On Foulkes' conjecture
type: Item
instance of: scholarly article
title: On Foulkes' conjecture
zbMATH Open document ID: 0937.20004
DOI: 10.1016/S0022-4049(97)00087-X
author: William F. IV Doran
published in: Journal of Pure and Applied Algebra
publication date: 2000-03-13T00:00:00Z
review text: \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' …
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20B30
Mathematics Subject Classification ID: 20B35
zbMATH DE Number: 1308206
zbMATH Keywords: symmetric groups
zbMATH Keywords: Foulkes' conjecture
zbMATH Keywords: Specht modules
zbMATH Keywords: group representations
zbMATH Keywords: tableaux
zbMATH Keywords: induced representations
zbMATH Keywords: Young subgroups
zbMATH Keywords: irreducible modules
zbMATH Keywords: multiplicities
Wikidata QID: Q122999974
reviewed by: Bethany R. Marsh
MaRDI profile type: Publication
cites work: A note on plethysm
cites work: A problem related to Foulkes's conjecture
cites work: Concomitants of the Quintic and Sextic Up To Degree Four in the Coefficients of the Ground Form
cites work: Q4002278
cites work: Characters of the Weyl group of SU(n) on zero weight spaces and centralizers of permutation representations
cites work: The representation theory of the symmetric groups
cites work: Q3952291
cites work: Q3947818
cites work: Q4001520
description: scientific article; zbMATH DE number 1308206