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()
.
AUTHORS:
Travis Scrimshaw (2010-09-26): Initial version
EXAMPLES:
Type \(A_n^{(1)}\) 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 \(D_n^{(1)}\) 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
- class sage.combinat.rigged_configurations.tensor_product_kr_tableaux.HighestWeightTensorKRT(tp_krt)¶
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!
- cardinality()¶
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
- class sage.combinat.rigged_configurations.tensor_product_kr_tableaux.TensorProductOfKirillovReshetikhinTableaux(cartan_type, B)¶
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 \(B^{r,1}\) or \(B^{1,s}\) and in general for types \(A_n^{(1)}\) and \(D_n^{(1)}\).For more information see [OSS2011] and
KirillovReshetikhinTableaux
.For more information on KR crystals, see
sage.combinat.crystals.kirillov_reshetikhin
.INPUT:
cartan_type
– a Cartan typeB
– an (ordered) list of pairs \((r,s)\) which give the dimension of a rectangle with \(r\) rows and \(s\) columns and corresponds to a Kirillov-Reshetikhin tableaux factor of \(B^{r,s}\).
REFERENCES:
- OSS2011
Masato Okado, Reiho Sakamoto, Anne Schilling, Affine crystal structure on rigged configurations of type \(D_n^{(1)}\), J. Algebraic Combinatorics 37(3) (2013) 571-599 (arXiv 1109.3523 [math.QA])
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
- Element¶
- rigged_configurations()¶
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))
- tensor(*crystals, **options)¶
Return the tensor product of
self
withcrystals
.If
crystals
is a list of (a tensor product of) KR tableaux, this returns aTensorProductOfKirillovReshetikhinTableaux
.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)]
- tensor_product_of_kirillov_reshetikhin_crystals()¶
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)]