Realization of affine type \(A\) Kirillov-Reshetikhin crystals via polytopes (Q388728)

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.NEWLINENEWLINEThe 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 ([\textit{S.-J. Kang} et al., Duke Math. J. 68, No. 3, 499--607 (1992; Zbl 0774.17017)], [\textit{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 \textit{E.Feigin, G. Fourier} and \textit{P. Littelmann} [Transform. Groups 16, No. 1, 71--89 (2011; Zbl 1237.17011)].NEWLINENEWLINEThe construction is outlined in detail and a number of very helpful examples are provided.