Séminaire Lotharingien de Combinatoire, 93B.21 (2025), 12 pp.
Samuel Spellman and Cristian Lenart
A Combinatorial Model for Affine Demazure Crystals of Levels Zero and One
Abstract.
Certain affine crystals, known as Kirillov-Reshetikhin (KR) crystals,
were realized in a uniform way (for all untwisted affine types) in
terms of the quantum alcove model, and their graded characters were
shown to coincide with the specialization of symmetric Macdonald
polynomials at t=0. We generalize these results by introducing a
"non-symmetric" quantum alcove model for certain level zero
Demazure-type crystals inside tensor products of KR crystals, whose
graded characters coincide with non-symmetric Macdonald polynomials at
t=0. Moreover, in type A, we construct an explicit affine
crystal isomorphism between the non-symmetric quantum alcove model and
the semi-standard key tabloid model for level one affine Demazure
crystals due to Assaf and Gonzáe;lez.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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