Type $D_n^{(1)}$ rigged configuration bijection

TL;DR

This paper constructs a bijection between rigged configurations and tensor products of Kirillov--Reshetikhin crystals for type D^{(1)}_n, proving invariance under the combinatorial R-matrix and establishing the fermionic formula.

Contribution

It introduces a general bijection for type D^{(1)}_n, demonstrating invariance, preserving statistics, and confirming the fermionic formula and crystal isomorphism.

Findings

Bijection between rigged configurations and crystals established

Invariance under the combinatorial R-matrix proven

Fermionic formula for type D^{(1)}_n confirmed

Abstract

We establish a bijection between the set of rigged configurations and the set of tensor products of Kirillov--Reshetikhin crystals of type $D^{(1)}_n$ in full generality. We prove the invariance of rigged configurations under the action of the combinatorial $R$-matrix on tensor products and show that the bijection preserves certain statistics (cocharge and energy). As a result, we establish the fermionic formula for type $D_n^{(1)}$. In addition, we establish that the bijection is a classical crystal isomorphism.