Kostant's Weight Multiplicity Formula and the Fibonacci and Lucas Numbers

TL;DR

This paper explores the zero-weight multiplicity in certain Lie algebra representations, revealing connections to Fibonacci and Lucas numbers through Kostant's formula.

Contribution

It establishes a novel link between weight multiplicities in Lie algebras and Fibonacci and Lucas numbers, providing explicit enumeration formulas.

Findings

Number of contributing terms in type A and B equals Fibonacci numbers.

Number of contributing terms in type C and D equals Lucas numbers.

Provides explicit combinatorial formulas for zero-weight multiplicities.

Abstract

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.