A crystal to rigged configuration bijection and the filling map for type $D_4^{(3)}$

TL;DR

This paper constructs a bijection between rigged configurations and Kirillov--Reshetikhin crystals for type D4(3), preserving key statistics and extending to a crystal model with virtualization compatibility.

Contribution

It introduces a new bijection for type D4(3) that preserves statistics and extends to a crystal model, including virtualization properties.

Findings

Bijection between rigged configurations and crystals established

Statistics correspondence between cocharge and energy shown

Virtualization map compatibility demonstrated

Abstract

We give a bijection $\Phi$ from rigged configurations to a tensor product of Kirillov--Reshetikhin crystals of the form $B^{r,1}$ and $B^{1,s}$ in type $D_4^{(3)}$. We show that the cocharge statistic is sent to the energy statistic for tensor products $\bigotimes_{i=1}^N B^{r_i,1}$ and $\bigotimes_{i=1}^N B^{1,s_i}$. We extend this bijection to a single $B^{r,s}$, show that it preserves statistics, and obtain the so-called Kirillov--Reshetikhin tableaux model for $B^{r,s}$. Additionally, we show that $\Phi$ commutes with the virtualization map and that $B^{1,s}$ is naturally a virtual crystal in type $D_4^{(1)}$, thus defining an affine crystal structure on rigged configurations corresponding to $B^{1,s}$.