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Representation Theory of the Symmetric Groups: The Okounkov-Vershik Approach, Character Formulas, and Partition Algebras

Tullio Ceccherini-Silberstein, Fabio Scarabotti and Filippo Tolli
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Studies in Advanced Mathematics 121
[Reviewed by
Thomas R. Hagedorn
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How does one begin to write about the representation theory of the finite symmetric group? The study of this apparently very specific problem has grown into an enormous area of algebra and combinatorics since its introduction over a hundred years ago in the works of Frobenius, Schur, and Young. Not only are there important connections to the representation theory of other groups, but in recent years, deep connections to other areas of mathematics and physics have been discovered. Even more amazing is the fact that there has been significant recent progress in understanding fundamental questions such as the construction of the irreducible representations and the derivation of branching laws. Much of this work is due to A. Vershik, G. Olshanskii, and A. Okounkov and the text under review seeks to “present the representation theory of the symmetric group along the new lines developed” by these and other authors.

The book is a comprehensive and self-contained description of the representation theory of the symmetric group, suitable for graduate students and faculty. It is easy to read and the authors have achieved their goal of writing a book that would “remain at an elementary level, without introducing the notions in their wider generality and avoiding too many technicalities.”

The book consists of eight chapters. Chapter one is an introduction to the representation theory of finite groups and chapter two introduces the theory of Gelfand-Tsetlin bases. The core of the book is chapter three, 77 pages long and focused on the Okounkov-Vershik approach to the representation theory of the symmetric group. Later chapters build upon this work and discuss symmetric polynomial functions (chapter four), the character theory of the symmetric group (chapter five), the Littlewood-Richardson rule (chapter 6, following James’s approach), finite-dimensional *-algebras and reciprocity laws (chapter 7), and Schur-Weyl duality (chapter eight).

The book is a quite an accomplishment. The authors have achieved their goal of writing a self-contained, comprehensive and accessible book that introduces the reader “to an active area of research.” But the next edition of this book would be even more successful if the authors would include more of the historical context for the work presented. For example, what was the state of the field prior to the ground-breaking work of Vershik, Olshanskii, and Okounkov and how has their work revolutionized the field? The motivated reader can find this information on the internet through suitable searches, but given the book’s goal of being self-contained, it would be valuable to include this material as part of the book. In a similar vein, it would be very helpful for the authors to have indicated open problems remaining to be solved.

Tom Hagedorn is Professor of Mathematics at The College of New Jersey.

Preface; 1. Representation theory of finite groups; 2. The theory of Gelfand–Tsetlin bases; 3. The Okounkov–Vershik approach; 4. Symmetric functions; 5. Content evaluation and character theory; 6. The Littlewood–Richardson rule; 7. Finite dimensional *-algebras; 8. Schur–Weyl dualities and the partition algebra; Bibliography; Index.