### An introduction to plethysms of symmetric functions and representations of general linear groups

Algebra Seminar

16th November 2022, 4:00 pm – 5:00 pm

Fry Building, LG.02

The plethysm product on symmetric functions corresponds to composition of polynomial representations of the general linear group. A typical example is Sym^n Sym^m E, where E is the d-dimensional natural representation of GL_d(C). Stanley's Problem 9, open since 2000, asks for the decomposition of this representation into irreducibles, while Foulkes' conjecture, open since 1949, states that when n >= m, each irreducible representation of GL_d(C) appears at least as often in Sym^n Sym^m E as in Sym^m Sym^n E. I will give an introduction to these mathematical objects, emphasising these and other open problems. I will finish with some recent results on stable constituents of plethysms, proved as part of joint work with Rowena Paget (University of Kent). No technical background will be assumed.

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