Between ``very large'' and ``infinite'': the asymptotic representation theory. (Q2874363)

In various situations we have infinite algebraic systems which are limits of finite systems of the same type. A well-known example is the infinite (finitary) symmetric group \(S_\infty\) which is a limit of the finite symmetric groups. The idea of asymptotic representation theory (ART) is to investigate relations between properties of the infinite object and properties of the finite objects, involving functional analysis, representation theory, probability theory and asymptotic combinatorics.NEWLINENEWLINE Inspired by early work of \textit{E. Thoma} [Math. Z. 85, 40-61 (1964; Zbl 0192.12402)], the author together with S. Kerov obtained properties of \(S_\infty\) and its representations as asymptotic properties (see, for example, [\textit{A. M. Vershik} and \textit{S. V. Kerov}, Funct. Anal. Appl. 15, 246-255 (1982); translation from Funkts. Anal. Prilozh. 15, No. 4, 15-27 (1981; Zbl 0507.20006)]). These results have been extended to other systems by the author and others, and an unexpected outcome of this work is a beautiful new approach to the representation theory of the finite symmetric groups (see [\textit{A. M. Vershik} and \textit{A. Yu. Okun'kov}, Russ. Math. Surv. 51, No. 2, 355-356 (1996); translation from Usp. Mat. Nauk 51, No. 2, 153-154 (1996; Zbl 0895.20011)]).NEWLINENEWLINE The present historical-expository paper describes work by \textit{J. von Neumann} [Port. Math. 3, 1-62 (1942; Zbl 0026.23302; JFM 68.0029.02)] on matrix theory and presents statements by Hermann Weyl on combinatorics which the author credits with developing his interest in ART. It then gives a brief description of examples of ART, concluding with some open problems and a brief bibliography. A longer version of this paper was presented at the Conference on Non-commutative Harmonic Analysis in Będlewo (Poland), September 24, 2012.