An element of a tensor product of Kirillov-Reshetikhin tableaux

A tensor product of KirillovReshetikhinTableauxElement.

AUTHORS:

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

EXAMPLES:

Type A_n^{(1)} examples:

sage: KRT = 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 = 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 = 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]]
class sage.combinat.rigged_configurations.tensor_product_kr_tableaux_element.TensorProductOfKirillovReshetikhinTableauxElement(parent, *path, **options)

Bases: sage.combinat.crystals.tensor_product.TensorProductOfRegularCrystalsElement

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

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

e(i)

Return the action of e_i on self.

EXAMPLES:

sage: KRT = TensorProductOfKirillovReshetikhinTableaux(['D', 4, 1], [[2,1]])
sage: T = KRT(pathlist=[[4,3]])
sage: T.e(1)
sage: T.e(2)
[[2], [4]]
f(i)

Return the action of f_i on self.

EXAMPLES:

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

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

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

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