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**The symmetric group. Representations, combinatorial algorithms, and symmetric functions.**
*(English)*
Zbl 0823.05061

Wadsworth & Brooks/Cole Mathematics Series. Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software. xviii, 197 p. (1991).

This book is written as background for those who are interested in what is currently happening in algebraic combinatorics. What makes this field so exciting is that the problems are often very easy to state, but the really effective tools are still being fashioned. The tools that are proving useful are coming in from fields as widely dispersed as classical analysis, modern representation theory, and pure combinatorics. Researchers from these diverse areas are also being drawn in, creating a need for a quick introduction to the basic tools that others are using. One function of this book is to provide a quick introduction to some of these other fields. In particular, it introduces the reader to representation theory and especially the repesentations of the symmetry group, to combinatorial arguments that have proven insightful such as insertion algorithms and lattice path counting, and to the theory of symmetric functions and especially the role of Schur functions. A second audience served by this book consists of graduate students who can use it to begin to find their way into a very fertile area in which they may find their own research projects. Finally, it is written so as to be accessible to good undergraduates. Sections of this book could be incorporated into a senior seminar or an advanced topics course. As the author states in the Preface, “The purpose of this monograph is to bring together, for the first time under one cover, many of the important results in this field.” These results are as diverse as the theory of Specht modules, Viennot’s lattice paths, and the Murnaghan-Nakayama rule. The exposition is exceptionally clear and well-written.

Reviewer: D.M.Bressoud (St.Paul)

### MSC:

05E10 | Combinatorial aspects of representation theory |

05E05 | Symmetric functions and generalizations |

05-02 | Research exposition (monographs, survey articles) pertaining to combinatorics |

20C20 | Modular representations and characters |