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. We formulate the atomic decomposition in arbitrary type, and view it as a strengthening of the monotonicity of K_{lambda,mu}(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_{lambda,mu}(t). We also give a geometric interpretation.

中文翻译：

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字符和晶体的原子分解

Lascoux 指出，A 类 Kostka-Foulkes 多项式 K_{lambda,mu}(t) 在所谓的原子多项式方面正向扩展。对于任何半单李代数，前多项式是最高权重 lambda 的不可约表示中主导权重 mu 的多重性的 t 类比。我们将原子分解公式化为任意类型，并将其视为对 K_{lambda,mu}(t) 单调性的加强。我们还将原子分解的组合版本定义为修改后的晶体图的分解。我们证明这个更强的版本适用于类型 A（它为 Lascoux 的陈述提供了一种新的概念性方法）、类型 B、C 和 D 在 t=1 的稳定范围内，以及在其他一些情况下，而我们推测它更普遍地成立。源自我们工作的另一个猜想导致了 K_{lambda,mu}(t) 的有效计算。我们还给出了几何解释。