Atomic decomposition of characters and crystals

Abstract Lascoux stated that the type A Kostka-Foulkes polynomials K λ , μ ( 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 μ in the irreducible representation of highest weight λ. We formulate the atomic decomposition in arbitrary type, and view it as a strengthening of the monotonicity of K λ , μ ( t ) . We also define a combinatorial version of the atomic decomposition, as a decomposition of a modified crystal graph. We prove that this stronger version holds in type A (which provides a new, conceptual approach to Lascoux's statement), in types B, C, and D in a stable range for t = 1 , as well as in some other cases, while we conjecture that it holds more generally. Another conjecture stemming from our work leads to an efficient computation of K λ , μ ( t ) . We also give a geometric interpretation.