University of Minnesota Combinatorics Seminar: 2020-2021

Date: 04/16/2021

Speaker: Nicholle Gonzalez

Affine Demazure Crystals for Nonsymmetric Macdonald Polynomials
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 as well as a filtration by finite Demazure crystals built over a family of tableaux. The crystal filtration is achieved 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 well as a new crystal-theoretic proof of the results of Sanderson. As a consequence, we derive a new combinatorial formula for the Kostka-Foulkes polynomials. This is joint work with Sami Assaf.

Affine Demazure Crystals for Nonsymmetric Macdonald Polynomials

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 as well as a filtration by finite Demazure crystals built over a family of tableaux. The crystal filtration is achieved 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 well as a new crystal-theoretic proof of the results of Sanderson. As a consequence, we derive a new combinatorial formula for the Kostka-Foulkes polynomials. This is joint work with Sami Assaf.