# 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. 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].

AUTHORS:

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

EXAMPLES:

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

sage: KRT = TensorProductOfKirillovReshetikhinTableaux(['A',3,1], [[3,1], [2,1]])
sage: KRT
Tensor product of Kirillov-Reshetikhin tableaux of type ['A', 3, 1] and tableau shape(s) [[1, 1, 1], [1, 1]]
sage: KRT.cardinality()
24
sage: HW = HighestWeightTensorProductOfKirillovReshetikhinTableaux(['A',3,1], [[1,1], [2,1], [3,1]])
sage: HW
Highest weight tensor product of Kirillov-Reshetikhin tableaux of type ['A', 3, 1] and tableau shape(s) [[1], [1, 1], [1, 1, 1]]
sage: HW.cardinality()
5
sage: len(HW)
5
sage: KRT = TensorProductOfKirillovReshetikhinTableaux(['A',3,1], [[1,1], [2,1], [3,1]])
sage: KRT.cardinality()
96


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

sage: KRT = TensorProductOfKirillovReshetikhinTableaux(['D', 4, 1], [[1, 1], [2, 1], [1, 1]])
sage: KRT
Tensor product of Kirillov-Reshetikhin tableaux of type ['D', 4, 1] and tableau shape(s) [[1], [1, 1], [1]]
sage: HW = HighestWeightTensorProductOfKirillovReshetikhinTableaux(['D', 4, 1], [[1, 1], [2, 1], [1, 1]])
sage: T = HW(pathlist=[[1], [-2, 2], [1]])
sage: T
[[1]] (X) [[2], [-2]] (X) [[1]]
sage: T2 = HW(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.AbstractTensorProductOfKRTableaux(cartan_type, B, biject_class)

Abstract class for all of tensor product of KR tableaux of a given Cartan type.

See TensorProductOfKirillovReshetikhinTableaux. This class should never be created directly.

list()

Create a list of the elements by using the iterator.

TESTS:

sage: HW = HighestWeightTensorProductOfKirillovReshetikhinTableaux(['A',3,1], [[3,1], [2,1]])
sage: HW.list()
[[[1], [2], [3]] (X) [[1], [2]], [[1], [3], [4]] (X) [[1], [2]]]

rigged_configurations()

Return the corresponding set of rigged configurations.

EXAMPLES:

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

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

Container class of all highest weight tensor product of KR tableaux.

A tensor product of KR tableaux is highest weight if the action of $$e_i$$ for $$i \in I \setminus \{0\}$$ are all undefined.

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

Element

alias of TensorProductOfKirillovReshetikhinTableauxElement

cardinality()

Return the number of highest weight tensor product of Kirillov-Reshetikhin tableaux.

EXAMPLES:

sage: HW = HighestWeightTensorProductOfKirillovReshetikhinTableaux(['A', 4, 1], [[2, 1]])
sage: HW.cardinality()
1

module_generators()

Module generators for this tensor product of KR tableaux.

EXAMPLES:

sage: HWKR = HighestWeightTensorProductOfKirillovReshetikhinTableaux(['A', 4, 1], [[2,2]])
sage: for x in HWKR.module_generators: x
...
[[1, 1], [2, 2]]

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

A tensor product of KirillovReshetikhinTableaux.

Through the bijection with rigged configurations, the tableaux that are produced in the Kirillov-Reshetikhin model for type $$D_n^{(1)}$$ 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 two tableaux models is given by a filling map.

REFERENCES:

 [OSS2011] Masato Okado, Reiho Sakamoto, Anne Schilling Affine crystal structure on rigged configurations of type $$D_n^{(1)}$$ J. Algebraic Combinatorics, to appear, doi:10.1007/s10801-012-0383-z (arXiv:1109.3523 [math.QA])

INPUT:

• cartan_type – An affine Cartan type
• B – An (ordered) list of dimensions.

The dimensions (i.e. B) is a list whose entries are lists of the form [r, s] which correspond to Kirillov-Reshetikhin tableaux with r rows and s columns.

EXAMPLES:

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

sage: KRT = 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

-2[ ][ ]-2

0[ ][ ]0

(/)

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 = KirillovReshetikhinCrystal(['A',3,1], 3,1)
sage: KR2 = KirillovReshetikhinCrystal(['A',3,1], 2,2)
sage: T = TensorProductOfCrystals(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

Element

alias of TensorProductOfKirillovReshetikhinTableauxElement

tensor_product_of_Kirillov_Reshetikhin_crystals()

Return the corresponding tensor product of Kirillov-Reshetikhin crystals.

EXAMPLES:

sage: KRT = 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 = TensorProductOfKirillovReshetikhinTableaux(['D', 4, 1], [[4,1], [3,3]])
sage: KR1 = KirillovReshetikhinCrystal(['D', 4, 1], 4, 1)
sage: KR2 = KirillovReshetikhinCrystal(['D', 4, 1], 3, 3)
sage: T = TensorProductOfCrystals(KR1, KR2)
sage: T == KRT.tensor_product_of_Kirillov_Reshetikhin_crystals()
True
sage: T is KRT.tensor_product_of_Kirillov_Reshetikhin_crystals()
True


#### Previous topic

A specific rigged configuration

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An element of a tensor product of Kirillov-Reshetikhin tableaux