Asymptotic representation theory of symmetric
groups deals with two types of problems: asymptotic properties of
representations of symmetric groups of large order and
representations of the limiting object, i.e., the infinite symmetric
group. The author contributed significantly in the development of
problems of both types, and his book presents an account of these
contributions, as well as those of other researchers.
Among the problems of the first type, the author discusses the properties of
the distribution of the normalized cycle length in a random permutation, and
the limiting shape of a random (with respect to the Plancherel measure)
Young diagram. He also studies stochastic properties of the deviations of
random diagrams from the limiting curve.
Among the problems of the second type, the author studies an important
problem of computing irreducible characters of the infinite symmetric group.
This leads him to the study of a continuous analog of the notion of Young
diagram, and, in particular, to a continuous analogue of the hook walk
algorithm, which is well known in the combinatorics of finite Young
diagrams. In turn, this construction provides a completely new description
of the relation between the classical moment problems of Hausdorff and
Markov.
Readership
Graduate students and research mathematicians interested in
representation theory and combinatorics.
