On the derived Picard group of the Brauer star algebra

TL;DR

This paper characterizes the structure of the derived Picard group of Brauer star algebras, showing it is generated by specific autoequivalences and braid relations, with variations depending on the multiplicity.

Contribution

It provides a detailed description of the generators and relations of the derived Picard group for Brauer star algebras, extending previous results to the multiplicity free case.

Findings

Generated by shift, Picard group, and specific autoequivalences for t>1

Relations satisfy the braid group on affine diagram rac{A}_{n-1}

Extended generators in the multiplicity free case

Abstract

In this paper we show that the derived Picard group $TrPic(A)$ of the Brauer star algebra of type $(n,t)$ is generated by shift, $Pic(A)$ and equivalences $\{H_i\}_{i=1}^n$ in the case $t>1$, where $H_i$ were shown to satisfy the relations of the braid group on the affine diagram $\widetilde{A}_{n-1}$ by Schaps and Zakay-Illouz. In the multiplicity free case we show that $TrPic(A)$ is generated by a slightly bigger set.