Admissible sequences, preprojective modules, and reduced words in the Weyl group of a quiver

TL;DR

This paper explores the relationship between preprojective modules, admissible sequences, and reduced words in the Weyl group of a quiver, revealing new connections and strengthening existing results in the theory of Coxeter groups.

Contribution

It establishes a unique correspondence between preprojective modules and shortest admissible sequences, linking these sequences to reduced words in the Weyl group, and applies this to properties of Coxeter elements.

Findings

Shortest admissible sequences uniquely correspond to preprojective modules.

A sequence is shortest iff its associated product is a reduced word.

The Weyl group is infinite iff powers of a Coxeter element are reduced words.

Abstract

This paper studies connections between the preprojective modules over the path algebra of a finite connected quiver without oriented cycles, the (+)-admissible sequences of vertices, and the Weyl group. For each preprojective module, there exists a unique up to a certain equivalence shortest (+)-admissible sequence annihilating the module. A (+)-admissible sequence is the shortest sequence annihilating some preprojective module if and only if the product of simple reflections associated to the vertices of the sequence is a reduced word in the Weyl group. These statements have the following application that strengthens known results of Howlett and Fomin-Zelevinsky. For any fixed Coxeter element of the Weyl group associated to an indecomposable symmetric generalized Cartan matrix, the group is infinite if and only if the powers of the element are reduced words.