Exceptional sequences and clusters

TL;DR

This paper links exceptional sequences in hereditary algebras to Coxeter elements in Weyl groups, providing a new combinatorial way to characterize cluster tilting sets in finite type cases.

Contribution

It establishes a novel characterization of exceptional sequences via reflection products and applies this to describe cluster tilting sets combinatorially.

Findings

Exceptional sequences correspond to inverse Coxeter elements.

New combinatorial characterization of cluster tilting sets.

Applicable to hereditary algebras of finite type.

Abstract

We show that exceptional sequences for hereditary algebras are characterized by the fact that the product of the corresponding reflections is the inverse Coxeter element in the Weyl group. We use this result to give a new combinatorial characterization of clusters tilting sets in the cluster category in the case where the hereditary algebra is of finite type.