Affine Demazure crystals for nonsymmetric Macdonald polynomials.

Abstract: Macdonald polynomials have long been hailed as a breakthrough in algebraic combinatorics as they simultaneously generalize both Hall-Littlewood and Jack symmetric polynomials. The nonsymmetric Macdonald polynomials $E_a(X;q,t)$ are a further generalization which contain the symmetric versions as special cases. When specialized at $t =0$ the nonsymmetric Macdonald polynomials were shown by Bogdon and Sanderson to arise as characters of affine Demazure modules, which are certain truncations of highest weight modules. In this talk, I will describe a type A combinatorial crystal which realizes the affine Demazure module structure and recovers the results of Bogdon and Sanderson crystal-theoretically. The construction yields a filtration of these affine crystals by finite Demazure crystals via certain embedding operators that model those of Knop and Sahi for nonsymmetric Macdonald polynomials. Thus, we obtain an explicit combinatorial expansion of the specialized nonsymmetric Macdonald polynomials as graded sums of key polynomials. As a consequence, we derive a new combinatorial formula for the Kostka-Foulkes polynomials. This is joint work with Sami Assaf.

combinatorics

Audience: researchers in the topic

York University Applied Algebra Seminar