A UNIFORM MODEL FOR KIRILLOV–RESHETIKHIN CRYSTALS III: NONSYMMETRICMACDONALD POLYNOMIALS AT t = 0 AND DEMAZURE CHARACTERS

By Cristian Lenart, S. Naito, Daisuke Sagaki, Anne Schilling, Mark Shimozono. Transformation groups, 2015

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We establish the equality of the specialization Ewλ(x ; q; 0) of the nonsymmetric Macdonald polynomial Ewλ(x ; q; t) at t = 0 with the graded character gch Uw+(λ) of a certain Demazure-type submodule Uw+(λ) of a tensor product of “single-column” Kirillov–Reshetikhin modules for an untwisted affine Lie algebra, where λ is a dominant integral weight and w is a (finite) Weyl group element; this generalizes our previous result, that is, the equality between the specialization Pλ(x ; q; 0) of the symmetric Macdonald polynomial Pλ(x ; q; t) at t = 0 and the graded character of a tensor product of single-column Kirillov–Reshetikhin modules. We also give two combinatorial formulas for the mentioned specialization of nonsymmetric Macdonald polynomials: one in terms of quantum Lakshmibai–Seshadri paths and the other in terms of the quantum alcove model.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

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