Stability of plethysm coefficients and modified polynomial induction
Abstract
Let hn denote the complete homogeneous symmetric function of degree n, sν the Schur function indexed by a partition ν, and ⟨·,·⟩ the Hall inner product. The plethysm coefficient ⟨ hn[hm], sν⟩ is the multiplicity of the Weyl module Wν(ℂN) in Symn(Symm(ℂN)) whenever ν has at most N parts. For a partition λ, let λ[mn]=(mn−|λ|,λ1,λ2,…). Bowman and Paget proved that ⟨ hn[hm], sλ[mn]⟩ is independent of m and n once m,n≥|λ|. We prove that this coefficient is already constant for m≥λ1 and n≥⌊(|λ|+λ1)/2⌋, and that its stable value equals ⟨ sλ, H[H−1−h1]⟩, where H=∑i≥0hi; 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 Wλ(ℂk), where k=⌊|λ|/2⌋, in an explicit representation of GLk(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 λ1≤ 4.
Related
- On evaluation of polynomial coefficients
- Moments of random multiplicative functions with polynomial coefficients
- Combinatorial interpretation of the coefficients of the order polynomial of fence posets
- Combinatorial Goussarov-Polyak-Viro Formulas for the Linking Number and Low Degree Coefficients of the Conway Polynomial
- Norm bounds on Fourier series with polynomial spectra and constrained coefficients
BibTeX
@misc{sarkar2026stability,
title = {{Stability of plethysm coefficients and modified polynomial induction}},
author = {Soumyadip Sarkar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07929},
eprint = {2610.07929},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07929},
}
RIS
TY - UNPB
TI - Stability of plethysm coefficients and modified polynomial induction
AU - Soumyadip Sarkar
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.07929
UR - https://arxiv.org/abs/2610.07929
AB - Let h_n denote the complete homogeneous symmetric function of degree n, s_ν the Schur function indexed by a partition ν, and ⟨·,·⟩ the Hall inner product. The plethysm coefficient ⟨ h_n[h_m], s_ν⟩ is the multiplicity of the Weyl module W_ν(ℂ^N) in Sym^n(Sym^m(ℂ^N)) whenever ν has at most N parts. For a partition λ, let λ[mn]=(mn−|λ|,λ_1,λ_2,…). Bowman and Paget proved that ⟨ h_n[h_m], s_λ[mn]⟩ is independent of m and n once m,n≥|λ|. We prove that this coefficient is already constant for m≥λ_1 and n≥⌊(|λ|+λ_1)/2⌋, and that its stable value equals ⟨ s_λ, H[H−1−h_1]⟩, where H=∑_i≥0h_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_λ(ℂ^k), where k=⌊|λ|/2⌋, in an explicit representation of GL_k(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 λ_1≤ 4.
ER -