For a simple, connected Lie group $G$, let $r_G(n)$ denote the number of $n$-dimensional representations of $G$. Find an asymptotic formula for $r_G(n)$ as $n\to\infty$.

An asymptotic formula is known in the cases $G=SU(2)$ and $G=SU(3)$. For the case of $SU(2)$, $r_{SU(2)} = p(n)$ is simply the number of integer partitions of $n$, whose asymptotic behavior is described by the well-known Hardy-Ramanujan formula: $$ p(n) \sim \frac{1}{4\sqrt{3}n} e^{\pi \sqrt{2n/3} }. $$ For the case of $SU(3)$, Romik (2017) derived an asymptotic formula of the form $$ r_{SU(3)} \sim \frac{K}{n^{3/5}} \exp\left(A_1 n^{2/5} - A_2 n^{3/10} - A_3 n^{1/5} - A_4 n^{1/10} \right), $$ where $K, A_1, A_2, A_3, A_4$ are certain constants described explicitly in terms of special values of the gamma and zeta functions.

In his paper, Romik posed the problem of finding asymptotics for the numbers $r_G(n)$ for more general Lie groups.

• D. Romik. On the number of n-dimensional representations of $SU(3)$, the Bernoulli numbers, and the Witten zeta function. Acta Arith. 180 (2017), 111–159.