Combinatorial problems related to Kostant's weight multiplicity formula.

摘要

It is well known that the dimension of a weight space for a finite-dimensional representation of a simple Lie algebra is given by Kostant's weight multiplicity formula, which consists of an alternation of a partition function over the Weyl group. We take a combinatorial approach to address the question of how many terms in the alternation contribute to the multiplicity of the zero weight for any semi-simple Lie algebra of rank 2 and provide diagrams associated to the contributing sets in these low rank examples.;We then consider the multiplicity of the zero weight for certain, very special, highest weights. Specifically, we consider the case where the highest weight is equal to the sum of all simple roots. This weight is dominant only in Lie types A and B. We prove that in all such cases the number of contributing terms is a Fibonacci number. Combinatorial consequences of this fact are provided.