Sequences of reflection functors and the preprojective component of a valued quiver

TL;DR

This paper provides an explicit geometric formula for identifying minimal reflection sequences that annihilate preprojective representations of valued quivers, revealing structural insights into their Auslander-Reiten components and related Weyl groups.

Contribution

It introduces a geometric approach to determine shortest reflection sequences for preprojective representations, linking quiver theory with Weyl group combinatorics.

Findings

Explicit formula for shortest (+)-admissible sequences

Partial order structure on equivalence classes of sequences

Applications to Weyl group reduced words

Abstract

This paper concerns preprojective representations of a finite connected valued quiver without oriented cycles. For each such representation, an explicit formula in terms of the geometry of the quiver gives a unique, up to a certain equivalence, shortest (+)-admissible sequence such that the corresponding composition of reflection functors annihilates the representation. The set of equivalence classes of the above sequences is a partially ordered set that contains a great deal of information about the preprojective component of the Auslander-Reiten quiver. The results apply to the study of reduced words in the Weyl group associated to an indecomposable symmetrizable generalized Cartan matrix.