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The crossing model for regular $A_n$-crystals

V. I. Danilov·A. V. Karzanov·G. A. Koshevoy
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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 \cite{DKK-07}, we present a new combinatorial construction, the so-called {\em 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.

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V. I. Danilov, A. V. Karzanov, G. A. Koshevoy. 2008-08-08. The crossing model for regular $A_n$-crystals. https://arxiv.org/abs/math/0612360

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