A tensor product of KirillovReshetikhinTableaux which are tableaux of
rows and
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
, 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:
EXAMPLES:
Type 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 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
Bases: sage.combinat.crystals.tensor_product.FullTensorProductOfRegularCrystals
Abstract class for all of tensor product of KR tableaux of a given Cartan type.
See TensorProductOfKirillovReshetikhinTableaux. This class should never be created directly.
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]]]
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))
Bases: sage.combinat.rigged_configurations.tensor_product_kr_tableaux.AbstractTensorProductOfKRTableaux
Container class of all highest weight tensor product of KR tableaux.
A tensor product of KR tableaux is highest weight if the action of
for
are all undefined.
For more on tensor product of Kirillov-Reshetikhin tableaux, see TensorProductOfKirillovReshetikhinTableaux.
alias of TensorProductOfKirillovReshetikhinTableauxElement
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 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]]
Bases: sage.combinat.rigged_configurations.tensor_product_kr_tableaux.AbstractTensorProductOfKRTableaux
A tensor product of KirillovReshetikhinTableaux.
Through the bijection with rigged configurations, the tableaux that are
produced in the Kirillov-Reshetikhin model for type 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.
For more information see [OSS2011] and KirillovReshetikhinTableaux.
REFERENCES:
[OSS2011] | Masato Okado, Reiho Sakamoto, Anne Schilling
Affine crystal structure on rigged configurations of type ![]() |
For more information on KR crystals, see sage.combinat.crystals.kirillov_reshetikhin.
INPUT:
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
alias of TensorProductOfKirillovReshetikhinTableauxElement
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