Rigid reflections and Kac--Moody algebras

TL;DR

This paper introduces the concepts of rigid reflections and roots in Coxeter groups via geometric curves, linking them to quiver representations and Kac--Moody algebra roots, with conjectures on their bijective correspondence.

Contribution

It defines rigid reflections and roots geometrically, relates them to quiver representations, and establishes a surjective map from Kac--Moody roots to rigid reflections in specific cases.

Findings

Rigid reflections and roots are defined via non-self-intersecting curves.

A surjective map from rank 2 Kac--Moody roots to rigid reflections is established.

Conjecture: this map is bijective for certain Coxeter groups.

Abstract

Given any Coxeter group, we define rigid reflections and rigid roots using non-self-intersecting curves on a Riemann surface with labeled curves. When the Coxeter group arises from an acyclic quiver, they are related to the rigid representations of the quiver. For a family of rank 3 Coxeter groups, we show that there is a surjective map from the set of reduced positive roots of a rank 2 Kac--Moody algebra onto the set of rigid reflections. We conjecture that this map is bijective.