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

TL;DR

This paper derives a closed formula for the $q$-analog of Kostant's partition function for an infinite family of weights and proves that the distribution of the number of positive roots in their decompositions converges to a Gaussian as the Lie algebra rank increases.

Contribution

It provides a new closed-form expression for the $q$-analog of Kostant's partition function and establishes a Gaussian limit law for the distribution of positive roots in decompositions.

Findings

Closed formula for the $q$-analog of Kostant's partition function.

Gaussian distribution convergence for the number of positive roots.

Results extend to the highest root of classical Lie algebras.

Abstract

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. Our main result considers 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 (normalized) distribution of the number of positive roots in the decomposition of any of these weights converges to a Gaussian distribution as the rank of the Lie algebra goes to infinity. We also extend these results to the highest root of the classical Lie algebras and we end our analysis with some directions for future research.