Uniform description of the rigged configuration bijection

Travis Scrimshaw

doi:10.1007/s00029-020-00564-8

paper

Uniform description of the rigged configuration bijection

arXiv:1703.08945 · doi:10.1007/s00029-020-00564-8

Abstract

We give a uniform description of the bijection $Φ$ from rigged configurations to tensor products of Kirillov--Reshetikhin crystals of the form $⨂_{i = 1}^{N} B^{r_{i}, 1}$ in dual untwisted types: simply-laced types and types $A_{2 n - 1}^{(2)}$ , $D_{n + 1}^{(2)}$ , $E_{6}^{(2)}$ , and $D_{4}^{(3)}$ . We give a uniform proof that $Φ$ is a bijection and preserves statistics. We describe $Φ$ uniformly using virtual crystals for all remaining types, but our proofs are type-specific. We also give a uniform proof that $Φ$ is a bijection for $⨂_{i = 1}^{N} B^{r_{i}, s_{i}}$ 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 $r$ .

60 pages, 5 figures, 3 tables; v2 incorporated changes from referee

Uniform description of the rigged configuration bijection

Abstract

References in corpus (13)

Cited by in corpus (3)