Geodesic
A generalization of the alcove model and its applications
Journal of Algebraic Combinatorics, Tome 41 (2015) no. 3, pp. 751-783.

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The alcove model of the first author and Postnikov uniformly describes highest weight crystals of semisimple Lie algebras. We construct a generalization, called the quantum alcove model. In joint work of C. Lenart et al. [Int. Math. Res. Not. 2015, No. 7, 1848--1901 (2015; Zbl 06435174)], this was shown to uniformly describe tensor products of column shape Kirillov-Reshetikhin crystals in all untwisted affine types; moreover, an efficient formula for the corresponding energy function is available. In the second part of this paper, we specialize the quantum alcove model to types $A$ and $C$. We give explicit affine crystal isomorphisms from the specialized quantum alcove model to the corresponding tensor products of column shape Kirillov-Reshetikhin crystals, which are realized in terms of Kashiwara-Nakashima columns.
Classification : 05E10, 05C50, 20G42
Keywords: Kirillov-Reshetikhin crystals, energy function, alcove model, quantum Bruhat graph, Kashiwara-Nakashima columns 
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     author = {Lenart, Cristian and Lubovsky, Arthur},
     title = {A generalization of the alcove model and its applications},
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Lenart, Cristian; Lubovsky, Arthur. A generalization of the alcove model and its applications. Journal of Algebraic Combinatorics, Tome 41 (2015) no. 3, pp. 751-783. http://geodesic.mathdoc.fr/item/JAC_2015__41_3_a6/