo

    Implement a direct action of \(f_0\) without moving to KR crystals.

    EXAMPLES:

        sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 2, 2, model='KR')
        sage: KRT.module_generators[0].f(0)
        [[1, 1], [2, -1]]

  `left_split`()

    Return the image of `self` under the left column splitting map.

    EXAMPLES:

        sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 2, 3, model='KR')
        sage: mg = KRT.module_generators[1]; mg.pp()
          1 -2  1
          2 -1  2
        sage: ls = mg.left_split(); ls.pp()
          1 (X)  -2  1
          2      -1  2
        sage: ls.parent()
        Tensor product of Kirillov-Reshetikhin tableaux of type ['D', 4, 1] and factor(s) ((2, 1), (2, 2))

  `lusztig_involution`()

    Return the result of the classical Lusztig involution on `self`.

    EXAMPLES:

        sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 2, 3, model='KR')
        sage: mg = KRT.module_generators[1]
        sage: mg.lusztig_involution()
        [[-2, -2, 1], [-1, -1, 2]]
        sage: elt = mg.f_string([2,1,3,2]); elt
        [[3, -2, 1], [4, -1, 2]]
        sage: elt.lusztig_involution()
        [[-4, -2, 1], [-3, -1, 2]]

  `phi`(i)

    Compute \(\phi_i\) of `self`.

    Todo

    Compute \(\phi_0\) without moving to KR crystals.

    EXAMPLES:

        sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 2, 2, model='KR')
        sage: KRT.module_generators[0].phi(0)
        2

  `pp`()

    Pretty print `self`.

    EXAMPLES:

        sage: KRT = crystals.KirillovReshetikhin(['A', 4, 1], 2, 2, model='KR')
        sage: elt = KRT(2, 1, 4, 3); elt
        [[1, 3], [2, 4]]
        sage: elt.pp()
        1  3
        2  4

  `right_split`()

    Return the image of `self` under the right column splitting map.

    Let \(\ast\) denote the `Lusztig involution`, and \(\mathrm{ls}\) as the `left splitting map`. The right splitting map is defined as \(\mathrm{rs} := \ast \circ \mathrm{ls} \circ \ast\).

    EXAMPLES:

        sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 2, 3, model='KR')
        sage: mg = KRT.module_generators[1]; mg.pp()
          1 -2  1
          2 -1  2
        sage: ls = mg.right_split(); ls.pp()
         -2  1 (X)   1
         -1  2       2
        sage: ls.parent()
        Tensor product of Kirillov-Reshetikhin tableaux of type ['D', 4, 1] and factor(s) ((2, 2), (2, 1))

  `to_array`(rows=True)

    Return a 2-dimensional array representation of this Kirillov-Reshetikhin element.

    If the output is in rows, then it outputs the top row first (in the English convention) from left to right.

    For example: if the reading word is \([2, 1, 4, 3]\), so as a \(2 \times 2\) tableau:

        1 3
        2 4

    we output `[[1, 3], [2, 4]]`.

    If the output is in columns, then it outputs the leftmost column first with the bottom element first. In other words this parses the reading word into its columns.

    Continuing with the previous example, the output would be `[[2, 1], [4, 3]]`.

    INPUT:

    - `rows` – (Default: `True`) Set to `True` if the resulting array is by row, otherwise it is by column.

    EXAMPLES:

        sage: KRT = crystals.KirillovReshetikhin(['A', 4, 1], 2, 2, model='KR')
        sage: elt = KRT(2, 1, 4, 3)
        sage: elt.to_array()
        [[1, 3], [2, 4]]
        sage: elt.to_array(False)
        [[2, 1], [4, 3]]

  `to_classical_highest_weight`(index_set=None)

    Return the classical highest weight element corresponding to `self`.

    INPUT:

    - `index_set` – (Default: `None`) Return the highest weight with respect to the index set. If `None` is passed in, then this uses the classical index set.

    OUTPUT:

    A pair `[H, f_str]` where `H` is the highest weight element and `f_str` is a list of \(a_i\) of \(f_{a_i}\) needed to reach `H`.

    EXAMPLES:

        sage: KRTab = crystals.KirillovReshetikhin(['D',4,1], 2, 2, model='KR')
        sage: elt = KRTab(3,2,-1,1); elt
        [[2, 1], [3, -1]]
        sage: elt.to_classical_highest_weight()
        [[[1, 1], [2, -1]], [1, 2]]

  `to_kirillov_reshetikhin_crystal`()

    Construct a `KashiwaraNakashimaTableaux()` element from `self`.

    We construct the Kirillov-Reshetikhin crystal element as follows:

    1. Determine the shape \(\lambda\) of the KR crystal from the weight.
    2. Determine a path \(e_{i_1} e_{i_2} \cdots e_{i_k}\) to the highest weight.
    3. Apply \(f_{i_k} \cdots f_{i_2} f_{i_1}\) to a highest weight KR crystal of shape \(\lambda\).

    EXAMPLES:

        sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 2, 2, model='KR')
        sage: elt = KRT(3,2,-1,1); elt
        [[2, 1], [3, -1]]
        sage: elt.to_kirillov_reshetikhin_crystal()
        [[2], [3]]

    TESTS:

    Spinor tests:

        sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 4, 3, model='KR')
        sage: KRC = crystals.KirillovReshetikhin(['D',4,1], 4, 3, model='KN')
        sage: elt = KRT(-3,-4,2,1,-3,-4,2,1,-2,-4,3,1); elt
        [[1, 1, 1], [2, 2, 3], [-4, -4, -4], [-3, -3, -2]]
        sage: ret = elt.to_kirillov_reshetikhin_crystal(); ret
        [++--, [[1], [3], [-4], [-3]]]
        sage: test = KRT(ret); test
        [[1, 1, 1], [2, 2, 3], [-4, -4, -4], [-3, -3, -2]]
        sage: test == elt
        True

  `to_tableau`()

    Return a `Tableau` object of `self`.

    EXAMPLES:

        sage: KRT = crystals.KirillovReshetikhin(['A', 4, 1], 2, 2, model='KR')
        sage: elt = KRT(2, 1, 4, 3); elt
        [[1, 3], [2, 4]]
        sage: t = elt.to_tableau(); t
        [[1, 3], [2, 4]]
        sage: type(t)
        <class 'sage.combinat.tableau.Tableaux_all_with_category.element_class'>

  `weight`()

    Return the weight of `self`.

    EXAMPLES:

        sage: KR = crystals.KirillovReshetikhin(['D',4,1], 2, 2, model='KR')
        sage: KR.module_generators[1].weight()
        -2*Lambda[0] + Lambda[2]

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