Cluster automorphism group of braid varieties

TL;DR

This paper investigates the automorphism groups of cluster structures on braid varieties, providing a new description and explicit computations to deepen understanding of their symmetries.

Contribution

It offers a novel description of the cluster automorphism group for braid varieties and demonstrates its action through explicit examples.

Findings

Explicit computation of automorphism groups for specific braid varieties

New description of the cluster automorphism group in this context

Enhanced understanding of symmetries in braid varieties

Abstract

The cluster automorphism group of a cluster variety was defined by Gekhtman--Shapiro--Vainshtein, and later studied by Lam--Speyer. Braid varieties are interesting affine algebraic varieties indexed by positive braid words. It was proved recently that braid varieties are cluster varieties. In this paper, we propose a description of the cluster automorphism group and its action on braid varieties, and compute several examples.