On the asymptotic behavior of the $q$-analog of Kostant's partition function

Kostant's partition function counts the number of distinct ways to express a weight of a classical Lie algebra $\mathfrak{g}$ as a sum of positive roots of $\mathfrak{g}$. We refer to each of these expressions as decompositions of a weight and our main result establishes that the (normalized) distribution of the number of positive roots in the decomposition of the highest root of a classical Lie algebra of rank $r$ converges to a Gaussian distribution as $r\to\infty$. We extend these results to an infinite family of weights, irrespective of Lie type, for which we establish a closed formula for the $q$-analog of Kostant's partition function and then prove that the analogous distribution also converges to a Gaussian distribution as the rank of the Lie algebra goes to infinity. We end our analysis with some directions for future research.