The symmetric group Smn 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 before presenting a new approach to studying plethysm: using the Schur-Weyl duality between the symmetric group and the partition algebra. This method allows us to study stability properties of certain plethysm coefficients. This is joint work with Chris Bowman. If time permits, I will also discuss some new results with Chris Bowman and Mark Wildon.
This video was produced by the Okinawa Institute of Science and Technology, as part of their OIST Representation Theory Seminar series.
