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# Journal of Combinatorics

## Volume 13 (2022)

### Number 2

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

Pages: 167 – 199

DOI: https://dx.doi.org/10.4310/JOC.2022.v13.n2.a1

#### Authors

#### 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.

#### Keywords

Kostant's partition function, $q$-analog of Kostant's partition function, Gaussian distribution

#### 2010 Mathematics Subject Classification

Primary 05E10, 22E60. Secondary 05A15.

Received 16 January 2020

Accepted 3 March 2021

Published 30 March 2022