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Friday, August 9, 2019

**Abstract:** A. Lascoux stated that the type $A$ Kostka-Foulkes polynomials $K_{\lambda,\mu}(t)$ expand positively in terms of so-called atomic polynomials. For any semisimple Lie algebra, the former polynomial is a $t$-analogue of the multiplicity of the dominant weight $\mu$ in the irreducible representation of highest weight $\lambda$. I formulate the atomic decomposition in arbitrary type, and also define a combinatorial version of it, as a decomposition of a modified version of the Kashiwara crystal graph encoding the representation. This stronger version is shown to hold in type $A$ (which provides a new, conceptual approach to Lascoux's statement), as well as in types $B$, $C$, and $D$ in a stable range for $t=1$. Some applications are also discussed. This is joint work with Cedric Lecouvey, and the presentation will be largely self-contained.