On the number of irreducible representations of $\so(5)$

Abstract

Let $d(n)$ be the divisor function and it is well known that $\sum_{1\leq n \leq x}d(n) = x\log x+(2\gamma-1)x +\mathcal{O}\left(x^{\theta+\epsilon}\right)$ where $\gamma$ is the Euler constant, $\epsilon>0$ and $1/4<\theta<1/3$. In this paper, we obtain an asymptotic formula for the number of irreducible representations of $\mathfrak{so}(5)$. More precisely, the irreducible representations of the Lie algebra $\mathfrak{so}(5)$ are a family of representations of dimension $jk(j+k)(j+2k)/6$ for $j, k\in \mathbb{N}_{0}$ and suppose that $\varrho_{\mathfrak{so}(5)}(n)$ is the number of irreducible $\mathfrak{so}(5)$ representations of dimension $n$. We obtain an asymptotic formula for the summatory function $\sum_{1\leq n \leq x}\varrho_{\mathfrak{so}(5)}(n)$.