The crossing model for regular An-crystals

A regular An-crystal is an edge-colored directed graph, with n colors, related to an irreducible highest weight integrable module over Uq(sln+1). Based on Stembridgeʹs local axioms for regular simply-laced crystals and a structural characterization of regular A2-crystals in [V.I. Danilov, A.V. Karzanov, G.A. Koshevoy, Combinatorics of regular A2-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 An-crystals. Using the model, we obtain a series of results on the combinatorial structure of such crystals and properties of their subcrystals.