Stability of plethysm coefficients and modified polynomial induction

Abstract

The plethysm coefficient $\langle h_n[h_m], s_λ\rangle$ is the multiplicity of the Weyl module $W_λ(\mathbb{C}^N)$ in the representation $\mathrm{Sym}^n(\mathrm{Sym}^m(\mathbb{C}^N))$ of $GL_N(\mathbb{C})$. We give short proofs of two stability results: the theorem of Bowman and Paget that $\langle h_n[h_m], s_{λ[mn]} \rangle$ is constant for $m, n \geq |λ|$, and Brion's theorem that $\langle h_n[h_{m+d}],\allowbreak s_{λ+(nd)} \rangle$ stabilizes as $d \to \infty$. A key step is the stability of vector partition functions. We show that the stable value in the theorem of Bowman and Paget equals $\langle h_{\lfloor|λ|/2\rfloor}[H-h_1], s_λ\rangle$. Our main new result connects this stable value to the multiplicity of the Weyl module in a representation of $GL_{|λ|}(\C)$. We give a formula for the stable Foulkes' coefficient in terms of a certain vector-partition function.