Asymptotic Representation Theory of the Symmetric Group and its Applications in Analysis
About this Title
S. V. Kerov. Translated by N. V. Tsilevich
Publication: Translations of Mathematical Monographs
Publication Year: 2003; Volume 219
ISBNs: 978-0-8218-3440-4 (print); 978-1-4704-4643-7 (online)
MathSciNet review: MR1984868
MSC: Primary 20C30; Secondary 05E05, 05E10, 20C32, 20P05, 60C05
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.
Graduate students and research mathematicians interested in representation theory and combinatorics.
Table of Contents
- Boundaries and dimension groups of certain graphs
- The boundary of the Young graph and MacDonald polynomials
- The Plancherel measure of the symmetric group
- Young diagrams in problems of analysis