On Foulkes' conjecture
\textit{H.~O.~Foulkes} [in J. Lond. Math. Soc. 25, 205-209 (1950; Zbl 0037.14902)] made a conjecture which is one of the longest standing open problems in the representation theory of the symmetric group. If \(G\) is a group, denote by \(\text{id}\) the trivial representation of \(G\). If \(H\) is a subgroup of \(G\) and \(V\) a representation of \(H\), denote by \(V \uparrow^G_H\) the induced representation of \(G\) on \(V\). Denote the symmetric group on \(a\) letters by \(S_a\). Then it can be shown that if \(a\leq b\), the wreath product \(S_a\text{ wr }S_b\) is isomorphic to the normaliser of the Young subgroup \(S_a\times S_a\times\cdots\times S_a\) (with \(b\) copies of \(S_a\)) in \(S_{ab}\). Foulkes conjectured that in this situation, \(\text{id} \uparrow_{S_a\text{ wr }S_b}^{S_{ab}}\) is a subrepresentation of \(\text{id} \uparrow_{S_b\text{ wr }S_a}^{S_{ab}}\). The idea of this paper is to provide a new approach to proving this conjecture by defining, in a combinatorial fashion, a certain linear map. If it can be shown that this linear map is injective, it will follow, from results in this paper, that Foulkes' conjecture holds. The author considers a certain representation \(S^{\lambda,a}\) of \(S_a\), where \(\lambda\) is a partition of \(ab\) (this is a generalisation of the Specht module). \textit{D.~A.~Gay} showed [in Rocky Mt. J. Math. 6, 449-455 (1976; Zbl 0334.22010)] that the multiplicity of the irreducible module \(S^\lambda\) in \(\text{id} \uparrow^{S_{ab}}_{S_a\text{ wr }S_b}\) is equal to the multiplicity of the trivial representation in \(S^{\lambda,a}\). The author embeds \(S^{\lambda,a}\) into a larger representation \(V^{\lambda,a}\), and calculates the sum of all the trivial subrepresentations \(M^{\lambda,a}\) in \(V^{\lambda,a}\). It follows from Gay's result that the dimension of \(S^{\lambda,a}\cap M^{\lambda,a}\) equals the multiplicity of \(S^\lambda\) in \(\text{id} \uparrow_{S_a\text{ wr }S_b}^{S_{ab}}\). Thus, to prove Foulkes' conjecture, it is sufficient to show that for \(a\leq b\), \(\dim(S^{\lambda,a}\cap M^{\lambda,a})\leq\dim(S^{\lambda,b}\cap M^{\lambda,b})\). The author then defines a linear map from \(S^{\lambda,a}\cap M^{\lambda,a}\) to \(S^{\lambda,b}\cap M^{\lambda,b}\). He conjectures that this map is injective (Foulkes' conjecture would follow from this by the above argument). Finally, the author studies this linear map on various extensions of these spaces. In particular, on \(M^{\lambda,a}\), it is shown that its matrix is fairly sparse and close to triangular. A number of related open problems are listed.
scientific article; zbMATH DE number 1308206
scientific article