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# Sami Assaf, USC – Geometric Methods in Representation Theory Seminar

## September 7, 2018 @ 4:00 pm - 5:00 pm

**367 Phillips Hall, 4:00 pm – 5:00 pm**

**Speaker:** Sami Assaf, USC

**Title:*** **Nonsymmetric Macdonald polynomials and Demazure characters*

**Abstract: **Macdonald introduced symmetric functions in two parameters that simultaneously generalize Hall—Littlewood symmetric functions and Jack symmetric functions. Opdam and Macdonald independently introduced nonsymmetric polynomial versions of these that Cherednik then generalized to any root system. Ion showed that these nonsymmetric Macdonald polynomials with one parameter specialized to 0 arise as characters for affine Demazure modules. Recently, I used the Haglund—Haiman—Loehr combinatorial formula for nonsymmetric Macdonald polynomials in type A to show that, in fact, the specialized nonsymmetric Macdonald polynomials are graded sums of finite Demazure characters in type A. In this talk, I’ll present joint work with Nicolle Gonzalez where we construct an explicit Demazure crystal for specialized nonsymmetric Macdonald polynomials, giving rise to an explicit formula for the Demazure expansion in terms of certain lowest weight elements. Connecting back with the symmetric case, this gives a refinement of the Kostka—Foulkes polynomials defined by the Schur expansion of Hall—Littlewood symmetric functions.

This talk assumes no prior knowledge of Macdonald polynomials, Demazure characters, or crystals.