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  • Tuesday, March 12, 7.527, 14:00

    Mark Wildon (Royal Holloway, London), Plethysms and polynomial representations of SL2(C)

  • Abstract:
    Let E be a 2-dimensional complex vector space. The irreducible polynomial representations of SL(E) are the symmetric powers Symr(E), for r ∈ N0. In this talk on joint work with Rowena Paget (University of Kent), I will show how compositions of these representations, such as Sym2 Symn 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 Sym2Symn(E) ≅ ⋀2Sym(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 SL2(Fq).