Uniform description of the rigged configuration bijection

TL;DR

This paper provides a uniform description and proof of the bijection between rigged configurations and tensor products of Kirillov--Reshetikhin crystals across various affine types, using virtual crystals and tableaux representations.

Contribution

It introduces a uniform framework for the rigged configuration bijection across multiple types, including new tableau descriptions for crystals.

Findings

Proves bijection and statistic preservation uniformly across types.

Describes crystals using tableaux of fixed height and shape.

Provides a conjectural crystal basis description for certain modules.

Abstract

We give a uniform description of the bijection $\Phi$ from rigged configurations to tensor products of Kirillov--Reshetikhin crystals of the form $\bigotimes_{i=1}^N B^{r_i,1}$ in dual untwisted types: simply-laced types and types $A_{2n-1}^{(2)}$, $D_{n+1}^{(2)}$, $E_6^{(2)}$, and $D_4^{(3)}$. We give a uniform proof that $\Phi$ is a bijection and preserves statistics. We describe $\Phi$ uniformly using virtual crystals for all remaining types, but our proofs are type-specific. We also give a uniform proof that $\Phi$ is a bijection for $\bigotimes_{i=1}^N B^{r_i,s_i}$ when $r_i$, for all $i$, map to $0$ under an automorphism of the Dynkin diagram. Furthermore, we give a description of the Kirillov--Reshetikhin crystals $B^{r,1}$ using tableaux of a fixed height $k_r$ depending on $r$ in all affine types. Additionally, we are able to describe crystals $B^{r,s}$ using $k_r \times s$ shaped tableaux that are conjecturally the crystal basis for Kirillov--Reshetikhin modules for various nodes $r$.