ALGEBRAIC COMBINATORICS

no. 5
Affine Demazure crystals for specialized nonsymmetric Macdonald polynomials
Assaf, Sami1; González, Nicolle2
1 Department of Mathematics, University of Southern California, 3620 S. Vermont Ave., Los Angeles, CA 90089-2532, U.S.A.
2 UCLA Department of Mathematics, Los Angeles CA 90095-1555, U.S.A.
Algebraic Combinatorics, Volume 4 (2021) no. 5, pp. 777-793.

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.178
Classification: 05E10, 05E05, 05A05, 05E18, 17B37
Keywords: Affine Demazure crystal, affine Demazure character, nonsymmetric Macdonald polynomial
Assaf, Sami 1; González, Nicolle 2

1 Department of Mathematics, University of Southern California, 3620 S. Vermont Ave., Los Angeles, CA 90089-2532, U.S.A.
2 UCLA Department of Mathematics, Los Angeles CA 90095-1555, U.S.A.
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Assaf, Sami; González, Nicolle. Affine Demazure crystals for specialized nonsymmetric Macdonald polynomials. Algebraic Combinatorics, Volume 4 (2021) no. 5, pp. 777-793. doi : 10.5802/alco.178. https://alco.centre-mersenne.org/articles/10.5802/alco.178/

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