Reflection groups and quiver mutation: Diagrammatics

TL;DR

This paper classifies diagrams associated with reduced reflection factorizations in finite Weyl groups, revealing their connection to quiver mutation and providing new presentations of these groups.

Contribution

It introduces a complete classification of diagrams linked to reflection factorizations and connects them to quiver mutation and Weyl group presentations.

Findings

Diagrams are cyclically orientable iff they are mutation-equivalent to Dynkin quivers.

Each diagram encodes a natural presentation of the Weyl group.

The classification extends previous work on reflection groups and quiver mutations.

Abstract

We extend Carter's notion of admissible diagrams and attach a "Dynkin-like" diagram to each reduced reflection factorization of an element in a finite Weyl group. We give a complete classification for the diagrams attached to reduced reflection factorizations. Remarkably, such a diagram turns out to be cyclically orientable if and only if it is isomorphic to the underlying graph of a quiver which is mutation-equivalent to a Dynkin quiver. Furthermore we show that each diagram encodes a natural presentation of the Weyl group as reflection group. The latter one extends work of Cameron, Seidel and Tsaranov as well as Barot and Marsh.