The crossing model for regular An-crystals
Abstract A regular A n -crystal is an edge-colored directed graph, with n colors, related to an irreducible highest weight integrable module over U q ( sl n + 1 ) . Based on Stembridge's local axioms for regular simply-laced crystals and a structural characterization of regular A 2 -crystals in [V.I. Danilov, A.V. Karzanov, G.A. Koshevoy, Combinatorics of regular A 2 -crystals, J. Algebra 310 (2007) 218–234], we present a new combinatorial construction, the so-called crossing model, and prove that this model generates precisely the set of regular A n -crystals. Using the model, we obtain a series of results on the combinatorial structure of such crystals and properties of their subcrystals.