# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1901.07520