Mutating ordered $\tau$-rigid modules with applications to Nakayama algebras

TL;DR

This paper studies mutation operations on $ au$-rigid modules over finite-dimensional algebras, providing a combinatorial description for Nakayama algebras and extending the understanding of $ au$-exceptional sequences.

Contribution

It introduces a new interpretation of mutation in terms of TF-ordered $ au$-rigid modules and proves transitivity of mutation specifically for Nakayama algebras.

Findings

Mutation is transitive for Nakayama algebras.

Provides explicit combinatorial description of mutation.

Connects $ au$-exceptional sequences with $ au$-rigid modules.

Abstract

A mutation operation for $\tau$-exceptional sequences of modules over any finite-dimensional algebra was recently introduced, generalising the mutation for exceptional sequences of modules over hereditary algebras. We interpret this mutation in terms of TF-ordered $\tau$-rigid modules, which are in bijection with $\tau$-exceptional sequences. As an application we show that the mutation is transitive for Nakayama algebras, by providing an explicit combinatorial description of mutation over this class of algebras.