Computing the $q$-Multiplicity of the Positive Roots of ${\mathfrak{sl}}_{r+1}(\mathbb{C})$ and Products of Fibonacci Numbers

Abstract

Using Kostant's weight multiplicity formula, we describe and enumerate the terms contributing a nonzero value to the multiplicity of a positive root $\mu$ in the adjoint representation of ${\mathfrak{sl}}_{r+1}(\mathbb{C})$, which we denote $L(\tilde{\alpha })$, where $\tilde{\alpha }$ is the highest root of ${\mathfrak{sl}}_{r+1}(\mathbb{C})$. We prove that the number of terms contributing a nonzero value in the multiplicity of the positive root $\mu ={\alpha }_i+{\alpha }_{i+1}+\cdots +{\alpha }_j$ with $1\le i\le j\le r$ in $L(\tilde{\alpha })$ is given by the product $F_i\cdot F_{r-j+1}$, where $F_n$ is the $n^{\text{th}}$ Fibonacci number. Using this result, we show that the $q$-multiplicity of the positive root $\mu ={\alpha }_i+{\alpha }_{i+1}+\cdots +{\alpha }_j$ with $1\le i\le j\le r$ in the representation $L(\tilde{\alpha })$ is precisely $q^{r-h(\mu )}$, where $h(\mu )=j-i+1$ is the height of the positive root $\mu$. Setting $q=1$ recovers the known result that the multiplicity of a positive root in the adjoint representation of ${\mathfrak{sl}}_{r+1}(\mathbb{C})$ is one.

16 pages, 0 figures