BETWEEN “VERY LARGE” AND “INFINITE”: THE ASYMPTOTIC
REPRESENTATION THEORY
Abstract: I illustrate the historical roots of the theory which I called later “Asymptotic
Representation Theory” – the theory which can be considered as a part functional analysis,
representation theory, and more general – probability theory, asymptotic combinatorics, the
theory of random matrices, dynamics, etc. The first and very concrete example is a
remarkable (and forgotten) paper by J. von Neumann, which I try here to connect with the
modern theory of random matrices; the second example is a quote of an important thought of
H. Weyl about the theory of symmetric groups. In the last section I give a short review of the
ideas of the asymptotic representation theory, which was developed starting from the 1970s,
and now became very popular. I mention several important problems, and give a list
(incomplete) of references. But the reader must remember that this is just a synopsis of the
“baby talk”.
2000 AMS Mathematics Subject Classification: Primary: 47L15; Secondary:
15B52.
Keywords and phrases: Matrix algebras, invariant subspaces, random matrices,
characters.