The product of any key polynomial with a single-row Schur polynomial expands into key polynomials with coefficients in {-1,0,1}, via a weight-preserving bijection on Kohnert diagrams.
Demazure crystals for specialized nonsymmetric Macdonald polynomials
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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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A Pieri rule for Demazure characters of the general linear group
The product of any key polynomial with a single-row Schur polynomial expands into key polynomials with coefficients in {-1,0,1}, via a weight-preserving bijection on Kohnert diagrams.