# When is the $q$-multiplicity of a weight a power of $q$?

## TL;DR

This paper investigates when the $q$-multiplicity of a weight in Lie algebra representations is a power of $q$, using Kostant's formula and Fibonacci numbers to analyze specific weight pairs.

## Contribution

It describes and enumerates contributing terms to weight multiplicities for certain pairs, and conjectures that their $q$-multiplicities are always powers of $q$.

## Key findings

- Contributing sets are enumerated by Fibonacci numbers.
- Computed $q$-multiplicities for specific weight pairs.
- Conjecture that $q$-multiplicities are powers of $q$ in all cases.

## Abstract

Berenshtein and Zelevinskii provided an exhaustive list of pairs of weights $(\lambda,\mu)$ of simple Lie algebras $\mathfrak{g}$ (up to Dynkin diagram isomorphism) for which the multiplicity of the weight $\mu$ in the representation of $\mathfrak{g}$ with highest weight $\lambda$ is equal to one. Using Kostant's weight multiplicity formula we describe and enumerate the contributing terms to the multiplicity for subsets of these pairs of weights and show that, in these cases, the cardinality of these contributing sets is enumerated by (multiples of) Fibonacci numbers. We conclude by using these results to compute the associated $q$-multiplicity for the pairs of weights considered, and conjecture that in all cases the $q$-multiplicity of such pairs of weights is given by a power of $q$.