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Realization of affine type $$A$$ Kirillov-Reshetikhin crystals via polytopes. (English) Zbl 1300.17012
To an affine Lie algebra $$\mathfrak{g}$$, we may associate a quantum affine algebra $$U_{q}'(\mathfrak{g})$$ (without derivation). Amongst the finite-dimensional irreducible modules for this quantum algebra, the class of Kirillov-Reshetikhin modules are particularly important and they have received substantial attention. As with quantized enveloping algebras of finite-dimensional semisimple Lie algebras, one of the most important methods of study is the construction of crystal bases, following Kashiwara.
The main aim of the present work is to provide a realization of crystals for Kirillov-Reshetikhin modules in type $$A_n^{(1)}$$. Previous realizations involving Young tableaux and the Robinson-Schensted-Knuth correspondence ([S.-J. Kang et al., Duke Math. J. 68, No. 3, 499–607 (1992; Zbl 0774.17017)], [M. Shimozono, J. Algebr. Comb. 15, No. 2, 151–187 (2002; Zbl 1106.17305)]) are known, and the approach here is intended to complement these and provide an alternative method to compute these crystals via the polytope originally defined by E.Feigin, G. Fourier and P. Littelmann [Transform. Groups 16, No. 1, 71–89 (2011; Zbl 1237.17011)].
The construction is outlined in detail and a number of very helpful examples are provided.

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations
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##### References:
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