Prof. Wildon's paper “Plethysms of symmetric functions and highest weight representations” written jointly with Melanie de Boeck and Rowena Paget (University of Kent) was recently accepted by the top journal Transactions of the American Mathematical Society.
Prof Wildon and Prof Paget
The paper “Plethysms of symmetric functions and highest weight representations” concerns symmetric functions: functions that stay the same when one switches around the variables. For example xy+yz+zx is a symmetric function of the variables x, y, z.
A basic result is that the sum of two symmetric functions is still symmetric. Prof. Wildon's paper concerns a different operation: the plethysm product. This product is defined not by normal multiplication, but instead by a version of function composition. Understanding the plethysm product is a fundamental open problem in algebraic combinatorics with close links to the representation theory of the symmetric group and general linear groups. There is also a connection to computational complexity theory: one programme aims to prove the algebraic version of the famous P versus NP Conjecture by reducing it to a question about plethysms and related operations on symmetric functions.
Prof. Wildon said 'We started work on this paper back in 2018 so I am delighted that it has finally been accepted. The problems it solves are technical to describe, but of broad interest to pure mathematicians and also computer scientists. I look forward to continuing my collaboration with Dr Paget in the coming academic year, when hopefully we will, at last, be able to meet face-to-face.'