Rigged configuration bijection and proof of the X = M conjecture for nonexceptional affine types

Abstract

We establish a bijection between rigged configurations and highest weight elements of a tensor product of Kirillov-Reshetikhin crystals for all nonexceptional types. A key idea for the proof is to embed both objects into bigger sets for simply-laced types A (1) n or D (1) n , whose bijections have already been established. As a consequence we settle the X = M conjecture in full generality for nonexceptional types. Furthermore, the bijection extends to a classical crystal isomorphism and sends the combinatorial R-matrix to the identity map on rigged configurations.

A Uniform Approach to Soliton Cellular Automata Using Rigged Configurations

“…B] as "decoupling" rules on the level of rigged configurations by forgetting ν (r) and instead using ν (r) to describe the "decoupled" KR crystals. Moreover, our results subsume the results on the scattering and phase shift of SCA (see Conjecture 2.10) from [bM12, FOY00, HHI + 01, HKO + 02a, HKT00, MOW12, MW13, TNS99, Yam04, Yam07] as the requisite properties of Φ are known [KSS02,OSSS17,OSS18,Scr17c]. Our results also include other types that have not previously been considered; e.g., E…”

An Explicit Algorithm of Rigged Configuration Bijection for the Adjoint Crystal of Type $G_{2}^{(1)}$

“…i − 1, and p (a) i − 2, respectively. Here, "q-singular" is the abbreviation of "quasi singular" firstly introduced in [19] and commonly used in the literature [29,35]. The "qqsingular" strings are firstly introduced in this paper.…”

“…Until now, this has been extended in more general setting and in various types [3,14,25,26,27,28,23,24,30,31,34,35,32]. For nonexceptional types, the generalization of the rigged configuration bijection has been established [29]. On the other hand, there remain unsolved problems in exception types.…”

Higher order Kirillov-Reshetikhin modules, Imaginary modules and Monoidal Categorification for $U_q(A_n^{(1)})$

“…Subsequently these modules were also defined for the quantum loop algebras associated to a simple Lie algebra and these are now called the Kirillov-Reshetikhin (KR) modules. The modules have many nice properties and there is an extensive literature on the subject, which includes their connections with integrable systems, the combinatorics of crystal bases, the fermionic formula [8,9,13,22,28,30,31], and more recently [18,19,20] they are connected with the monoidal categorification of cluster algebras.…”