Asymptotics of Plethysm

We study multiplicities $a_{\mu ,(dk)}^{d\lambda }$ of highest weight representations ${\mathbb{S}}_{d\lambda }({\mathbb{C}}^n)$, $\lambda ⊢pk$, of length at most $p$, in ${\mathbb{S}}_{\mu }(S^{dk}({\mathbb{C}}^n))$, $\mu ⊢p$, so called plethysm coefficients, as $d$ tends to $\mathrm{\infty }$. These are given by quasi-polynomials, which in the case of $S^p(S^{dk}({\mathbb{C}}^n))$ can explicitly be computed by Pieri’s rule. We show that for all but a finite, explicit list of $\lambda$’s the leading term is in fact constant and that $$a_{\mu ,(dk)}^{d\lambda }\sim \frac{\dim V_{\mu }}{p!}{c}_{p,dk}^{d\lambda }$$ as $d\to \mathrm{\infty }$. In particular, we answer a conjecture of Kahle and Michałek, going back to Howe.