Combinatorial auslander-reiten quivers and reduced expressions

In this paper, we introduce the notion of combinatorial Auslander-Reiten (AR) quivers for commutation classes [ ˜w] of w in a finite Weyl group. This combinatorial object is the Hasse diagram of the convex partial order ≺ _{[ ˜w]} on the subset Φ(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 ₀ ] of the longest element w ₀ of any finite type.