**Translations of Mathematical Monographs**

2003;
201 pp;
Hardcover

MSC: Primary 20; 22;

Print ISBN: 978-0-8218-3440-4

Product Code: MMONO/219

List Price: $94.00

AMS Member Price: $75.20

MAA Member Price: $84.60

**Electronic ISBN: 978-1-4704-4643-7
Product Code: MMONO/219.E**

List Price: $94.00

AMS Member Price: $75.20

MAA Member Price: $84.60

#### Supplemental Materials

# Asymptotic Representation Theory of the Symmetric Group and its Applications in Analysis

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*S. V. Kerov*

This book reproduces the doctoral thesis written by a remarkable mathematician,
Sergei V. Kerov. His untimely death at age 54 left the mathematical community
with an extensive body of work and this one-of-a-kind monograph. In it, he gives
a clear and lucid account of results and methods of asymptotic representation
theory. The book is a unique source of information on the important topic of
current research.

Asymptotic representation theory of symmetric groups deals with problems of two
types: 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 both
directions. 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, Kerov studies an important problem of
computing irreducible characters of the infinite symmetric group. This leads 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.

The book is suitable for graduate students and research mathematicians
interested in representation theory and combinatorics.

#### Readership

Graduate students and research mathematicians interested in representation theory and combinatorics.

#### Table of Contents

# Table of Contents

## Asymptotic Representation Theory of the Symmetric Group and its Applications in Analysis

- Cover Cover11
- Title page iii4
- Contents v6
- Foreword vii8
- Publications of Sergei Kerov xiii14
- Introduction 118
- Boundaries and dimension groups of certain graphs 5976
- The boundary of the Young graph and MacDonald polynomials 7390
- The Plancherel measure of the symmetric group 109126
- Young diagrams in problems of analysis 145162
- References 189206
- Comments on Kerov’s thesis by G. Olshanski 197214
- Additional references 199216
- Back Cover Back Cover1222