A tensor product of KirillovReshetikhinTableauxElement.
AUTHORS:
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.
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 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 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]]
Return the classical weight of self.
EXAMPLES:
sage: KRT = 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 = 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)
Pretty print self.
EXAMPLES:
sage: TPKRT = 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
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
and
, as well as
and
for general affine types.
INPUT:
OUTPUT:
The rigged configuration corresponding to self.
EXAMPLES:
Type example:
sage: KRT = 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 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 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
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: ret = KRT(tp_krc); ret
[[-1]] (X) [[2, 1], [-1, -1]]
sage: ret == elt
True