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

Abstract: 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 \(Sym^r E\). Composing polynomial representations, for example to form \(Sym^4 Sym^2 E\), 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 \(Sym^m Sym^r E \cong Sym^r Sym^m E\), 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 Ph.D student Eoghan McDowell.

combinatoricsquantum algebrarings and algebrasrepresentation theory

Audience: researchers in the topic

OIST representation theory seminar

Series comments: Timings of this seminar may vary from week to week.