Séminaire Lotharingien de Combinatoire, B76d (2019), 27 pp.

Per Alexandersson

Non-Symmetric Macdonald Polynomials and Demazure-Lusztig Operators

Abstract. We extend the family non-symmetric Macdonald polynomials and define permuted-basement Macdonald polynomials. We show that these also satisfy a triangularity property with respect to the monomial basis and behave well under the Demazure-Lusztig operators. The symmetric Macdonald polynomials Pλ are expressed as a sum of permuted-basement Macdonald polynomials via an explicit formula.

By letting q=0, we obtain t-deformations of key polynomials and Demazure atoms and we show that the Hall-Littlewood polynomials expand positively into these deformations. This generalizes a result by Haglund, Luoto, Mason and van Willigenburg. As a corollary, the Schur polynomials decompose with non-negative coefficients into t-deformations of general Demazure atoms and thus generalize the t=0 case which was previously known. This gives a unified formula for the classical expansion of Schur polynomials in Hall-Littlewood polynomials and the expansion of Schur polynomials into Demazure atoms.

Received: April 15, 2016. Accepted: May 30, 2019. Final Version: July 30, 2019.

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