The Foulkes conjecture: monotonicity of plethysm multiplicities between $\mathrm{Sym}^a(\mathrm{Sym}^b U)$ and $\mathrm{Sym}^b(\mathrm{Sym}^a U)$ for $a \le b$

Foulkes' 1950 conjecture predicts that for all positive integers $a \le b$ and every partition $\lambda$, the multiplicity of the irreducible polynomial $\mathrm{GL}$-representation of highest weight $\lambda$ in $\mathrm{Sym}^a(\mathrm{Sym}^b U)$ is at most its multiplicity in $\mathrm{Sym}^b(\mathrm{Sym}^a U)$. Equivalently, there should exist a $\mathrm{GL}(U)$-equivariant injection $\mathrm{Sym}^a(\mathrm{Sym}^b U) \hookrightarrow \mathrm{Sym}^b(\mathrm{Sym}^a U)$. The conjecture is confirmed for $a \le 5$ (the cases $a = 4, 5$ by computer verification of injectivity of a canonical map followed by McKay's 2008 propagation theorem) and asymptotically for $b$ large relative to $a$ (Brion), but it remains open in general; the smallest unsettled value is $a = 6$, and surveys as recent as 2021 and primary papers as recent as July 2026 still record it as open. The associated open problem of describing the kernel of the canonical map (equivalently, degree-$a$ equations of the $b$-th Chow variety) beyond the computed special cases remains open as well.

Background

Let $U$ be a finite-dimensional complex vector space. A partition $\lambda = (\lambda_1 \ge \lambda_2 \ge \dots)$ is a weakly decreasing sequence of nonnegative integers; to each partition $\lambda$ of at most $\dim U$ parts corresponds an irreducible polynomial representation $\{\lambda\}$ of $\mathrm{GL}(U)$ (a Schur module), and every polynomial representation of $\mathrm{GL}(U)$ decomposes as a direct sum of copies of these irreducibles, with finite multiplicities. $\mathrm{Sym}^n$ denotes the $n$-th symmetric power. Iterating such constructions — e.g. forming $\mathrm{Sym}^a(\mathrm{Sym}^b U)$, symmetric tensors inside symmetric tensors — and decomposing the result into irreducibles $\{\lambda\}$ is the classical plethysm problem; the multiplicities are the plethysm coefficients, among the least understood numbers of algebraic combinatorics. In 1950 Foulkes conjectured, in the language of classical invariant theory, that these coefficients are monotone when the two symmetric powers are exchanged: for $a \le b$, every irreducible $\{\lambda\}$ occurs in $\mathrm{Sym}^a(\mathrm{Sym}^b U)$ with multiplicity at most its multiplicity in $\mathrm{Sym}^b(\mathrm{Sym}^a U)$. Via Schur–Weyl duality (the correspondence between representations of $\mathrm{GL}(U)$ and of symmetric groups acting on tensor powers) this is equivalent to a statement about the Foulkes modules: the permutation modules of the symmetric group $S_{ab}$ induced from the wreath-product Young subgroups $S_x \wr S_y$ (the semidirect product $S_x^y \rtimes S_y$), whose decompositions into irreducibles are sought. Under this convention, $\mathrm{Sym}^a(\mathrm{Sym}^b U)$ corresponds to $\mathrm{ind}_{S_b \wr S_a}^{S_{ab}} \mathbf{1}$ (invariants of $U^{\otimes ab}$ under $S_b^a \rtimes S_a$) and $\mathrm{Sym}^b(\mathrm{Sym}^a U)$ corresponds to $\mathrm{ind}_{S_a \wr S_b}^{S_{ab}} \mathbf{1}$, so the conjecture asserts that for every partition $\lambda$ the multiplicity of the Specht module of shape $\lambda$ in $\mathrm{ind}_{S_b \wr S_a}^{S_{ab}} \mathbf{1}$ is at most its multiplicity in $\mathrm{ind}_{S_a \wr S_b}^{S_{ab}} \mathbf{1}$. The conjecture is also equivalent to the existence of a $\mathrm{GL}(U)$-equivariant linear injection $\mathrm{Sym}^a(\mathrm{Sym}^b U) \hookrightarrow \mathrm{Sym}^b(\mathrm{Sym}^a U)$. A natural candidate, already considered by Hadamard in 1897 and endorsed by Howe in 1987, is the map $\Psi_{a \times b}$ obtained by embedding symmetric tensors into all tensors $U^{\otimes ab}$, reordering tensor factors, and projecting back; Hadamard conjectured $\Psi_{a \times b}$ injective for all $a \le b$. This stronger statement is false: Müller and Neunhöffer (2005) computed that $\Psi_{5 \times 5}$ has a nontrivial kernel, and Cheung–Ikenmeyer–Mkrtchyan (2015/2017) determined that kernel explicitly as a $\mathrm{GL}$-representation and showed $\Psi_{6 \times 6}$ is also non-injective — while the Foulkes multiplicity inequality itself, which does not single out this particular map, survives as an open problem. McKay (2008) proved a propagation theorem: if $\Psi_{a \times (b-1)}$ is injective, then $\Psi_{a \times c}$ is injective for all $c \ge b$; Ikenmeyer (2015) gave a short operator-theoretic proof via weight spaces (simultaneous eigenspaces for the diagonal torus) and projections onto invariant spaces. By this propagation, injectivity of the base-case map $\Psi_{4 \times 4}$ gives the conjecture for $a = 4$, and injectivity of $\Psi_{5 \times 6}$ gives it for $a = 5$.

The kernel of $\Psi_{a \times b}$ has a geometric meaning going back to Hadamard: it is the homogeneous degree-$a$ part of the vanishing ideal of the $b$-th Chow variety $\mathrm{Ch}_b$, the variety of completely decomposable forms (products of linear forms) inside $\mathrm{Sym}^b \mathbb{C}^b$. Prior partial results: the conjecture holds for $a \le 2$ (formulas of Thrall 1942), $a = 3$ (Dent–Siemons 2000), $a = 4, 5$ as above, and additionally by direct computation for $a \le 4$ with $b \le 14$ and for $a + b \le 17$ with $b \le 12$ (Müller–Neunhöffer 2005); for partitions $\lambda$ with at most two rows the two multiplicities are in fact equal, by classical Hermite reciprocity (the isomorphism $\mathrm{Sym}^a(\mathrm{Sym}^b \mathbb{C}^2) \simeq \mathrm{Sym}^b(\mathrm{Sym}^a \mathbb{C}^2)$ of Hermite 1854); Brion (1993) proved the conjecture for $b$ sufficiently large relative to $a$. The problem sits at the interface of symmetric-group representation theory, symmetric function algebra, invariant theory, and geometric complexity theory (the program that studies the permanent-versus-determinant complexity separation via representation theory and algebraic geometry), where plethysm multiplicities supply explicit obstruction candidates.

Problem Statement

Prove or disprove the Foulkes conjecture: for all integers $1 \le a \le b$, every complex vector space $U$ of dimension at least $b$, and every partition $\lambda$, the multiplicity of the irreducible polynomial $\mathrm{GL}(U)$-representation $\{\lambda\}$ in $\mathrm{Sym}^a(\mathrm{Sym}^b U)$ is less than or equal to its multiplicity in $\mathrm{Sym}^b(\mathrm{Sym}^a U)$. Equivalent formulations: (i) existence of a $\mathrm{GL}(U)$-equivariant injection $\mathrm{Sym}^a(\mathrm{Sym}^b U) \hookrightarrow \mathrm{Sym}^b(\mathrm{Sym}^a U)$ for all $a \le b$; (ii) for all $a \le b$, the multiplicity of the Specht module of shape $\lambda$ in the Foulkes module $\mathrm{ind}_{S_b \wr S_a}^{S_{ab}} \mathbf{1}$ (corresponding to $\mathrm{Sym}^a(\mathrm{Sym}^b U)$) is at most its multiplicity in $\mathrm{ind}_{S_a \wr S_b}^{S_{ab}} \mathbf{1}$ (corresponding to $\mathrm{Sym}^b(\mathrm{Sym}^a U)$). The smallest open value is $a = 6$: the instance $b = 6$ is trivial because the two modules coincide when $a = b$ (although $\Psi_{6 \times 6}$ itself is known not to be injective), and Müller–Neunhöffer's computational verification covers $a \le 4$ with $b \le 14$ and $a + b \le 17$ with $b \le 12$, so the remaining open instances with $a = 6$ begin at $b = 12$. The conjecture must be answered in full generality; the associated open description of the kernels $\ker \Psi_{a \times b}$ (equivalently, the degree-$a$ parts of the vanishing ideals of the Chow varieties $\mathrm{Ch}_b$) beyond the computed special cases is part of the same problem cluster but is not required for closure of the multiplicity inequality itself.

References

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