Master thesis defense David Møller Andersen

Master thesis defense, David Møller Andersen

Title: Representations of symmetric groups, Schur-Weyl duality, and the Temperley-Lieb algebra

Abstract: We study representations of the symmetric groups Sn over an algebraically closed field K of characteristic zero, and with an offset in these, we look at Schur-Weyl duality and at the Temperley-Lieb algebra and category. We start off by constructing the irreducible representations of Sn in two different ways. First, the classical approach with Young symmetrizers and secondly, by branching. After this, Schur-Weyl duality gives us a way to construct irreducible representations of GL(V ) from the irreducible representations of Sn. Lastly, we define the Templerley-Lieb algebra TLn(q) and the Temperley-Lieb category TL(q). We construct a map from TLn(−2) to EndSL2(K)((K2)⊗n) which we prove, using Sn and Schur-Weyl duality, is an isomorphism. We also give some intuition for a way to expand this isomorphism to a functor from TL(−2) to the category whose objects are (K2)⊗n and whose morphisms are SL2(K)-invariant linear maps, which restricts on endomorphisms to our isomorphism, and this functor turns out to be an equivalence of categories.