Quotient closed subcategories of quiver representations

TL;DR

This paper establishes a natural bijection between elements of the Coxeter group associated with a quiver and cofinite quotient-closed subcategories of its module category, linking algebraic and combinatorial structures.

Contribution

It introduces a novel correspondence connecting Coxeter group elements with quotient-closed subcategories in quiver representations, via ideals in the preprojective algebra.

Findings

Bijection between Coxeter group elements and subcategories

Linkage of subcategories to ideals in preprojective algebra

Characterization of quotient-closed subcategories

Abstract

Let Q be a finite quiver without oriented cycles, and let k be an algebraically closed field. The main result in this paper is that there is a natural bijection between the elements in the associated Coxeter group W_Q and the cofinite additive quotient-closed subcategories of the category of finite dimensional right modules over kQ. We prove this correspondence by linking these subcategories to certain ideals in the preprojective algebra associated to Q, which are also indexed by elements of W_Q.