Uniform description of the rigged configuration bijection

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.