REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

ABSTRACT: I will start by reviewing the Erdei-Thoma's classification of irreducible characters of the infinite symmetric group (1950-60s). A related problem arises when symmetric groups are replaced by general linear groups over a (fixed) finite field, but the classification of irreducibles in that case is still conjectural. As was understood by Vershik and Kerov in the 1980s, both problems - for symmetric and linear groups - lead to certain probability distributions on Young diagrams (connected to Schur and Hall-Littlewood symmetric polynomials, respectively), whose properties can be translated into the language of characters.

Moreover, distributions on Young diagrams in the symmetric groups case are deeply related to the classical Robinson-Schensted insertion algorithm from combinatorics. A corresponding construction for linear groups was recently discovered by the speaker (jointly with Borodin and Bufetov), which allows to establish a law of large numbers for random Young diagrams.

All relevant definitions will be given in the course of the talk.