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 Combinatorial Auslander-Reiten quivers and reduced expressions J. Korean Math. Soc. 2019 Vol. 56, No. 2, 353-385 Published online 2019 Mar 01 Se-jin Oh, Uhi Rinn Suh Ewha Womans University; Seoul National University Abstract : In this paper, we introduce the notion of combinatorial Aus\-lander-Reiten (AR) quivers for commutation classes $[\widetilde{w}]$ of $w$ in a finite Weyl group. This combinatorial object is the Hasse diagram of 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 functors, we can apply their properties to the representation theory of KLR algebras and dual PBW-basis associated to any commutation class $[\widetilde{w}_0]$ of the longest element $w_0$ of any finite type. Keywords : combinatorial AR-quiver, reduced expressions MSC numbers : 81R50, 05E10, 16T30, 17B37 Full-Text :