Demazure crystals for specialized nonsymmetric Macdonald 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.

Demazure crystals for Kohnert polynomials

“…Our presentation of the pairing rule and raising operators is taken from [4] where it is shown to be equivalent to that in [6]. Definition 3.2.5 ([6]).…”

“…In Theorem 4.2.5, we prove rectification embeds any set of diagrams connected under the crystal operators into a connected highest weight crystal in a way that interwines the crystal operators on diagrams with the Kashiwara crystal operators [17] on the highest weight crystal. In [4], Assaf and González use rectification in this same fashion to embed diagrams that generate specialized nonsymmetric Macdonald polynomials [16] into highest weight crystals, thus realizing a Demazure crystal structure for nonsymmetric Macdonald polynomials. In [5], Assaf and Quijada prove rectification specializes to Robinson-Schensted insertion [30,31] on semistandard Young tableaux and use it as a tool to prove a signed Pieri formula for Demazure characters.…”

Locks fit into keys: a crystal analysis of lock polynomials

“…Assaf and Schilling [AS18a](Definition 3.7) gave an explicit combinatorial construction of key crystals with raising and lowering operators that act on semistandard key tableaux, which can be translated into the language of Kohnert diagrams and tableaux, as presented in [AG19] by Assaf and González. In this paper, we focus specifically on these crystal operators on Kohnert diagrams and tableaux.…”

A Pieri rule for Demazure characters of the general linear group

“…5, which is a Kohnert diagram for the weak composition (4, 1, 5, 0, 4). We label the cells in row i with label 5 − i + 1 as shown, then raise the cells to the semistandard Young tableau of shape (5,4,4,1) shown on the right of Fig. 5.…”

“…To ease notation, given a weak composition a and a positive integer k, we denote the target space of the bijection by D(a, k), that is In Section 4.1, we give a simple insertion algorithm for the case k = 1, developing along the way several tools for understanding Kohnert diagrams that are essential for all cases. In Section 4.2, we review a generalization of the classical RSK insertion algorithm on tableaux to an insertion algorithm on diagrams [1,4]. In Section 4.3, we use rectification to construct a more subtle insertion algorithm for the case k ≥ ℓ(a), where ℓ(a) denotes the largest index i for which a i > 0.…”