Mutation of $\tau$-exceptional sequences for acyclic quivers over local algebras

TL;DR

This paper demonstrates that mutation of τ-exceptional sequences over certain local algebra-based quivers aligns with classical mutation, revealing the braid group's transitive action on these sequences.

Contribution

It establishes the equivalence between τ-exceptional sequence mutation and classical mutation for acyclic quivers over local algebras, extending known mutation theories.

Findings

Mutation of τ-exceptional sequences coincides with classical mutation.

The braid group acts transitively on complete τ-exceptional sequences.

The result applies to acyclic quivers over local algebras.

Abstract

Let $k$ be an algebraically closed field. Let $R$ be a local commutative finite dimensional $k$-algebra and let $Q$ be a quiver with no loops or oriented cycles. We show that mutation of $\tau$-exceptional sequences over $\Lambda = R\otimes_k kQ \cong RQ$ in the sense of Buan, Hanson, and Marsh coincides with the classical mutation of exceptional sequences defined by Crawley-Boevey and Ringel. In particular, the braid group acts transitively on the set of complete $\tau$-exceptional sequences in $\text{mod }{\Lambda}$.