Representation theory of the symmetric groups. The Okounkov-Vershik approach, character formulas, and partition algebras.

*(English)*Zbl 1230.20002
Cambridge Studies in Advanced Mathematics 121. Cambridge: Cambridge University Press (ISBN 978-0-521-11817-0/hbk). xv, 412 p. (2010).

The aim of this book is to provide an account of the representation theory of the symmetric group in characteristic zero, accessible to non-experts. The novelty is to emphasise the recent approach of Okounkov and Vershik, rather than the traditional Young-James approach.

The authors are very careful to make sure that as little background material as possible is required. They spend quite a lot of time in the first section on undergraduate representation theory, and then later on symmetric functions and then binomial coefficients; in between, they treat more advanced topics such as Gel’fand-Tsetlin bases. In the reviewer’s view, this makes the book rather unbalanced; it lurches from the straightforward to the difficult, and the sequence of topics looks rather ad hoc. As a result, the novelty of the Okounkov-Vershik approach is rather buried. I think it would have been much better to relegate the background material on symmetric functions and binomial coefficients (which will be very well known to many readers) to appendices.

This book would probably function reasonably well as the text for a graduate course, but I think would not serve very well as a reference book. It is fairly clear from reading the book that the authors are not specialists in this area; while this can sometimes be a good thing (since authors expositing their own specialities can take things for granted and fail to anticipate difficulties), I think here the narrowness of the authors’ viewpoint is evident. This is evidenced in their use of the terminology “Young module” for Young’s permutation module. In fact, the terminology “Young module” is established as something different (an indecomposable summand of a permutation module) which only have significance in modular representation theory.

The authors’ English is rather shaky in places, and the typography of the book is very poor in places. This is disappointing from such a respected publisher.

To summarise, I don’t especially recommend this book, mainly because there are so many very good books on the representation theory of the symmetric groups. The novelty of this book doesn’t outweigh the shortcomings.

The authors are very careful to make sure that as little background material as possible is required. They spend quite a lot of time in the first section on undergraduate representation theory, and then later on symmetric functions and then binomial coefficients; in between, they treat more advanced topics such as Gel’fand-Tsetlin bases. In the reviewer’s view, this makes the book rather unbalanced; it lurches from the straightforward to the difficult, and the sequence of topics looks rather ad hoc. As a result, the novelty of the Okounkov-Vershik approach is rather buried. I think it would have been much better to relegate the background material on symmetric functions and binomial coefficients (which will be very well known to many readers) to appendices.

This book would probably function reasonably well as the text for a graduate course, but I think would not serve very well as a reference book. It is fairly clear from reading the book that the authors are not specialists in this area; while this can sometimes be a good thing (since authors expositing their own specialities can take things for granted and fail to anticipate difficulties), I think here the narrowness of the authors’ viewpoint is evident. This is evidenced in their use of the terminology “Young module” for Young’s permutation module. In fact, the terminology “Young module” is established as something different (an indecomposable summand of a permutation module) which only have significance in modular representation theory.

The authors’ English is rather shaky in places, and the typography of the book is very poor in places. This is disappointing from such a respected publisher.

To summarise, I don’t especially recommend this book, mainly because there are so many very good books on the representation theory of the symmetric groups. The novelty of this book doesn’t outweigh the shortcomings.

Reviewer: Matthew Fayers (London)

##### MSC:

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

20C30 | Representations of finite symmetric groups |

05E05 | Symmetric functions and generalizations |

05E10 | Combinatorial aspects of representation theory |

20B30 | Symmetric groups |