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

TL;DR

This paper proves the $X=M$ conjecture for all nonexceptional affine types by establishing a bijection between rigged configurations and crystal elements, extending known results from simply-laced types.

Contribution

It introduces a new bijection for nonexceptional types and proves the $X=M$ conjecture in full generality, connecting rigged configurations with crystal bases.

Findings

Established a bijection for all nonexceptional types.

Proved the $X=M$ conjecture universally for these types.

Extended the bijection to a classical crystal isomorphism.

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.