Combinatorics of regular \(A_{2}\)-crystals
The authors study regular \(2\)-colored simply laced crystals for the Cartan matrix of type \(A_2\), which are connected and in which all monochromatic paths are finite. The main structural result of the paper says that each such crystal can be produced using replicating and gluing together in a certain way, from two crystals of a very special form. As a consequence, the combinatorial structure of such crystals becomes rather transparent and the authors obtain several corollaries on combinatorial, polyhedral and other properties of such crystals. In particular, it turns out that such crystal is a Hasse diagram of a finite lattice. A structural characterization of some other class of crystals is also given.
scientific article; zbMATH DE number 5144390