Kevin Chang, Pamela Harris, Erik Insko
Published 2011-11-28, updated 2019-07-30Version 2
Consider the weight $\lambda$ which is the sum of all simple roots of a simple Lie algebra. Using Kostant's weight multiplicity formula we describe and enumerate the contributing terms to the multiplicity of the zero weight in the representation with highest weight $\lambda$. We prove that in Lie algebras of type $A$ and $B$, the number of contributing terms to the multiplicity of the zero-weight space in the representation with highest weight $\lambda$ is given by a Fibonacci number, and that in Lie algebras of type $C$ and $D$, the analogous result is given by a multiple of a Lucas number.
On the adjoint representation of $\mathfrak{sl}_n$ and the Fibonacci numbers
The adjoint representation of a Lie algebra and the support of Kostant's weight multiplicity formula
The support of Kostant's weight multiplicity formula is an order ideal in the weak Bruhat order
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