Rowena Paget

Plethysm via the partition algebra

The symmetric group S_{mn} acts naturally on the collection of set partitions of a set of size mn into n sets each of size m. The irreducible constituents of the associated ordinary character are largely unknown; in particular, they are the subject of the longstanding Foulkes Conjecture. There are equivalent reformulations using polynomial representations of infinite general linear groups or using plethysms of symmetric functions. I will review plethysm from these three perspectives then present a new approach to studying plethysm: using the Schur-Weyl duality between the symmetric group and the partition algebra. This is joint work with Chris Bowman (J.Alg.‘24) and with Chris Bowman and Mark Wildon (Adv.Math.’25).