Stability of plethysm coefficients and modified polynomial induction

Abstract

Let $h_n$ denote the complete homogeneous symmetric function of degree $n$, $s_{\nu }$ the Schur function indexed by a partition $\nu$, and $⟨\cdot ,\cdot ⟩$ the Hall inner product. The plethysm coefficient $⟨h_n[h_m],s_{\nu }⟩$ is the multiplicity of the Weyl module $W_{\nu }({\mathbb{C}}^N)$ in ${\mathrm{Sym}}^n({\mathrm{Sym}}^m({\mathbb{C}}^N))$ whenever $\nu$ has at most $N$ parts. For a partition $\lambda$, let $\lambda [mn]=(mn-|\lambda |,{\lambda }_1,{\lambda }_2,\dots )$. Bowman and Paget proved that $⟨h_n[h_m],s_{\lambda [mn]}⟩$ is independent of $m$ and $n$ once $m,n\ge |\lambda |$. We prove that this coefficient is already constant for $m\ge {\lambda }_1$ and $n\ge \lfloor (|\lambda |+{\lambda }_1)/2\rfloor$, and that its stable value equals $⟨s_{\lambda },H[H-1-h_1]⟩$, where $H={\sum }_{i\ge 0}{h}_i$; the proof uses the skewing operators adjoint to multiplication by complete homogeneous symmetric functions. We give a short new proof of the Bowman-Paget theorem based on stability of vector-partition functions, which count the ways of writing a vector of non-negative integers as an unordered sum of non-zero such vectors (its parts). Using a Vandermonde-type determinantal identity, we deduce that the stable value is a signed sum of numbers of vector partitions all of whose parts have entries summing to at least 2. The same method gives a short proof of Brion's stability theorem. We show that the stable value equals the multiplicity of $W_{\lambda }({\mathbb{C}}^k)$, where $k=\lfloor |\lambda |/2\rfloor$, in an explicit representation of $G{L}_k(\mathbb{C})$ independent of $m$ and $n$, obtained by modifying the polynomial induction of Narayanan, Paul, Prasad and Srivastava. Finally, we prove an explicit positive formula for the stable value, which counts the number of lattice points in a polytope, when ${\lambda }_1\le 4$.

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