Affine crystal structure on rigged configurations of type D_n^(1)

TL;DR

This paper constructs an affine crystal action on rigged configurations of type D_n^(1), establishing an isomorphism with Kirillov-Reshetikhin crystals and introducing a tableau-based representation for these crystals.

Contribution

It introduces a new affine crystal action on rigged configurations of type D_n^(1) and a tableau representation for Kirillov-Reshetikhin crystals, extending previous work.

Findings

Established isomorphism between rigged configurations and Kirillov-Reshetikhin crystals.

Developed a tableau representation for certain Kirillov-Reshetikhin crystals.

Constructed a bijection between tensor products of crystals and rigged configurations.

Abstract

Extending the work arXiv:math/0508107, we introduce the affine crystal action on rigged configurations which is isomorphic to the Kirillov-Reshetikhin crystal B^{r,s} of type D_n^(1) for any r,s. We also introduce a representation of B^{r,s} (r not equal to n-1,n) in terms of tableaux of rectangular shape r x s, which we coin Kirillov-Reshetikhin tableaux (using a non-trivial analogue of the type A column splitting procedure) to construct a bijection between elements of a tensor product of Kirillov-Reshetikhin crystals and rigged configurations.