Two lectures on the asymptotic representation theory and statistics of Young diagrams

Abstract

Investigation of classical groups of high ranks leads to two kinds of problems. Questions of the first kind deal with asymptotical properties of groups, their representations, characters and other attributes as group rank grows to infinity. Another kind of questions (in the spirit of infinite dimensional analysis) deal with properties of infinite dimensional analogues of classical groups. Let us discuss, for instanse, the most simple nontrivial example of classical group series, that is the series of symmetrical groups S _N.Typical question of the first kind is what is the structure of the symmetric group of high rank and its representations? A question of the second kind is what can be told about infinite symmetric group, that is a group of finite permutations of natural numbers? Both kinds of questions are closely connected, but it is appropriate mention here that while questions of the first kind seem to be more natural and their importance was emphasized as early as 1940s by H.Weyl [33] and J.von Neumann [14], nevertheless the functional analysis evolved mainly into investigation of infinite dimensional groups,which is certainly caused by its applications to physics.Questions of both kinds are parts of asymptotical representation and group theory,but proper asymptotical problems were, strangely enough,investigated much less and thus they make up the main part of the theory.In a wide context their study was started in the 1970s, but a lot of separate problems were considered earlier.It is important to emphasize from the very beginning that the questions considered deal with the structure of groups and their representations as a whole rather than with investigation of particular functionals on groups or its representations.For example, the question about the distribution of maximal cycle length in typical permutation was considered by V.L.Goncharov in 1940s [4] while the question about the general structure of typical conjugacy class in symmetric group was studied only in the middle of 1970s by A.Schmidt and myself [19],[20].The same can be said about representations of the symmetric group.