Mark Wildon: Plethysms, polynomial representations of linear groups and Hermite reciprocity over an arbitrary field

Let E be a 2-dimensional vector space. Over the complex numbers the irreducible polynomial representations of the special linear group SL(E) are the symmetric powers Sym^rE. Composing polynomial representations, for example to form Sym⁴ Sym²E, corresponds to the plethysm product on symmetric functions. Expressing such a plethysm as a linear combination of Schur functions has been identified by Richard Stanley as one of the fundamental open problems in algebraic combinatorics. In my talk I will use symmetric functions to prove some classical isomorphisms, such as Hermite reciprocity Sym^m Sym^rE ≅ Sym^rSym^mE, and some others discovered only recently in joint work with Rowena Paget. I will then give an overview of new results showing that, provided suitable dualities are introduced, Hermite reciprocity holds over arbitrary fields; certain other isomorphisms (we can prove) have no modular generalization.

The final part is joint work with my PhD student Eoghan McDowell.

This video was produced by the Okinawa Institute of Science and Technology, as part of their OIST Representation Theory Seminar series.