Preface

Summary

The purpose of this book is to provide an account of both the ordinary and the modular representation theory of the symmetric groups. The range of applications of this theory is vast, varying from theoretical physics through combinatorics to the study of polynomial identity algebras, and new uses are still being found. So diverse are the questions which arise that we feel justified in hoping the reader might find that some part of our text inspires him to undertake research of his own into one of the many unsolved problems in this elegant branch of mathematics.

There are several different ways of approaching symmetric group representations, and while we have tried to illuminate parts of the theory by giving more than one description of it, we have made no effort to cover every view of the subject.

The ordinary representation theory of the symmetric groups was first developed by Frobenius, but the greatest contribution to the early material came from Alfred Young. Since Young's main interest lay in quantitative substitutional analysis, it is difficult for a modern mathematican to understand his papers. The reader is referred to the book Substitutional Analysis by D. E. Rutherford for a pleasant account of a great part of Young's work. Both Frobenius's and Young's collected works are now available. We include an account of the group algebra and its idempotents, along the lines pursued by Young, since the symmetric group is one of the very rare cases where many aspects of general representation theory can be described explicitly.