The goal of the course is to give a new approach to the theory of representations of symmetric groups together with the combinatorics of Young diagrams and then to expose the asymptotic aspects of that theory together with representations of infinite symmetric and some other infinite dimensional groups.
Part 1. Finite theory.
1. Introduction. Background of the theory of complex representations of finite groups.
2. Inductive family of groups and algebras, Gel'fand-Zetlin algebras, induced representations, Bratteli diagrams.
3. Symmetric groups and Hecke algebras, spectral interpretation of Young tableau. Canonical form of the irreducible representations of symmetric groups. Dimensions and characters. Connection with symmetric functions.
Part 2. Asymptotic theory.
1. Thoma's theorem about characters of infinite symmetric groups. Factor-representations. Ergodic method and central measures on the Young graph.
2. Asymptotics of the Plancherel measure on diagrams and representations. Refinement of asymptotics and application to probability theory and combinatorics.
3. The infinite unitary and orthogonal groups. Characters and asymptotics.
4. Survey on representation theory of other infinite dimensional groups.
Ort: HG G 43 (Hermann-Weyl-Zimmer)