Kostant's weight multiplicity formula and the Fibonacci numbers

Details Kostant's weight multiplicity formula and the Fibonacci numbers Kostant's weight multiplicity formula and the Fibonacci numbers

2011-11-28

arxiv.org/abs/1111.6648v1

Mathematics

Representation Theory

9 pages

Scientific paper

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. We address the question of how many terms in the alternation contribute to 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.

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