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().
AUTHORS:
EXAMPLES:
Type examples:
sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['A',3,1], [[3,1], [2,1]])
sage: KRT
Tensor product of Kirillov-Reshetikhin tableaux of type ['A', 3, 1] and factor(s) ((3, 1), (2, 1))
sage: KRT.cardinality()
24
sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['A',3,1], [[1,1], [2,1], [3,1]])
sage: KRT
Tensor product of Kirillov-Reshetikhin tableaux of type ['A', 3, 1] and factor(s) ((1, 1), (2, 1), (3, 1))
sage: len(KRT.module_generators)
5
sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['A',3,1], [[1,1], [2,1], [3,1]])
sage: KRT.cardinality()
96
Type examples:
sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['D', 4, 1], [[1, 1], [2, 1], [1, 1]])
sage: KRT
Tensor product of Kirillov-Reshetikhin tableaux of type ['D', 4, 1] and factor(s) ((1, 1), (2, 1), (1, 1))
sage: T = KRT(pathlist=[[1], [-2, 2], [1]])
sage: T
[[1]] (X) [[2], [-2]] (X) [[1]]
sage: T2 = KRT(pathlist=[[1], [2, -2], [1]])
sage: T2
[[1]] (X) [[-2], [2]] (X) [[1]]
sage: T == T2
False
Bases: sage.structure.unique_representation.UniqueRepresentation
Class so we do not have to build the module generators for TensorProductOfKirillovReshetikhinTableaux at initialization.
Warning
This class is for internal use only!
Return the cardinality of self which is the number of highest weight elements.
EXAMPLES:
sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['D',4,1], [[2,2]])
sage: from sage.combinat.rigged_configurations.tensor_product_kr_tableaux import HighestWeightTensorKRT
sage: HW = HighestWeightTensorKRT(KRT)
sage: HW.cardinality()
3
sage: len(HW)
3
sage: len(KRT.module_generators)
3
Deprecated in trac ticket #13872. Use instead the attribute module_generators of TensorProductOfKirillovReshetikhinTableaux.
EXAMPLES:
sage: HighestWeightTensorProductOfKirillovReshetikhinTableaux(['A',2,1], [[1,1]])
doctest:...: DeprecationWarning: this class is deprecated.
Use TensorProductOfKirillovReshetikhinTableaux(cartan_type, B).module_generators instead
See http://trac.sagemath.org/13872 for details.
Highest weight elements of Tensor product of Kirillov-Reshetikhin tableaux of type ['A', 2, 1] and factor(s) ((1, 1),)
Bases: sage.combinat.crystals.tensor_product.FullTensorProductOfRegularCrystals
A tensor product of KirillovReshetikhinTableaux.
Through the bijection with rigged configurations, the tableaux that are produced in all nonexceptional types 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 elements of the Kirillov-Reshetikhin crystal, given by the Kashiwara-Nakashima tableaux, and the Kirillov-Reshetikhin tableaux is given by a filling map.
Note
The tableaux for all non-simply-laced types are provably correct if the
bijection with rigged configurations
holds. Therefore this is currently only proven for or
and in general for types
and
.
For more information see [OSS2011] and KirillovReshetikhinTableaux.
For more information on KR crystals, see sage.combinat.crystals.kirillov_reshetikhin.
INPUT:
REFERENCES:
[OSS2011] | Masato Okado, Reiho Sakamoto, Anne Schilling,
Affine crystal structure on rigged configurations of type ![]() |
EXAMPLES:
We can go between tensor products of KR crystals and rigged configurations:
sage: KRT = crystals.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 = crystals.KirillovReshetikhin(['A',3,1], 3,1)
sage: KR2 = crystals.KirillovReshetikhin(['A',3,1], 2,2)
sage: T = crystals.TensorProduct(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
We can get the highest weight elements by using the attribute module_generators:
sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['A',3,1], [[3,1], [2,1]])
sage: list(KRT.module_generators)
[[[1], [2], [3]] (X) [[1], [2]], [[1], [3], [4]] (X) [[1], [2]]]
To create elements directly (i.e. not passing in KR tableaux elements), there is the pathlist option will receive a list of lists which contain the reversed far-eastern reading word of the tableau. That is to say, in English notation, the word obtain from reading bottom-to-top, left-to-right.
sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['A',3,1], [[3,2], [1,2], [2,1]])
sage: elt = KRT(pathlist=[[3, 2, 1, 4, 2, 1], [1, 3], [3, 1]])
sage: elt.pp()
1 1 (X) 1 3 (X) 1
2 2 3
3 4
One can still create elements in the same way as tensor product of crystals:
sage: K1 = crystals.KirillovReshetikhin(['A',3,1], 3, 2, model='KR')
sage: K2 = crystals.KirillovReshetikhin(['A',3,1], 1, 2, model='KR')
sage: K3 = crystals.KirillovReshetikhin(['A',3,1], 2, 1, model='KR')
sage: eltlong = KRT(K1(3, 2, 1, 4, 2, 1), K2(1, 3), K3(3, 1))
sage: eltlong == elt
True
alias of TensorProductOfKirillovReshetikhinTableauxElement
Return the corresponding set of rigged configurations.
EXAMPLES:
sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['A',3,1], [[1,3], [2,1]])
sage: KRT.rigged_configurations()
Rigged configurations of type ['A', 3, 1] and factor(s) ((1, 3), (2, 1))
Return the tensor product of self with crystals.
If crystals is a list of (a tensor product of) KR tableaux, this returns a TensorProductOfKirillovReshetikhinTableaux.
EXAMPLES:
sage: TP = crystals.TensorProductOfKirillovReshetikhinTableaux(['A', 3, 1], [[1,3],[3,1]])
sage: K = crystals.KirillovReshetikhin(['A', 3, 1], 2, 2, model='KR')
sage: TP.tensor(K, TP)
Tensor product of Kirillov-Reshetikhin tableaux of type ['A', 3, 1]
and factor(s) ((1, 3), (3, 1), (2, 2), (1, 3), (3, 1))
sage: C = crystals.KirillovReshetikhin(['A',3,1], 3, 1, model='KN')
sage: TP.tensor(K, C)
Full tensor product of the crystals
[Kirillov-Reshetikhin tableaux of type ['A', 3, 1] and shape (1, 3),
Kirillov-Reshetikhin tableaux of type ['A', 3, 1] and shape (3, 1),
Kirillov-Reshetikhin tableaux of type ['A', 3, 1] and shape (2, 2),
Kirillov-Reshetikhin crystal of type ['A', 3, 1] with (r,s)=(3,1)]
Return the corresponding tensor product of Kirillov-Reshetikhin crystals.
EXAMPLES:
sage: KRT = crystals.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 = crystals.TensorProductOfKirillovReshetikhinTableaux(['D', 4, 1], [[4,1], [3,3]])
sage: KR1 = crystals.KirillovReshetikhin(['D', 4, 1], 4, 1)
sage: KR2 = crystals.KirillovReshetikhin(['D', 4, 1], 3, 3)
sage: T = crystals.TensorProduct(KR1, KR2)
sage: T == KRT.tensor_product_of_kirillov_reshetikhin_crystals()
True
sage: T is KRT.tensor_product_of_kirillov_reshetikhin_crystals()
True