Asymptotics of Plethysm

We study multiplicities $a^{d\lambda}_{\mu,(dk)}$ of highest weight representations $\mathbb S_{d\lambda}(\mathbb C^n)$, $\lambda\vdash pk$, of length at most $p$, in $\mathbb{S}_{\mu}(S^{dk}(\mathbb C^n))$, $\mu\vdash p$, so called plethysm coefficients, as $d$ tends to $\infty$. 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 $$a^{d\lambda}_{\mu,(dk)}\sim \frac{\dim V_\mu}{p!}c^{d\lambda}_{p,dk}$$ as $d\to\infty$. In particular, we answer a conjecture of Kahle and Micha\l ek, going back to Howe.