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arXiv · 1901.07520

Demazure crystals for specialized nonsymmetric Macdonald polynomials

Sami Assaf·Nicolle Gonzalez
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Abstract

We give an explicit, nonnegative formula for the expansion of nonsymmetric Macdonald polynomials specialized at $t=0$ in terms of Demazure characters. Our formula results from constructing Demazure crystals whose characters are the nonsymmetric Macdonald polynomials, which also gives a new proof that these specialized nonsymmetric Macdonald polynomials are positive graded sums of Demazure characters. Demazure crystals are certain truncations of classical crystals that give a combinatorial skeleton for Demazure modules. To prove our construction, we develop further properties of Demazure crystals, including an efficient algorithm for computing their characters from highest weight elements. As a corollary, we obtain a new formula for the Schur expansion of Hall--Littlewood polynomials in terms of a simple statistic on highest weight elements of our crystals.

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Sami Assaf, Nicolle Gonzalez. 2019-01-22. Demazure crystals for specialized nonsymmetric Macdonald polynomials. https://arxiv.org/abs/1901.07520

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