Séminaire Lotharingien de Combinatoire, 82B.40 (2019), 12 pp.

Cédric Lecouvey and Cristian Lenart

Atomic decomposition of characters and crystals

Abstract. Lascoux stated that the type A Kostka-Foulkes polynomials Kλ,μ(t) (Lusztig's t-analogue of weight multiplicity) expand positively in terms of so-called atomic polynomials. We define, in arbitrary type, a combinatorial version of the atomic decomposition, based on the connected components of a modified crystal graph. We prove this property in type A, as well as in types B, C, and D in a stable range for t = 1. We also discuss other cases, applications, and a geometric interpretation.

Received: November 15, 2018. Accepted: February 17, 2019. Final version: April 1, 2019.

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