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.

我们给出了一个专门针对非对称Macdonald多项式展开的显式非负公式 $Ť=0$就Demazure角色而言。我们的公式是通过构造特征为非对称Macdonald多项式的Demazure晶体而得出的，这也提供了新的证明，即这些专门的非对称Macdonald多项式是Demazure特征的正级和。Demazure晶体是经典晶体的某些截断，为Demazure模块提供了组合骨架。为了证明我们的构造，我们开发了Demazure晶体的更多特性，包括一种用于从重量最大的元素中计算其特征的有效算法。作为推论，我们获得了关于霍尔-利特伍德多项式的舒尔展开式的新公式，其依据是我们晶体重量最高的元素的简单统计量。