Abstract

The symmetric group S_mn acts on set partitions of a set of size mn into n sets each of size m. The character of this module is the subject of the long-standing Foulkes Conjecture. It corresponds to the representation Symⁿ (Sym^m (V)) of the infinite general linear group GL(V), obtained by composing the two symmetric-power functors, and to the plethystic product of the Schur functions s_(n) and s_(m). Decomposing an arbitrary plethysm into Schur functions has been identified by Richard Stanley as a key open problem in algebraic combinatorics. In this overview talk I will give a combinatorial description of all maximal and minimal partitions in the dominance order that label the Schur functions appearing in an arbitrary plethysm. If time permits I will also discuss relationships between plethysm coefficients, and how these may be proved via highest-weight vectors. This talk is on joint work with Rowena Paget.