BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Mark Wildon (Royal Holloway\, University of London) DTSTART:20210427T073000Z DTEND:20210427T083000Z DTSTAMP:20260423T021407Z UID:OISTRTS/15 DESCRIPTION:Title: Plethysms\, polynomial representations of linear groups and Hermite recip rocity over an arbitrary field\nby Mark Wildon (Royal Holloway\, Unive rsity of London) as part of OIST representation theory seminar\n\n\nAbstra ct\nLet \\(E\\) be a \\(2\\)-dimensional vector space. Over the complex nu mbers the irreducible polynomial representations of the special linear gro up \\(SL(E)\\) are the symmetric powers \\(Sym^r E\\). Composing polynomia l representations\, for example to form \\(Sym^4 Sym^2 E\\)\, corresponds to the plethysm product on symmetric functions. Expressing such a plethysm as a linear combination of Schur functions has been identified by Richard Stanley as one of the fundamental open problems in algebraic combinatoric s. In my talk I will use symmetric functions to prove some classical isomo rphisms\, such as Hermite reciprocity \\(Sym^m Sym^r E \\cong Sym^r Sym^m E\\)\, and some others discovered only recently in joint work with Rowena Paget. I will then give an overview of new results showing that\, provided suitable dualities are introduced\, Hermite reciprocity holds over arbitr ary fields\; certain other isomorphisms (we can prove) have no modular gen eralization. The final part is joint work with my Ph.D student Eoghan McDo well.\n LOCATION:https://researchseminars.org/talk/OISTRTS/15/ END:VEVENT END:VCALENDAR