On the combinatorics of Kostant's partition function
The author proves the following technical theorem. Let R be a reduced root system with base \(B=\{\alpha_ 1,...,\alpha_ n\}\) and let \(- C_{ji}=<\alpha_ j,\alpha_ i>\), the Cartan integer. Theorem. Let \(t_ i\) (1\(\leq i\leq n)\) be nonnegative integers. Then \(\sum t_ i \alpha_ i\) can be written as a sum of positive nonsimple roots with nonnegative coefficients iff \(t_ i\leq \sum_{j\neq i}C_{ji} t_ j\) for every \(i=1,...,n.\) One direction in the proof is quite delicate using a construction which handles the type \(G_ 2\) case separately from the other types. As an application of this theorem, there is a result in representation theory. Let U be the universal enveloping algebra of a complex semisimple Lie group and let V(\(\Lambda)\) be an irreducible U-module of highest weigth \(\Lambda\). The result is roughly that the multiplicity of any weight space in V(\(\Lambda)\) is determined by the multiplicities of the allbracket ones.
scientific article; zbMATH DE number 3973106