Combinatorial Auslander-Reiten quivers and reduced expressions

In this paper, we introduce the notion of combinatorial Auslander-Reiten(AR) quiver for commutation classes $[\tilde{w}]$ of $w$ in finite Weyl group. This combinatorial object visualizes the convex partial order ${\prec }_{[\tilde{w}]}$ on the subset $\mathrm{\Phi }(w)$ 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 $[{\tilde{w}}_0]$ of the longest element $w_0$.