Young tableau descriptions for the polyhedral realizations of crystal bases in type $A_n$

TL;DR

This paper establishes a combinatorial framework linking polyhedral realizations of crystal bases in type A_n to Young tableaux, providing explicit descriptions and applications in crystal embeddings and Lusztig data.

Contribution

It introduces a novel combinatorial correspondence between polyhedral crystal bases and Young tableaux, enhancing understanding of their structure and embeddings.

Findings

Explicit correspondence between polyhedral realizations and Young tableaux.

A crystal structure on Gelfand-Tsetlin patterns derived from the correspondence.

Concrete realizations of crystal embeddings and Lusztig data.

Abstract

By utilizing the combinatorial properties of various tableau models, we establish an explicit correspondence between the polyhedral realizations of the crystal bases $\mathcal B(\lambda)$ (resp. $\mathcal B(\infty)$) of type $A_n$ and the reverse semi-standard Young tableaux (resp. reverse marginally large tableaux), thereby providing a combinatorial description of the corresponding polyhedral realizations. Furthermore, a crystal structure on the set of Gelfand-Tsetlin patterns is obtained via the correspondence between the polyhedral realization of $\mathcal{B}(\lambda)$ and the reverse tableaux. As applications of our framework, we present concrete combinatorial realizations of the crystal embedding of $\mathcal B(\lambda)$ into $\mathcal B(\infty)$ and the set of Lusztig data.