Tuesday, March 12, 7.527, 14:00

Mark Wildon (Royal Holloway, London), Plethysms and polynomial representations
of SL_{2}(C)

Abstract:

Let *E* be a *2*-dimensional complex vector space. The irreducible
polynomial representations of *SL(E)* are the symmetric powers
*Sym*^{r}(E),
for *r ∈ N*_{0}. In this talk on joint work with Rowena Paget (University
of Kent), I will show how compositions of these representations, such as
*Sym*^{2} Sym^{n} E,
correspond to certain 'plethysms' of symmetric functions.
Decomposing these plethysms into Schur functions has been identified by
Richard Stanley as one of the fundamental open problems in algebraic
combinatorics. I will use symmetric functions to prove some classical
isomorphisms, such as
*Sym*^{2}Sym^{n}(E) ≅ ⋀^{2}Sym^{(n+1)}(E), some
more recent isomorphisms due to King and Manivel, and some entirely new
isomorphisms, whose existence was revealed by computer search. Our methods
also prove the converse of the King and Manivel results. I will end by
mentioning some recent work of my Ph.D student Eoghan McDowell on analogues
of these results for representations of the finite groups
*SL*_{2}(F_{q}).