Braid group actions on grassmannians and extended crystals of type $A$

TL;DR

This paper establishes the equivalence of braid group actions on infinite Grassmannian cluster algebras, quantum Grothendieck rings, and extended crystals in affine type A, unifying different algebraic structures.

Contribution

It proves that the braid actions from Grassmannian cluster algebras coincide with those on quantum Grothendieck rings and extended crystals in affine type A.

Findings

Braid actions $\sigma_i$ match $\mathsf{T}_i$ on Grothendieck rings.

Braid actions $\sigma_i$ match $\mathsf{R}_i$ on extended crystals.

Unified understanding of braid group actions across multiple algebraic frameworks.

Abstract

Let $\sigma_i$ be the braid actions on infinite Grassmannian cluster algebras induced from Fraser's braid group actions. Let $\mathsf{T}_i$ be the braid group actions on (quantum) Grothendieck rings of Hernandez-Leclerc category ${\mathscr C}_\mathfrak{g}^0$ of affine type $A_n^{(1)}$, and $\mathsf{R}_i$ the braid group actions on the corresponding extended crystals. In the paper, we prove that the actions $\sigma_i$ coincide with the braid group actions $\mathsf{T}_i$ and $\mathsf{R}_i$.