Tilting mutation of weakly symmetric algebras and stable equivalence

TL;DR

This paper explores tilting mutations of weakly symmetric algebras, detailing how these mutations alter the quiver and relations, and investigates their stable equivalences and implications for derived Picard groups.

Contribution

It provides a detailed description of tilting mutations at vertices of weakly symmetric algebras and studies their stable equivalences and effects on simple modules.

Findings

Explicit quiver and relation descriptions after mutation

Stable equivalences induced by tilting mutations

Answering a question on derived Picard groups

Abstract

We consider tilting mutations of a weakly symmetric algebra at a subset of simple modules, as recently introduced by T. Aihara. These mutations are defined as the endomorphism rings of certain tilting complexes of length 1. Starting from a weakly symmetric algebra A, presented by a quiver with relations, we give a detailed description of the quiver and relations of the algebra obtained by mutating at a single loopless vertex of the quiver of A. In this form the mutation procedure appears similar to, although significantly more complicated than, the mutation procedure of Derksen, Weyman and Zelevinsky for quivers with potentials. By definition, weakly symmetric algebras connected by a sequence of tilting mutations are derived equivalent, and hence stably equivalent. The second aim of this article is to study these stable equivalences via a result of Okuyama describing the images of the simple modules. As an application we answer a question of Asashiba on the derived Picard groups of a class of self-injective algebras of finite representation type. We conclude by introducing a mutation procedure for maximal systems of orthogonal bricks in a triangulated category, which is motivated by the effect that a tilting mutation has on the set of simple modules in the stable category.