Coxeter groups and quiver representations

TL;DR

This paper explores the connection between Coxeter groups and quiver representations, highlighting classical and recent developments, including a new elementary proof of a bijection in the general case.

Contribution

It provides an elementary proof of the bijection between torsion free classes with finitely many indecomposables and c-sortable elements in Weyl groups, extending previous Dynkin type results.

Findings

Torsion free classes correspond to c-sortable elements

Classical topics include roots and reflection functors

New proof applies to general quivers, not just Dynkin types

Abstract

In this expository note, I showcase the relevance of Coxeter groups to quiver representations. I discuss (1) real and imaginary roots, (2) reflection functors, and (3) torsion free classes and c-sortable elements. The first two topics are classical, while the third is a more recent development. I show that torsion free classes in rep Q containing finitely many indecomposables correspond bijectively to c-sortable elements in the corresponding Weyl group. This was first established in Dynkin type by Ingalls and Thomas; it was shown in general by Amiot, Iyama, Reiten, and Todorov. The proof in this note is elementary, essentially following the argument of Ingalls and Thomas, but without the assumption that Q is Dynkin.