Representation theory of symmetric groups

Prerequisite

Undergaduate Algebra, Probability and Functional Analysis

Abstract

Representation theory of symmetric groups is a classical topic with reach history and many intersections with other areas of mathematics and physics. The conventional approach to the theory, which goes back to Frobenius, Schur, and Young, is bases on ad hoc constructions, starting from the Young diagrams. We plan to present Vershik-Okounkov approach, where the corresponding objects appear naturally. The construction is based on the existence of the Gelfand-Tsetlin basis for the chain of symmetric groups, and Young diagrams appear as a convenient way to describe the spectrum of the Gelfand-Tsetlin algebra. The second part of the course will be devoted to the asymptotic representation theory. We will focus on the Plancherel measure, one of the best known and well studied objects in the field, and discuss the famous Vershik-Kerov-Logan-Shepp limit shape theorem, as well as fluctuations (the celebrated Baik-Deift-Johansson Theorem).