Combinatorial auslander-reiten quivers and reduced expressions

Abstract

In this paper, we introduce the notion of combinatorial Auslander-Reiten (AR) quivers for commutation classes [(w) over tilde] of w in a finite Weyl group. This combinatorial object is the Hasse diagram of the convex partial order (sic)([(w) over tilde]) on the subset Phi(w) of positive roots. By analyzing properties of the combinatorial AR-quivers with labelings and reflection functors, we can apply their properties to the representation theory of KLR algebras and dual PBW-basis associated to any commutation class [(w) over tilde (0)] of the longest element w(0) of any finite type.