Demazure crystals for specialized nonsymmetric Macdonald polynomials

Affine Demazure crystals for specialized nonsymmetric Macdonald polynomials

“…Kashiwara and Nakashima and, independently, Littelmann gave explicit tableaux models for finite type crystals [16,20], and in a similar spirit Assaf and Schilling gave an explicit tableaux model for Demazure crystals in type A [5]. Assaf and González [4] recently gave a crystal theoretic proof of Assaf's result [3] decomposing nonsymmetric Macdonald polynomials specialized at t = 0 as a nonnegative q-graded sum of finite Demazure characters, leading to more explicit formulas.…”

“…This gives a new combinatorial proof of Sanderson's result that the nonsymmetric Macdonald polynomial specialized at t = 0 is the affine Demazure character. Generalizing our earlier finite crystal construction [4], we define affine edges to the finite Demazure crystal on semistandard key tabloids. We provide a realization of the Bruhat filtration on Demazure crystals via embedding operators, which correspond to a combinatorial analogue of the Demazure operators and recover Knop and Sahi's operators (at t = 0) at the level of characters.…”

“…, x n ; q, 0), using notation from [11]. The specialized nonsymmetric Macdonald polynomials can be defined combinatorially as the generating polynomial of semi-standard key tabloids, defined below following notation in [3,4], and so this is the natural choice for the underlying set of the Demazure crystal basis B w (λ).…”

“…In [4] the authors define finite raising and lowering operators on semi-standard key tabloids of shape α giving rise to a finite Demazure crystal structure on SSKD(α). We recall these operators here; see [4,Section 5] for further details.…”

“…• if all entries i of T are i-paired or if the leftmost unpaired i is in row i and all columns to its left have an i in the same row and an i + 1 above, then f i (T ) = 0; • otherwise, f i changes the leftmost unpaired i to i + 1 and swaps the entries i and i + 1 in each of the consecutive columns left of this entry that have an i in the same row and an i + 1 above, and swaps the entries i and i + 1 in each of the consecutive columns right of this entry that have an i in the same row and an i + 1 below. 4]). For i 1, the raising operator e i acts on T ∈ SSKD(α) by • if all entries i + 1 of T are i-paired, then e i (T ) = 0;…”