Tensor Product of Kirillov-Reshetikhin Tableaux Elements

A tensor product of KirillovReshetikhinTableauxElement.

AUTHORS:

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

class sage.combinat.rigged_configurations.tensor_product_kr_tableaux_element.TensorProductOfKirillovReshetikhinTableauxElement(parent, list=[[]], **options)

Bases: TensorProductOfRegularCrystalsElement

An element in a tensor product of Kirillov-Reshetikhin tableaux.

For more on tensor product of Kirillov-Reshetikhin tableaux, see TensorProductOfKirillovReshetikhinTableaux.

The most common way to construct an element is to specify the option pathlist which is a list of lists which will be used to generate the individual factors of KirillovReshetikhinTableauxElement.

EXAMPLES:

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

sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['A', 3, 1], [[1,1], [2,1], [1,1], [2,1], [2,1], [2,1]])
sage: T = KRT(pathlist=[[2], [4,1], [3], [4,2], [3,1], [2,1]])
sage: T
[[2]] (X) [[1], [4]] (X) [[3]] (X) [[2], [4]] (X) [[1], [3]] (X) [[1], [2]]
sage: T.to_rigged_configuration()

0[ ][ ]0
1[ ]1

1[ ][ ]0
1[ ]0
1[ ]0

0[ ][ ]0

sage: T = KRT(pathlist=[[1], [2,1], [1], [4,1], [3,1], [2,1]])
sage: T
[[1]] (X) [[1], [2]] (X) [[1]] (X) [[1], [4]] (X) [[1], [3]] (X) [[1], [2]]
sage: T.to_rigged_configuration()

(/)

1[ ]0
1[ ]0

0[ ]0

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

sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['D', 4, 1], [[1,1], [1,1], [1,1], [1,1]])
sage: T = KRT(pathlist=[[-1], [-1], [1], [1]])
sage: T
[[-1]] (X) [[-1]] (X) [[1]] (X) [[1]]
sage: T.to_rigged_configuration()

0[ ][ ]0
0[ ][ ]0

0[ ][ ]0
0[ ][ ]0

0[ ][ ]0

0[ ][ ]0

sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['D', 4, 1], [[2,1], [1,1], [1,1], [1,1]])
sage: T = KRT(pathlist=[[3,2], [1], [-1], [1]])
sage: T
[[2], [3]] (X) [[1]] (X) [[-1]] (X) [[1]]
sage: T.to_rigged_configuration()

0[ ]0
0[ ]0
0[ ]0

0[ ]0
0[ ]0
0[ ]0

1[ ]0

1[ ]0

sage: T.to_rigged_configuration().to_tensor_product_of_kirillov_reshetikhin_tableaux()
[[2], [3]] (X) [[1]] (X) [[-1]] (X) [[1]]

classical_weight()

Return the classical weight of self.

EXAMPLES:

sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['D',4,1], [[2,2]])
sage: elt = KRT(pathlist=[[3,2,-1,1]]); elt
[[2, 1], [3, -1]]
sage: elt.classical_weight()
(0, 1, 1, 0)
sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['A',3,1], [[2,2],[1,3]])
sage: elt = KRT(pathlist=[[2,1,3,2],[1,4,4]]); elt
[[1, 2], [2, 3]] (X) [[1, 4, 4]]
sage: elt.classical_weight()
(2, 2, 1, 2)

left_split()

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

EXAMPLES:

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

lusztig_involution()

Return the result of the classical Lusztig involution on self.

EXAMPLES:

sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['A',3,1], [[2,2],[1,3]])
sage: elt = KRT(pathlist=[[2,1,3,2],[1,4,4]])
sage: li = elt.lusztig_involution(); li
[[1, 1, 4]] (X) [[2, 3], [3, 4]]
sage: li.parent()
Tensor product of Kirillov-Reshetikhin tableaux of type ['A', 3, 1] and factor(s) ((1, 3), (2, 2))

pp()

Pretty print self.

EXAMPLES:

sage: TPKRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['A',4,1], [[2,2],[3,1],[3,3]])
sage: TPKRT.module_generators[0].pp()
  1  1 (X)   1 (X)   1  1  1
  2  2       2       2  2  2
             3       3  3  3

right_split()

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

EXAMPLES:

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

Let \(\ast\) denote the Lusztig involution, we check that \(\ast \circ \mathrm{ls} \circ \ast = \mathrm{rs}\):

sage: all(x.lusztig_involution().left_split().lusztig_involution() == x.right_split() for x in KRT)
True

to_rigged_configuration(display_steps=False)

Perform the bijection from self to a rigged configuration which is described in [RigConBijection], [BijectionLRT], and [BijectionDn].

Note

This is only proven to be a bijection in types \(A_n^{(1)}\) and \(D_n^{(1)}\), as well as \(\bigotimes_i B^{r_i,1}\) and \(\bigotimes_i B^{1,s_i}\) for general affine types.

INPUT:

• display_steps – (default: False) Boolean which indicates if we want to output each step in the algorithm.

OUTPUT:

The rigged configuration corresponding to self.

EXAMPLES:

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

sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['A', 3, 1], [[2,1], [2,1], [2,1]])
sage: T = KRT(pathlist=[[4, 2], [3, 1], [2, 1]])
sage: T
[[2], [4]] (X) [[1], [3]] (X) [[1], [2]]
sage: T.to_rigged_configuration()

0[ ]0

1[ ]1
1[ ]0

0[ ]0

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

sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['D', 4, 1], [[2,2]])
sage: T = KRT(pathlist=[[2,1,4,3]])
sage: T
[[1, 3], [2, 4]]
sage: T.to_rigged_configuration()

0[ ]0

-1[ ]-1
-1[ ]-1

0[ ]0

(/)

Type \(D_n^{(1)}\) spinor example:

sage: CP = crystals.TensorProductOfKirillovReshetikhinTableaux(['D', 5, 1], [[5,1],[2,1],[1,1],[1,1],[1,1]])
sage: elt = CP(pathlist=[[-2,-5,4,3,1],[-1,2],[1],[1],[1]])
sage: elt
[[1], [3], [4], [-5], [-2]] (X) [[2], [-1]] (X) [[1]] (X) [[1]] (X) [[1]]
sage: elt.to_rigged_configuration()

2[ ][ ]1

0[ ][ ]0
0[ ]0

0[ ][ ]0
0[ ]0

0[ ]0

0[ ][ ]0

This is invertible by calling to_tensor_product_of_kirillov_reshetikhin_tableaux():

sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['D', 4, 1], [[2,2]])
sage: T = KRT(pathlist=[[2,1,4,3]])
sage: rc = T.to_rigged_configuration()
sage: ret = rc.to_tensor_product_of_kirillov_reshetikhin_tableaux(); ret
[[1, 3], [2, 4]]
sage: ret == T
True

to_tensor_product_of_kirillov_reshetikhin_crystals()

Return a tensor product of Kirillov-Reshetikhin crystals corresponding to self.

This works by performing the filling map on each individual factor. For more on the filling map, see KirillovReshetikhinTableaux.

EXAMPLES:

sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['D',4,1], [[1,1],[2,2]])
sage: elt = KRT(pathlist=[[-1],[-1,2,-1,1]]); elt
[[-1]] (X) [[2, 1], [-1, -1]]
sage: tp_krc = elt.to_tensor_product_of_kirillov_reshetikhin_crystals(); tp_krc
[[[-1]], [[2], [-1]]]

We can recover the original tensor product of KR tableaux:

sage: ret = KRT(tp_krc); ret
[[-1]] (X) [[2, 1], [-1, -1]]
sage: ret == elt
True