Braid group action on extended crystals

TL;DR

This paper establishes a braid group action on the extended crystal of finite type and explores its interpretation within the Hernandez-Leclerc category, advancing the understanding of crystal symmetries.

Contribution

It introduces a braid group action on the extended crystal $\, ext{hat} B( ext{infinity})$ and interprets it in the Hernandez-Leclerc category, linking crystal theory with categorical frameworks.

Findings

Existence of a braid group action on extended crystals of finite type.

Interpretation of the braid group action within the Hernandez-Leclerc category.

Enhanced understanding of crystal symmetry and categorical relations.

Abstract

In the paper, we prove that there exists a braid group action on the extended crystal $\widehat{B}(\infty)$ of finite type. The extended crystal $\widehat{B}(\infty)$ and its braid group action are investigated from the viewpoint of crystal similarity. We then interpret the braid group action on $\widehat{B}(\infty)$ in the Hernandez-Leclerc category $\mathscr{C}_\mathfrak{g}^0$.