Kirillov-Reshetikhin crystals $B^{1,s}$ for $\widehat{\mathfrak{sl}}_n$ using Nakajima monomials

TL;DR

This paper constructs explicit models of Kirillov-Reshetikhin crystals for affine sl_n using Nakajima monomials, connecting crystal structures with the Kyoto path model and extending to various crystal types.

Contribution

It provides a new realization of KR crystals via Nakajima monomials and describes their tensor products and alternative models.

Findings

Explicit Nakajima monomial models for $B^{1,s}$ and $B^{r,1}$ crystals.

Tensor product structure described via index shifts.

Connection established with Kyoto path model.

Abstract

We give a realization of the Kirillov--Reshetikhin crystal $B^{1,s}$ using Nakajima monomials for $\widehat{\mathfrak{sl}}_n$ using the crystal structure given by Kashiwara. We describe the tensor product $\bigotimes_{i=1}^N B^{1,s_i}$ in terms of a shift of indices, allowing us to recover the Kyoto path model. Additionally, we give a model for the KR crystals $B^{r,1}$ using Nakajima monomials.