Kirillov-Reshetikhin crystals $B^{7,s}$ for type $E_7^{(1)}$

TL;DR

This paper constructs a combinatorial crystal structure on Kirillov-Reshetikhin crystals for type E7^{(1)} and describes the associated combinatorial R-matrix, advancing understanding of their algebraic and combinatorial properties.

Contribution

It introduces a new combinatorial crystal structure for B^{7,s} in type E7^{(1)} and explicitly describes the combinatorial R-matrix for these crystals.

Findings

Constructed a crystal structure on B^{7,s} for E7^{(1)}.

Described the combinatorial R-matrix explicitly.

Provides tools for further algebraic and combinatorial analysis.

Abstract

We construct a combinatorial crystal structure on the Kirillov-Reshetikhin crystal $B^{7,s}$ in type $E_7^{(1)}$, where $7$ is the unique node in the orbit of $0$ in the affine Dynkin diagram. We then describe the combinatorial $R$-matrix $R \colon B^{7,s} \otimes B^{7,s'} \to B^{7,s'} \otimes B^{7,s}$.