Non-Symmetric Macdonald Polynomials and Demazure-Lusztig Operators

Per Alexandersson

Séminaire lotharingien de combinatoire, Tome 76 (2016-2018)

Résumé

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 P_λ 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.