Mathematics Colloquium

The problem of statistics of irreducible components in "large representations" of "large groups" was addressed in pioneering works by Logan and Shepp and by Vershik and Kerov in 1970's. They described the statistics of irreducible components of the regular representation of the symmetric group $S_N$ for large $N$ . Since then there has been steady progress in this direction. Now this area is known as the asymptotic representation theory. In the first part of the talk I will give an overview of results obtained in the last decades. Then I will report on some recent progress, and, if time permits, will mention some recent results about the statistics of indecomposable modules for quantum groups at roots of unity (based on a joint work with A. Lachowska, O. Postnova, D. Soloviev).