Combinatorial Auslander-Reiten quivers and reduced expressions

TL;DR

This paper introduces combinatorial Auslander-Reiten quivers for commutation classes in finite Weyl groups, providing a visual tool to understand partial orders on roots and applications to representation theory.

Contribution

It defines combinatorial AR-quivers for commutation classes, linking combinatorics with representation theory of KLR algebras and PBW generators.

Findings

Visualizes convex partial orders on roots

Connects combinatorial AR-quivers to KLR algebra representations

Provides tools for analyzing multiplication structures

Abstract

In this paper, we introduce the notion of combinatorial Auslander-Reiten(AR) quiver for commutation classes $[\widetilde{w}]$ of $w$ in finite Weyl group. This combinatorial object visualizes the convex partial order $\prec_{[\widetilde{w}]}$ on the subset $\Phi(w)$ of positive roots. By analyzing properties of the combinatorial AR-quivers with labelings and reflection maps, we can apply their properties to the representation theory of KLR algebras and multiplication structure of dual PBW generators associated to any commutation class $[\widetilde{w}_0]$ of the longest element $w_0$.