On Kostant's weight $q$-multiplicity formula for $\mathfrak{sp}_6(\mathbb{C})$

Kostant's weight $q$-multiplicity formula is an alternating sum over a finite group known as the Weyl group, whose terms involve the $q$-analog of Kostant's partition function. The $q$-analog of the partition function is a polynomial-valued function defined by $\wp_q(\xi)=\sum_{i=0}^k c_i q^i$, where $c_i$ is the number of ways the weight $\xi$ can be written as a sum of exactly $i$ positive roots of a Lie algebra $\mathfrak{g}$. The evaluation of the $q$-multiplicity formula at $q = 1$ recovers the multiplicity of a weight in an irreducible highest weight representation of $\mathfrak{g}$. In this paper, we specialize to the Lie algebra $\mathfrak{sp}_6(\mathbb{C})$ and we provide a closed formula for the $q$-analog of Kostant's partition function, which extends recent results of Shahi, Refaghat, and Marefat. We also describe the supporting sets of the multiplicity formula (known as the Weyl alternation sets of $\mathfrak{sp}_6(\mathbb{C})$), and use these results to provide a closed formula for the $q$-multiplicity for any pair of dominant integral weights of $\mathfrak{sp}_6(\mathbb{C})$. Throughout this work, we provide code to facilitate these computations.