Stability of plethysm coefficients and modified polynomial induction

The plethysm coefficient $⟨h_n[h_m],s_{\lambda }⟩$ is the multiplicity of the Weyl module $W_{\lambda }({\mathbb{C}}^N)$ in the representation ${\mathrm{S}\mathrm{y}\mathrm{m}}^n({\mathrm{S}\mathrm{y}\mathrm{m}}^m({\mathbb{C}}^N))$ of $G{L}_N(\mathbb{C})$. We give short proofs of two stability results: the theorem of Bowman and Paget that $⟨h_n[h_m],s_{\lambda [mn]}⟩$ is constant for $m,n\ge \mathrm{\mid }\lambda \mathrm{\mid }$, and Brion’s theorem that $⟨h_n[h_{m+d}],s_{\lambda +(nd)}⟩$ stabilizes as $d\to \mathrm{\infty }$. A key step is the stability of vector partition functions. We show that the stable value in the theorem of Bowman and Paget equals $⟨h_{\lfloor \mathrm{\mid }\lambda \mathrm{\mid }\mathrm{/}2\rfloor }[H-h_1],s_{\lambda }⟩$. Our main new result connects this stable value to the multiplicity of the Weyl module in a representation of $G{L}_{\mathrm{\mid }\lambda \mathrm{\mid }}(\text{C})$. We give a formula for the stable Foulkes’ coefficient in terms of a certain vector-partition function.