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Affine crystals, Macdonald polynomials and combinatorial models. (English) Zbl 1389.05168
Summary: Crystals are colored directed graphs encoding information about Lie algebra representations. Kirillov-Reshetikhin (KR) crystals correspond to certain finite-dimensional representations of affine Lie algebras. We present a combinatorial model which realizes tensor products of (column shape) KR crystals uniformly across untwisted affine types. A corollary states that the Macdonald polynomials (which generalize the irreducible characters of simple Lie algebras), upon a certain specialization, coincide with the graded characters of tensor products of KR modules. Some computational applications, as well as related work based on the present one, are also discussed.
05E05 Symmetric functions and generalizations
33D52 Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.)
20G42 Quantum groups (quantized function algebras) and their representations
05C20 Directed graphs (digraphs), tournaments
05E10 Combinatorial aspects of representation theory