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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 *K*_{λ,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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