Affine Demazure crystals for specialized nonsymmetric Macdonald polynomials

Sami Assaf,Nicolle González

doi:10.5802/alco.178

Abstract

We give a crystal-theoretic proof that nonsymmetric Macdonald polynomials specialized to $t=0$ are affine Demazure characters. We explicitly construct an affine Demazure crystal on semistandard key tabloids such that removing the affine edges recovers the finite Demazure crystals constructed earlier by the authors. We also realize the filtration on highest weight modules by Demazure modules by defining explicit embedding operators which, at the level of characters, parallels the recursion operators of Knop and Sahi for specialized nonsymmetric Macdonald polynomials. Thus we prove combinatorially in type A that every affine Demazure module admits a finite Demazure flag.

Highlights

Macdonald [21] defined an important family of polynomials that forms a basis of symmetric polynomials in C(q, t)[x1, . . . , xn]
Sanderson [27] made the connection between nonsymmetric Macdonald polynomials and affine Demazure modules [9] through the character formula stated by Demazure [10] and proved rigorously by Andersen [1]
Using the recurrence formula for nonsymmetric Macdonald polynomials discovered independently by Knop [17] and Sahi [26], Sanderson proved the nonsymmetric Macdonald polynomials specialized at t = 0 are affine Demazure characters

Summary

Introduction

Macdonald [21] defined an important family of polynomials that forms a basis of symmetric polynomials in C(q, t)[x1, . . . , xn]. 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. We give an explicit construction of affine Demazure crystals on semistandard key tabloids, the combinatorial objects that generate nonsymmetric Macdonald polynomials. This gives a new combinatorial proof of Sanderson’s result that the nonsymmetric Macdonald polynomial specialized at t = 0 is the affine Demazure character. Kumar conjectures our type A result holds in greater generality, in particular, that every affine Demazure module admits a finite Demazure flag

Demazure crystals

Tabloid crystals

Crystal filtration