BIRS Workshop Lecture Videos
Nonsymmetric Macdonald polynomials and Demazure characters Assaf, Sami
Nonsymmetric Macdonald polynomials are a polynomial generalization of their symmetric counterparts that exist for all root systems. The combinatorial formula for type A, due to Haglund, Haiman and Loehr, resembles the symmetric formula by the same authors, but with rational functions that complicate the combinatorics. By specializing one parameter to 0, the combinatorics simplifies and we are able to give an explicit formula for the expansion into Demazure characters, a basis for the polynomial ring that contains and generalizes the Schur basis for symmetric polynomials. The formula comes via an explicit Demazure crystal structure on semistandard key tabloids, constructed jointly with Nicolle Gonzalez. By taking stable limits, we return to the symmetric setting and obtain a new formula for the Schur expansion of Hall-Littlewood polynomials that uses a simple major index statistic computed from highest weights of the crystal.
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