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Transactions of the American Mathematical Society

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Nonsymmetric Macdonald polynomials and a refinement of Kostka-Foulkes polynomials

Author: Sami Assaf
Journal: Trans. Amer. Math. Soc. 370 (2018), 8777-8796
MSC (2010): Primary 33D52; Secondary 05E05
Published electronically: July 31, 2018
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Abstract: We study the specialization of the type A nonsymmetric Macdonald polynomials at $ t=0$ based on the combinatorial formula of Haglund, Haiman, and Loehr. We prove that this specialization expands nonnegatively into the fundamental slide polynomials, introduced by the author and Searles. Using this and weak dual equivalence, we prove combinatorially that this specialization is a positive graded sum of Demazure characters. We use stability results for fundamental slide polynomials to show that this specialization stabilizes and to show that the Demazure character coefficients give a refinement of the Kostka-Foulkes polynomials.

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Sami Assaf
Affiliation: Department of Mathematics, University of Southern California, 3620 S. Vermont Avenue, Los Angeles, California 90089-2532

Keywords: Macdonald polynomials, Demazure characters, Kostka--Foulkes polynomials
Received by editor(s): March 7, 2017
Received by editor(s) in revised form: March 9, 2017, and August 16, 2017
Published electronically: July 31, 2018
Article copyright: © Copyright 2018 by Sami Assaf

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