--- title: Foulkes' Conjecture in Representation Theory url: https://www.emergentmind.com/topics/foulkes-conjecture type: topic --- # Foulkes' Conjecture in Representation Theory Foulkes’ conjecture is a plethystic comparison principle in the representation theory of \(\mathrm{GL}(V)\), symmetric functions, and symmetric groups. In its classical form, it predicts that whenever \(m\le n\), the plethysm \(\Sym^m(\Sym^n V)\) embeds into \(\Sym^n(\Sym^m V)\); equivalently, the symmetric-function difference \(s_m[s_n]-s_n[s_m]\) is Schur-positive, or, in permutation-module language, each irreducible constituent of the Foulkes module for \(S_m\wr S_n\) occurs with multiplicity at most that of the transposed wreath-product construction [2503.21108]. The conjecture has generated a large body of work on plethysm coefficients, permutation modules, propagation phenomena, stable ranges, combinatorial models, and \(q\)- and \(\mathfrak{sl}_2(\mathbb C)\)-generalizations [1509.04957], [2110.01055], [2507.06220]. ## 1. Classical statement and equivalent formulations Let \(\Lambda\) be the ring of symmetric functions, and for partitions \(\lambda\vdash n\), \(\mu\vdash m\), write \[ s_\lambda[s_\mu]=\sum_{\nu\vdash nm} a_{\lambda,\mu}^\nu\, s_\nu, \] where the integers \(a_{\lambda,\mu}^\nu\ge 0\) are the plethysm coefficients [2503.21108]. In the rectangular case \(\lambda=(n)\), \(\mu=(m)\), Foulkes’ conjecture asserts that if \(n\le m\), then \[ s_m[s_n]-s_n[s_m] \] is Schur-positive; equivalently, for every partition \(\nu\vdash nm\), \[ a_{m,n}^\nu\ge a_{n,m}^\nu \] [2503.21108]. In \(\mathrm{GL}(V)\)-language, the same assertion is the existence of an injective equivariant map \[ \Sym^m(\Sym^n V)\longhookrightarrow \Sym^n(\Sym^m V) \] whenever \(m\le n\) [1202.0067]. A standard equivalent formulation uses permutation characters of the symmetric group. If \(\phi^{(m^n)}=\Ind_{S_m\wr S_n}^{S_{mn}}\mathbf 1\), then the conjecture becomes \[ \langle \phi^{(m^n)},\chi^\lambda\rangle \ge \langle \phi^{(n^m)},\chi^\lambda\rangle \] for every partition \(\lambda\vdash mn\) [1202.0067]. In the notation of generalized Foulkes modules, the same statement is expressed as an \(S_{ab}\)-equivariant embedding \(F_b^a\hookrightarrow F_a^b\) for \(a\le b\) [2110.01055]. These formulations emphasize different aspects of the problem. The symmetric-function form isolates Schur positivity, the \(\mathrm{GL}(V)\)-form emphasizes functorial plethysm, and the symmetric-group form connects the conjecture to induction from wreath products and Specht-module multiplicities. ## 2. Foulkes modules and structural representation theory A central object is the Foulkes module. In one convention, \(H^{(m^n)}\) is the permutation module for \(S_{mn}\) on the set of set-partitions of \(\{1,2,\dots,mn\}\) into \(n\) blocks each of size \(m\), equivalently \[ H^{(m^n)}\cong \Ind_{S_m\wr S_n}^{S_{mn}}\mathbf 1 \] [1007.2946]. In another convention, \(F_b^a\) denotes the permutation module on ordered set-partitions of type \((a^b)\), again realized as an induced module from the wreath product [2110.01055]. These are the symmetric-group incarnations of the plethysms appearing in Foulkes’ conjecture. Structural analysis of these modules has produced several refined results. Wildon constructs homomorphisms from Specht modules \(S^\lambda\) into Foulkes modules using closed set-families, and gives a combinatorial description of the minimal partitions, in dominance order, that label irreducible constituents of Foulkes modules [1007.2946]. For odd \(m\), a partition \(\lambda\) is a minimal constituent of \(\phi^{(m^n)}\) precisely when there exists a minimal set-family of shape \((m^n)\) and type \(\lambda\); for even \(m\), the only minimal constituent is \(\chi^{(m^n)}\), with multiplicity one [1007.2946]. This isolates the “bottom layer” of the character and shows that any counterexample would have to occur at or above that level. Hegedüs and Madireddi extend the framework to generalized Foulkes modules \[ F_\nu^a=\Ind_{S_a\wr S_b}^{S_{ab}} \Inf_{S_b}^{S_a\wr S_b}(S^\nu), \] where \(\nu\vdash b\) [2110.01055]. Restricting \(F_\nu^a\) to the maximal Young subgroup \(S_b\times S_{ab-b}\), they identify a distinguished component \(U_{\nu,a}\), and prove that \[ U_{\nu,a}\cong \bigoplus_{\mu,\lambda\vdash b} c_{\mu,\lambda}^\nu\cdot (S^\mu\boxtimes F_\lambda^{\,a-1}), \] where \(c_{\mu,\lambda}^\nu\) are Kronecker coefficients [2110.01055]. This “peels off” one plethystic layer and links generalized Foulkes modules to Kronecker-theoretic data. ## 3. Verified cases, propagation, and stable behavior The conjecture remains open in general, but several finite families are known. Hegedüs–Madireddi record that it is known for \(a\le 5\) by work of Dent, McKay, Müller–Neunhöffer, and others [2110.01055]. The case \(a=5\) was established by Cheung, Ikenmeyer, and Mkrtchyan through a computer-assisted combinatorial computation proving that \[ Y_{5,6}:\Sym^5(\Sym^6 V)\to \Sym^6(\Sym^5 V) \] is injective; McKay’s propagation theorem then yields the conjecture for all \(b\ge 5\) [1509.03944]. McKay’s propagation theorem is a major reduction principle. It states that if \(Y_{a,b-1}\) is injective, then \(Y_{a,c}\) is injective for every \(c\ge b\) [1509.04957]. Since \(Y_{a,a}\) is trivially an isomorphism, proving the conjecture for fixed \(a\) reduces to checking \(Y_{a,a+1}\) [1509.03944]. Ikenmeyer recasts McKay’s argument in terms of weight spaces of tensor powers and commuting raising operators \(P_{i,j}\), yielding a short proof of the propagation phenomenon [1509.04957]. A different simplification occurs in the stable range. Bowman and Paget use the partition algebra and Schur–Weyl duality to prove that if \(\lambda\) has size \(r\) and \(m,n\ge r\), then for all \(p,q\ge r\), \[ p((n),(m),\lambda_{[mn]})=p((q),(p),\lambda_{[pq]}), \] where \(p((n),(m),\nu)\) denotes the relevant plethysm coefficient [1809.08128]. In particular, for \(m,n>r\), the coefficient depends only on the product \(mn\), not on the ordered pair \((m,n)\). In that stable range, the original and strengthened Foulkes inequalities become equalities [1809.08128]. ## 4. Combinatorial and computational approaches Several lines of work replace direct plethysm computation by more tractable combinatorial or recursive models. Evseev, Paget, and Wildon introduce a deflation map from characters of \(S_{mn}\) to characters of \(S_n\), obtained by restricting to \(S_m\wr S_n\) and projecting onto constituents on which the base group acts trivially [1202.0067]. Their combinatorial deflation formula generalizes the Murnaghan–Nakayama rule and produces a recursion for the multiplicities \(\langle \phi^{(m^n)},\chi^\lambda\rangle\). Implemented computationally, this verified Foulkes’ conjecture for all \(m,n\) with \(m+n\le 19\) [1202.0067]. Cheung–Ikenmeyer–Mkrtchyan attack the \(a=5\) case through highest-weight vectors and symmetrized tableaux. They reduce injectivity of \(Y_{a,b}\) to the kernels of maps on highest-weight spaces, construct bases by \((a,b)\)-symmetrized tableaux, and evaluate these vectors at generic points in \(\Sym^b V\) using determinant precomputation and several search-tree speed-ups [1509.03944]. This approach is explicitly representation-theoretic but also heavily combinatorial. Lim studies a coarser statistic: the total number of irreducible constituents, counted with multiplicity, in \(s_\lambda[s_m]\). For fixed \(n,m\), he defines a character \(N^m\) of \(S_n\) by counting matrices in \[ M(n,m)=\{A\in \mathbb Z_{\ge 0}^{n\times n}\mid \text{all row-sums and column-sums equal }m\} \] satisfying \(\sigma A^T=A\), and proves \[ \sum_{\nu\vdash nm} a_{\lambda,m}^\nu=\langle \chi^\lambda,N^m\rangle_{S_n} \] [2503.21108]. Each value \(N^m(\rho)\) is an Ehrhart–quasipolynomial counting lattice points in a rational polytope \(P(\sigma)\) [2503.21108]. In the single-row case \(\lambda=(n)\), the resulting total multiplicity becomes the cardinality of a matrix quotient set \(T(n,m)/\sim\), leading to a weaker “cardinality-only” form of Foulkes’ conjecture [2503.21108]. ## 5. Generalizations, \(q\)-analogues, and variants One major generalization is Bergeron’s \(q\)-analogue. Replacing complete homogeneous symmetric functions \(h_n\) by Hall–Littlewood polynomials \(H_n(x;q)\), Bergeron defines \[ F_{a,b}(x;q)=\frac{H_b[H_a](x;q)-H_a[H_b](x;q)}{1-q} \] and conjectures that its Schur expansion has coefficients in \(\mathbb N[q]\) [1602.08134]. At \(q=0\), this recovers the classical Foulkes difference; at \(q=1\), the conjecture is proved, and the specialization lies in \(\mathbb N[h_1,h_2,e_2]\) [1602.08134]. The conjecture was verified computationally for all \(1m\ge 2\) and every partition \(\lambda\) with more than two parts, \[ \langle h_n[h_1h_{m-1}],s_\lambda\rangle \ge \langle h_m[h_1h_{n-1}],s_\lambda\rangle, \] and they prove the case \(m=2\), with an exceptional behavior at the two-row shape \((2,2n-2)\) [1708.02995]. ## 6. Infinite families, consequences, and boundaries of the conjecture A recent advance is due to Gutiérrez and Szwej, who prove the \(\mathfrak{sl}_2(\mathbb C)\)-version, and hence the Gaussian-coefficient \(q\)-Foulkes statement, whenever \(a\mid c\) or equivalently \(a\mid d\) [2507.06220]. Their main theorem states that if \[ a\le b,\qquad c\le d,\qquad ab=cd, \] and \(a\mid c\), then there is an \(\mathfrak{sl}_2\)-equivariant injection \[ \Sym^a(\Sym^b\mathbb C^2)\longhookrightarrow \Sym^c(\Sym^d\mathbb C^2) \] [2507.06220]. This is the first proof in this family valid for infinitely many values of \(a\); previously only the cases \(a=2\) and \(a=3\) were known, so every prime \(a\) is now covered [2507.06220]. Their proof uses a polynomial-ring model \[ \Sym^m(\Sym^n\mathbb C^2)\cong \Lambda_{\le n}[x_1,\dots,x_m], \] where the \(\mathfrak{sl}_2\) raising operator becomes the total derivative \(\nabla=\sum_i \partial/\partial x_i\) [2507.06220]. If \(c=ka\), they define a block-repetition algebra map \[ \Phi_k:\Lambda[\mathbf x_b]\to \Lambda[\mathbf y_d] \] by repeating each \(y_i\) exactly \(k\) times in the substitution; \(\Phi_k\) is injective, sends \(\Lambda_{\le a}[\mathbf x_b]\) into \(\Lambda_{\le c}[\mathbf y_d]\), and commutes with \(\nabla\), hence with the full \(\mathfrak{sl}_2\)-action [2507.06220]. Hermite reciprocity then converts the resulting injection into the desired plethystic inclusion. By taking torus characters, the proof yields \[ \binom{c+d}{c}_q-\binom{a+b}{a}_q\in \mathbb N[q], \] and semisimplicity arguments further imply symmetric and unimodal complementary weight multiplicities, settling Zanello’s unimodality conjecture in these infinitely many cases [2507.06220]. At the same time, the scope of the conjecture has clear limits. Gill proves that a naïve modular analogue is false: if \(\mathbb F\) has odd prime characteristic \(p\) and \(a