Torsion pairs for quivers and the Weyl groups

TL;DR

This paper connects Coxeter group elements, $c$-sortable elements, and torsion pairs via representation theory of preprojective algebras, providing new insights and proofs of existing conjectures.

Contribution

It offers a new interpretation of Reading's map $\pi^c$ in terms of representation theory and describes cofinite torsion classes explicitly, advancing understanding of the structure of Coxeter groups.

Findings

Interpretation of $\pi^c$ in representation theory

Explicit description of cofinite torsion classes

Proofs of conjectures by Oppermann, Reiten, and others

Abstract

We give an interpretation of the map $\pi^c$ defined by Reading, which is a map from the elements of a Coxeter group to the $c$-sortable elements, in terms of the representation theory of preprojective algebras. Moreover, we study a close relationship between $c$-sortable elements and torsion pairs, and give an explicit description of the cofinite torsion classes in the context of the Coxeter group. As a consequence, we give a proof of some conjectures proposed by Oppermann, Reiten, and the second author.