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 $A_n^{(1)}$ or $D_n^{(1)}$, 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.

30 pages, 2 figures; v2 Referenced Naoi's work in the introduction, clarified some notation; v3 Various additions for more self-containment (e.g., the signature rule) and typos fixed

References in corpus (15)

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• X=M for symmetric powers
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• KKR type bijection for the exceptional affine algebra E_6^{(1)}
• Simplicity and similarity of Kirillov-Reshetikhin crystals

Cited by in corpus (8)

• On higher level Kirillov--Reshetikhin crystals, Demazure crystals, and related uniform models
• Simplicity of tensor products of Kirillov--Reshetikhin modules: nonexceptional affine and G types
• A uniform approach to soliton cellular automata using rigged configurations
• Uniform description of the rigged configuration bijection
• Kirillov-Reshetikhin modules of generalized quantum group of type $A$
• Quantum Q-systems and fermionic sums -- the non-simply laced case
• Kirillov-Reshetikhin crystals $B^{7,s}$ for type $E_7^{(1)}$
• An Explicit Algorithm of Rigged Configuration Bijection for the Adjoint Crystal of Type $G_2^{(1)}$