Mutating signed $\tau$-exceptional sequences

TL;DR

This paper explores properties of $ au$-exceptional sequences in finite-dimensional algebras, focusing on their structure, symmetry actions, and how mutations of support $ au$-tilting modules affect these sequences.

Contribution

It establishes a bijection between support $ au$-tilting modules and signed $ au$-exceptional sequences, and describes the symmetric group's action and mutation effects on these sequences.

Findings

Bijection between support $ au$-tilting modules and signed $ au$-exceptional sequences.

Description of symmetric group action on $ au$-exceptional sequences.

Analysis of mutation effects on $ au$-exceptional sequences.

Abstract

We establish some properties of $\tau$-exceptional sequences for finite-dimensional algebras. In an earlier paper we established a bijection between the set of ordered support $\tau$-tilting modules and the set of complete signed $\tau$-exceptional sequences. We describe the action of the symmetric group on the latter induced by its natural action on the former. Similarly, we describe the effect on a $\tau$-exceptional sequence obtained by mutating the corresponding ordered support $\tau$-tilting module via a construction of Adachi-Iyama-Reiten.