On the $q$-multiplicity of sums of distinct simple roots of ${\mathfrak{sl}}_{r+1}(\mathbb{C})$

Abstract

In combinatorial representation theory, Kostant's weight multiplicity formula $m(\lambda ,\mu )$ is a tool that provides a means of determining the multiplicity of a weight $\mu$ in the adjoint representation of a simple Lie algebra $\mathfrak{g}$, and in this work we consider the case of $\mathfrak{g}={\mathfrak{sl}}_{r+1}(\mathbb{C})$. In practice, performing calculations of Kostant's weight multiplicity formula is computationally intense, as the number of terms in this alternating sum grows factorially as the rank $r$ increases, of which most terms provide zero contribution to the overall sum. In this work, we determine the Weyl alternation set, that is the terms in the alternating sum with nonzero contribution, for integral weights $\lambda$ the highest root of ${\mathfrak{sl}}_{r+1}(\mathbb{C})$, and $\mu$ any nonempty collection of distinct simple roots. We show that the alternation set is enumerated by a product of Fibonacci numbers, with the product being dependent on the choice of distinct simple roots. Then we compute the weight $q$-multiplicity for any nonempty collection of distinct simple roots.