Exceptional small parameters in the stability of Foulkes plethysm coefficients

For a partition $\lambda$ and integers $m,n$ , let $p((n),(m),\lambda_{[mn]})$ be the coefficient of the Schur function $s_{\lambda_{[mn]}}$ in the plethysm product $s_{(n)}\circ s_{(m)}$ , equivalently the multiplicity of the corresponding Specht module in the Foulkes module induced from the wreath product $\mathfrak S_m\wr\mathfrak S_n$ . Bowman and Paget proved that this double sequence is constant, equal to a stable value $p_{\infty,\lambda}$ , whenever $m,n\ge|\lambda|$ , and the sequel by Bowman, Paget and Wildon proved the analogous bounds sharp in infinitely many cases. The problem asks for the completion of this stability picture: for each fixed $\lambda$ , exactly which sub-stable pairs $(m,n)$ with $\min(m,n)<|\lambda|$ are exceptional and what the coefficient values are there; the same determination below the sharp bounds in the ramified setting (arbitrary outer partition, one-row inner partition); and whether the stability equality extends to inner partitions with more than one row, where only an upper bound is currently known. A complete answer would determine all rectangular plethysm coefficients and decide Foulkes' 1950 conjecture.

Background

The irreducible complex representations of the symmetric group $\mathfrak S_N$ are the Specht modules $S^{(\alpha)}$ , labelled by the partitions $\alpha$ of $N$ (weakly decreasing sequences of positive integers summing to $N$ ); their characters are encoded by the Schur functions $s_\alpha$ , the standard basis of the ring of symmetric functions. The plethysm coefficients are the structure constants for the composition of Schur functors: for partitions $\mu\vdash m$ and $\nu\vdash n$ , the plethysm product $s_\nu\circ s_\mu$ expands as $\sum_\lambda p(\nu,\mu,\lambda)s_\lambda$ , and $p(\nu,\mu,\lambda)$ is at once the multiplicity of the irreducible polynomial representation of the general linear group of highest weight $\lambda$ in the Schur functor of shape $\nu$ applied to that of shape $\mu$ , and — in the case at issue — a symmetric-group branching multiplicity: for the one-row partitions $\nu=(n)$ and $\mu=(m)$ , it is the multiplicity of $S^{(\alpha)}$ in the Foulkes module $\mathrm{ind}_{\mathfrak S_m\wr\mathfrak S_n}^{\mathfrak S_{mn}}\mathbb C$ , where $\mathfrak S_m\wr\mathfrak S_n$ is the wreath product, i.e. the subgroup of $\mathfrak S_{mn}$ preserving a decomposition of $\{1,\dots,mn\}$ into $n$ blocks of size $m$ . Foulkes' 1950 conjecture asserts $p((m),(n),\alpha)\le p((n),(m),\alpha)$ for all $m\le n$ and all $\alpha\vdash mn$ ; it is known only when $m$ is much smaller than $n$ (Brion) and for $m\le 5$ by computations recorded in the source paper. Computing and understanding plethysm coefficients, ideally through positive combinatorial rules, is a central open problem of algebraic combinatorics, and the coefficients also govern multiplicities that are central to geometric complexity theory and quantum information.

A basic phenomenon is stability. For a partition $\lambda$ write $\lambda_{[mn]}=(mn-|\lambda|,\lambda_1,\lambda_2,\dots)$ ; every partition of $mn$ arises this way. Brion, and independently Carré–Thibon, showed that for fixed $\lambda$ the coefficients $p((n),(m),\lambda_{[mn]})$ eventually stabilize along one-parameter families. Bowman–Paget (Part I) proved much more: for every fixed $\lambda$ the double sequence is constant as soon as $m,n\ge|\lambda|$ , with stable value $p_{\infty,\lambda}=\sum_{\mu\in\mathcal P_1(|\lambda|)}p_\mu(\lambda)$ , where $\mathcal P_1(r)$ is the finite set of partitions of $r$ with no singleton parts and the generalized plethysm coefficients $p_\mu(\lambda)$ are explicit combinations of plethysm coefficients of smaller degree and Littlewood–Richardson coefficients (the Littlewood–Richardson rule being the classical combinatorial rule for the structure constants of ordinary products of Schur functions). The method realizes Foulkes modules through the partition algebra $P_r(d)$ , the diagram algebra spanned by set partitions of $\{1,\dots,r,\bar 1,\dots,\bar r\}$ that arises as the centralizer algebra of $\mathfrak S_d$ on tensor space $(\mathbb C^d)^{\otimes r}$ under Schur–Weyl duality.

The sequel by Bowman–Paget–Wildon (Part II) extends the framework to arbitrary plethysm coefficients $p(\beta_{[n]},\alpha_{[m]},\kappa_{[mn]})$ using the ramified partition algebra $R_r(m,n)$ , a diagram algebra on pairs of set partitions, one refining the other, which is Schur–Weyl dual to the wreath product $\mathfrak S_m\wr\mathfrak S_n$ : every such coefficient is the multiplicity of a simple module on restriction from $R_r(m,n)$ to $P_r(mn)$ . Part II proves that when the inner partition is a single row (the case $\alpha=\emptyset$ in its notation) the coefficient is constant for $m\ge r-|\beta|+[\beta\neq\emptyset]$ and $n\ge r+\beta_1$ , where $r=|\kappa|$ and $[\beta\neq\emptyset]$ is $1$ if $\beta\neq\emptyset$ and $0$ otherwise, and gives an explicit positive formula; moreover, by symmetric-function arguments, these bounds cannot be weakened in infinitely many cases — for instance $p((r),(r-1),(r^2-2r,r))$ is exactly one less than its stable value. For inner partitions with more than one row ( $\alpha\neq\emptyset$ ) the algebra $R_r(m,n)$ is never semisimple in the relevant parameter range, and only an upper bound, by the so-called ramified branching coefficients, is proven.

What no part of this literature determines is the behaviour below the stability thresholds: for fixed $\lambda$ , the set of exceptional pairs $(m,n)$ with $\min(m,n)<|\lambda|$ at which the stable value fails and the coefficient values there, and the analogous questions below the sharp bounds of Part II. This sub-stable region is exactly where the still-open Foulkes conjecture lives.

Problem Statement

The problem has one objective — completing the sharp stability picture for Foulkes-type plethysm coefficients by exact determination below the stable range.

(1) Foulkes case. For each fixed partition $\lambda$ , determine exactly the exceptional set $E(\lambda)=\{(m,n)\in\mathbb N^2 : mn-|\lambda|\ge\lambda_1,\ p((n),(m),\lambda_{[mn]})\neq p_{\infty,\lambda}\},$ where the condition $mn-|\lambda|\ge\lambda_1$ only records that $\lambda_{[mn]}=(mn-|\lambda|,\lambda_1,\lambda_2,\dots)$ is a partition; by Part I the set $E(\lambda)$ is contained in the region $\min(m,n)<|\lambda|$ — a region that is itself infinite, since one of $m,n$ may be arbitrarily large, and $E(\lambda)$ may be infinite as well (for example, for $\lambda=(2)$ , $s_{(n)}\circ s_{(1)}=s_{(n)}$ gives $p((n),(1),(n-2,2))=0$ while $p_{\infty,(2)}=1$ , so every pair $(1,n)$ with $n\ge 4$ is exceptional) — and by Part II it is nonempty for infinitely many $\lambda$ . Equivalently, determine the exact values $p((n),(m),\lambda_{[mn]})$ for all pairs with $\min(m,n)<|\lambda|$ for which $\lambda_{[mn]}$ is a partition. In particular, decide for which of these small pairs Foulkes' inequality $p((m),(n),\lambda_{[mn]})\le p((n),(m),\lambda_{[mn]})$ for $m\le n$ holds. Since every partition of $mn$ is of the form $\lambda_{[mn]}$ , a complete answer for all $\lambda$ is an exact determination of all rectangular plethysm coefficients $s_{(n)}\circ s_{(m)}$ and would decide Foulkes' 1950 conjecture.

(2) Ramified extension. Determine the values $p(\beta_{[n]},(m),\kappa_{[mn]})$ for arbitrary fixed $\beta,\kappa$ (with $r=|\kappa|$ ) and all pairs $(m,n)$ below the sharp bounds $m\ge r-|\beta|+[\beta\neq\emptyset]$ , $n\ge r+\beta_1$ for which the padded partitions $\beta_{[n]}$ and $\kappa_{[mn]}$ exist (that is, $n-|\beta|\ge\beta_1$ when $\beta\neq\emptyset$ , and $mn-|\kappa|\ge\kappa_1$ ); and settle whether the stability equality with the ramified branching coefficients extends from one-row inner partitions to inner partitions with more than one row ( $\alpha\neq\emptyset$ ), where currently only the upper bound of Part II, Theorem C is known.

The two components ask the same question — where exactly stable behaviour begins, and what the coefficients are before it — in the original and in the generalized setting of the partition-algebra framework.

References

Chris Bowman, Rowena Paget, The partition algebra and the plethysm coefficients I: Stability and Foulkes' conjecture, Journal of Algebra 655 (2024) 110–138, DOI 10.1016/j.jalgebra.2023.08.042, https://doi.org/10.1016/j.jalgebra.2023.08.042 (preprint arXiv:1809.08128)
Chris Bowman, Rowena Paget, Mark Wildon, The partition algebra and the plethysm coefficients II: Ramified plethysm, Advances in Mathematics 462 (2025) 110090, DOI 10.1016/j.aim.2024.110090, https://doi.org/10.1016/j.aim.2024.110090 (preprint arXiv:2311.02721)
H. O. Foulkes, Concomitants of the quintic and sextic up to degree four in the coefficients of the ground form, Journal of the London Mathematical Society s1-25 (1950) 205–209, DOI 10.1112/jlms/s1-25.3.205, https://doi.org/10.1112/jlms/s1-25.3.205
Michel Brion, Stable properties of plethysm: on two conjectures of Foulkes, Manuscripta Mathematica 80 (1993) 347–371, DOI 10.1007/BF03026558, https://doi.org/10.1007/BF03026558
Christophe Carré, Jean-Yves Thibon, Plethysm and vertex operators, Advances in Applied Mathematics 13 (1992) 390–403, DOI 10.1016/0196-8858(92)90018-R, https://doi.org/10.1016/0196-8858(92)90018-R
Ming Yean Lim, The number of irreducibles in the plethysm $s_\lambda[s_m]$ , arXiv:2503.21108 (2025), https://arxiv.org/abs/2503.21108
Álvaro Gutiérrez, Rosa Orellana, Franco Saliola, Anne Schilling, Mike Zabrocki, A geometric and generating function approach to plethysm, arXiv:2511.02649 (2025), https://arxiv.org/abs/2511.02649