Demazure crystals for specialized nonsymmetric Macdonald polynomials

TL;DR

This paper constructs Demazure crystals for specialized nonsymmetric Macdonald polynomials at t=0, providing explicit formulas and new combinatorial proofs for their positivity and expansions in terms of Demazure characters.

Contribution

It introduces a new crystal construction for nonsymmetric Macdonald polynomials at t=0, enabling explicit formulas and combinatorial proofs of their positivity.

Findings

Explicit nonnegative expansion formulas for specialized nonsymmetric Macdonald polynomials.

Development of an efficient algorithm to compute Demazure crystal characters.

New combinatorial formula for the Schur expansion of Hall--Littlewood polynomials.

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.