Eoghan McDowell successfully defended his thesis “Representations of the general linear group with multilinear constructions” last Friday.

Eoghan McDowell in Stuttgart in 2019

A representation is a vector space with the additional structure of a group action. The study of representations – as well as being a rich theory in itself – has connections with group theory, combinatorics and theoretical physics. Eoghan’s thesis, written under the supervision of Professor Mark Wildon, establishes a variety of results on representations of the general linear group. The examiners were Professor Matthew Fayers (QMUL) and Dr Chris Bowman (York). 

Several of Eoghan’s results address the problem of plethysm in prime characteristic: he generalises a number of “plethystic” isomorphisms, known to hold over the complex numbers, to arbitrary fields. Another result identifies how the inverse Schur functor – which gives a connection between representations of the general linear group and of the symmetric group – acts on the Specht module in all characteristics. Switching focus to representations of the finite special linear group, Eoghan defines a random walk driven by taking tensor products of representations, and describes its properties in correspondence with the representation theory. 

In November Eoghan begins a postdoc at the Okinawa Institute of Science and Technology, Japan, in the Representation Theory and Algebraic Combinatorics Unit under Professor Liron Speyer.