Rigged Configurations

AUTHORS:

  • Travis Scrimshaw (2010-09-26): Initial version
sage.combinat.rigged_configurations.rigged_configurations.KirillovReshetikhinCrystal(cartan_type, r, s)

Return the KR crystal \(B^{r,s}\) using rigged configurations.

This is the rigged configuration \(RC(B^{r,s})\) or \(RC(L)\) with \(L = (L_i^{(a)})\) and \(L_i^{(a)} = \delta_{a,r} \delta_{i,s}\).

EXAMPLES:

sage: K1 = crystals.kirillov_reshetikhin.RiggedConfigurations(['A',6,2], 2, 1)
sage: K2 = crystals.kirillov_reshetikhin.LSPaths(['A',6,2], 2, 1)
sage: K1.digraph().is_isomorphic(K2.digraph(), edge_labels=True)
True
sage.combinat.rigged_configurations.rigged_configurations.RCNonSimplyLaced

Rigged configurations in non-simply-laced types.

These are rigged configurations which lift to virtual rigged configurations in a simply-laced type.

For more on rigged configurations, see RiggedConfigurations.

sage.combinat.rigged_configurations.rigged_configurations.RCTypeA2Dual

Rigged configurations of type \(A_{2n}^{(2)\dagger}\).

For more on rigged configurations, see RiggedConfigurations.

EXAMPLES:

sage: RC = RiggedConfigurations(CartanType(['A',4,2]).dual(), [[1,2],[1,1],[2,1]])
sage: RC
Rigged configurations of type ['BC', 2, 2]^* and factor(s) ((1, 2), (1, 1), (2, 1))
sage: RC.cardinality()
750
sage: RC.virtual
Rigged configurations of type ['A', 3, 1] and factor(s) ((1, 2), (3, 2), (1, 1), (3, 1), (2, 1), (2, 1))
sage: RC = RiggedConfigurations(CartanType(['A',2,2]).dual(), [[1,1]])
sage: RC.cardinality()
3
sage: RC = RiggedConfigurations(CartanType(['A',2,2]).dual(), [[1,2],[1,1]])
sage: TestSuite(RC).run() # long time
sage: RC = RiggedConfigurations(CartanType(['A',4,2]).dual(), [[2,1]])
sage: TestSuite(RC).run() # long time
sage.combinat.rigged_configurations.rigged_configurations.RCTypeA2Even

Rigged configurations for type \(A_{2n}^{(2)}\).

For more on rigged configurations, see RiggedConfigurations.

EXAMPLES:

sage: RC = RiggedConfigurations(['A',4,2], [[2,1], [1,2]])
sage: RC.cardinality()
150
sage: RC = RiggedConfigurations(['A',2,2], [[1,1]])
sage: RC.cardinality()
3
sage: RC = RiggedConfigurations(['A',2,2], [[1,2],[1,1]])
sage: TestSuite(RC).run() # long time
sage: RC = RiggedConfigurations(['A',4,2], [[2,1]])
sage: TestSuite(RC).run() # long time
sage.combinat.rigged_configurations.rigged_configurations.RiggedConfigurations

Rigged configurations as \(U_q^{\prime}(\mathfrak{g})\)-crystals.

Let \(\overline{I}\) denote the classical index set associated to the Cartan type of the rigged configurations. A rigged configuration of multiplicity array \(L_i^{(a)}\) and dominant weight \(\Lambda\) is a sequence of partitions \(\{ \nu^{(a)} \mid a \in \overline{I} \}\) such that

\[\sum_{\overline{I} \times \ZZ_{>0}} i m_i^{(a)} \alpha_a = \sum_{\overline{I} \times \ZZ_{>0}} i L_i^{(a)} \Lambda_a - \Lambda\]

where \(\alpha_a\) is a simple root, \(\Lambda_a\) is a fundamental weight, and \(m_i^{(a)}\) is the number of rows of length \(i\) in the partition \(\nu^{(a)}\).

Each partition \(\nu^{(a)}\), in the sequence also comes with a sequence of statistics \(p_i^{(a)}\) called vacancy numbers and a weakly decreasing sequence \(J_i^{(a)}\) of length \(m_i^{(a)}\) called riggings. Vacancy numbers are computed based upon the partitions and \(L_i^{(a)}\), and the riggings must satisfy \(\max J_i^{(a)} \leq p_i^{(a)}\). We call such a partition a rigged partition. For more, see [RigConBijection] [CrysStructSchilling06] [BijectionLRT].

Rigged configurations form combinatorial objects first introduced by Kerov, Kirillov and Reshetikhin that arose from studies of statistical mechanical models using the Bethe Ansatz. They are sequences of rigged partitions. A rigged partition is a partition together with a label associated to each part that satisfy certain constraints. The labels are also called riggings.

Rigged configurations exist for all affine Kac-Moody Lie algebras. See for example [HKOTT2002]. In Sage they are specified by providing a Cartan type and a list of rectangular shapes \(B\). The list of all (highest weight) rigged configurations for given \(B\) is computed via the (virtual) Kleber algorithm (see also KleberTree and VirtualKleberTree).

Rigged configurations in simply-laced types all admit a classical crystal structure [CrysStructSchilling06]. For non-simply-laced types, the crystal is given by using virtual rigged configurations [OSS03]. The highest weight rigged configurations are those where all riggings are nonnegative. The list of all rigged configurations is computed from the highest weight ones using the crystal operators.

Rigged configurations are conjecturally in bijection with TensorProductOfKirillovReshetikhinTableaux of non-exceptional affine types where the list \(B\) corresponds to the tensor factors \(B^{r,s}\). The bijection has been proven in types \(A_n^{(1)}\) and \(D_n^{(1)}\) and when the only non-zero entries of \(L_i^{(a)}\) are either only \(L_1^{(a)}\) or only \(L_i^{(1)}\) (corresponding to single columns or rows respectively) [RigConBijection], [BijectionLRT], [BijectionDn].

KR crystals are implemented in Sage, see KirillovReshetikhinCrystal(), however, in the bijection with rigged configurations a different realization of the elements in the crystal are obtained, which are coined KR tableaux, see KirillovReshetikhinTableaux. For more details see [OSS2011].

Note

All non-simply-laced rigged configurations have not been proven to give rise to aligned virtual crystals (i.e. have the correct crystal structure or ismorphic as affine crystals to the tensor product of KR tableaux).

INPUT:

  • cartan_type – a Cartan type
  • B – a list of positive integer tuples \((r,s)\) corresponding to the tensor factors in the bijection with tensor product of Kirillov-Reshetikhin tableaux or equivalently the sequence of width \(s\) and height \(r\) rectangles

REFERENCES:

[HKOTT2002]G. Hatayama, A. Kuniba, M. Okado, T. Takagi, Z. Tsuboi. Paths, Crystals and Fermionic Formulae. Prog. Math. Phys. 23 (2002) Pages 205-272.
[CrysStructSchilling06](1, 2) Anne Schilling. Crystal structure on rigged configurations. International Mathematics Research Notices. Volume 2006. (2006) Article ID 97376. Pages 1-27.
[RigConBijection](1, 2) Masato Okado, Anne Schilling, Mark Shimozono. A crystal to rigged configuration bijection for non-exceptional affine algebras. Algebraic Combinatorics and Quantum Groups. Edited by N. Jing. World Scientific. (2003) Pages 85-124.
[BijectionDn]Anne Schilling. A bijection between type \(D_n^{(1)}\) crystals and rigged configurations. J. Algebra. 285 (2005) 292-334
[BijectionLRT](1, 2) Anatol N. Kirillov, Anne Schilling, Mark Shimozono. A bijection between Littlewood-Richardson tableaux and rigged configurations. Selecta Mathematica (N.S.). 8 (2002) Pages 67-135. (MathSciNet MR1890195).

EXAMPLES:

sage: RC = RiggedConfigurations(['A', 3, 1], [[3, 2], [1, 2], [1, 1]])
sage: RC
Rigged configurations of type ['A', 3, 1] and factor(s) ((3, 2), (1, 2), (1, 1))

sage: RC = RiggedConfigurations(['A', 3, 1], [[2,1]]); RC
Rigged configurations of type ['A', 3, 1] and factor(s) ((2, 1),)
sage: RC.cardinality()
6
sage: len(RC.list()) == RC.cardinality()
True
sage: RC.list() # random
[

                                         0[ ]0
(/)  (/)      (/)      -1[ ]-1  -1[ ]-1
                                         -1[ ]-1
(/)  -1[ ]-1  0[ ]0    0[ ]0    1[ ]1    -1[ ]-1

(/)  (/)      -1[ ]-1  (/)      -1[ ]-1  0[ ]0
   ,        ,        ,        ,        ,
]

A rigged configuration element with all riggings equal to the vacancy numbers can be created as follows:

sage: RC = RiggedConfigurations(['A', 3, 1], [[3,2], [2,1], [1,1], [1,1]]); RC
Rigged configurations of type ['A', 3, 1] and factor(s) ((3, 2), (2, 1), (1, 1), (1, 1))
sage: elt = RC(partition_list=[[1],[],[]]); elt

0[ ]0

(/)

(/)

If on the other hand we also want to specify the riggings, this can be achieved as follows:

sage: RC = RiggedConfigurations(['A', 3, 1], [[3, 2], [1, 2], [1, 1]])
sage: RC(partition_list=[[2],[2],[2]])

1[ ][ ]1

0[ ][ ]0

0[ ][ ]0
sage: RC(partition_list=[[2],[2],[2]], rigging_list=[[0],[0],[0]])

1[ ][ ]0

0[ ][ ]0

0[ ][ ]0

A larger example:

sage: RC = RiggedConfigurations(['D', 7, 1], [[3,3],[5,2],[4,3],[2,3],[4,4],[3,1],[1,4],[2,2]])
sage: elt = RC(partition_list=[[2],[3,2,1],[2,2,1,1],[2,2,1,1,1,1],[3,2,1,1,1,1],[2,1,1],[2,2]],
....:          rigging_list=[[2],[1,0,0],[4,1,2,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0],[0,0]])
sage: elt

3[ ][ ]2

1[ ][ ][ ]1
2[ ][ ]0
1[ ]0

4[ ][ ]4
4[ ][ ]1
3[ ]2
3[ ]1

2[ ][ ]1
2[ ][ ]0
0[ ]0
0[ ]0
0[ ]0
0[ ]0

0[ ][ ][ ]0
2[ ][ ]1
0[ ]0
0[ ]0
0[ ]0
0[ ]0

0[ ][ ]0
0[ ]0
0[ ]0

0[ ][ ]0
0[ ][ ]0

To obtain the KR tableau under the bijection between rigged configurations and KR tableaux, we can type the following. This example was checked against Reiho Sakamoto’s Mathematica program on rigged configurations:

sage: output = elt.to_tensor_product_of_kirillov_reshetikhin_tableaux(); output
[[1, 1, 1], [2, 3, 3], [3, 4, -5]] (X) [[1, 1], [2, 2], [3, 3], [5, -6], [6, -5]] (X)
[[1, 1, 2], [2, 2, 3], [3, 3, 7], [4, 4, -7]] (X) [[1, 1, 1], [2, 2, 2]] (X)
[[1, 1, 1, 3], [2, 2, 3, 4], [3, 3, 4, 5], [4, 4, 5, 6]] (X) [[1], [2], [3]] (X) [[1, 1, 1, 1]] (X) [[1, 1], [2, 2]]
sage: elt.to_tensor_product_of_kirillov_reshetikhin_tableaux().to_rigged_configuration() == elt
True
sage: output.to_rigged_configuration().to_tensor_product_of_kirillov_reshetikhin_tableaux() == output
True

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

sage: RC = RiggedConfigurations(['D', 4, 1], [[2, 1]])
sage: elt = RC(partition_list=[[1],[1,1],[1],[1]])
sage: tp_krc = elt.to_tensor_product_of_kirillov_reshetikhin_crystals(); tp_krc
[[]]
sage: ret = RC(tp_krc)
sage: ret == elt
True
sage: RC = RiggedConfigurations(['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 = T[1]; t
[[++++, []], [+++-, [[1], [2], [4], [-4]]]]
sage: ret = RC(t)
sage: ret.to_tensor_product_of_kirillov_reshetikhin_crystals()
[[++++, []], [+++-, [[1], [2], [4], [-4]]]]