Xl l INTRODUCTION A good example of this philosophy is the proof in Chapter VIII that all maximal tori in a compact Lie group are conjugate and the union of all the maximal tori is the entire group. The standard proofs either go through the conjugacy of the Cartan subalgebras and considerable additional argument, or else uses Weil's approach of using the Lefschitz fixed point theorem, a sledgehammer for what is a rather simple result. Instead, I use a simple argument inductive in the dimension of G. The first 90% of the proof is that used by Varadarajan [20], but at a critical point, he appeals to the structure theory of the semisimple Lie algebras, which requires tens of pages of careful algebraic argument. Instead, I use the existence of an invariant inner product for the adjoint representation. This is one of dozens of places where the proofs are ones I found while polishing the book. Nevertheless, I am not so naive as to think that there are any proofs here that don't appear somewhere in the literature, which is vast. But I do claim a coherent, elementary approach. Individual chapters begin with brief summaries of what they contain. Chapter I sets the stage, focusing on counting principles as a leitmotif. The high point is the Klein-Weyl determination of the finite subgroups of three-dimensional rotations. Chapters II-VI discuss the representations of finite groups. Chapters II—III develop the general theory and Chapters IV-VI, the representations of specific families of groups: Abelian and Clifford groups in Chapter IV, semidirect products in Chapter V, and permutation groups in Chapter VI. The final three chapters discuss the representations of compact groups, primarily compact Lie groups. Chapter VII discusses the general theory of Lie groups and the analogs of the results of Chapter III. Chapter VIII discusses the structure theory of compact Lie groups: maximal tori, roots, and the Weyl group. It is preparation for the final chapter which presents Weyl's theory of the representations of the classical groups. The final section draws together the two halves of the book in a fascinating way by providing a proof of the Probenius character formula for the permutation group. By focusing on finite and compact groups, we can present the basics completely. Any attempt to go beyond this would yield a multiple-volume work (as it has in other cases!). Nevertheless, when I've given this as a one-year graduate course, I have spent five weeks discussing related topics from the representation theory of noncompact groups. This book is based on a course I first gave at Princeton in the mid-70's. Over the ensuing twenty years, I have given the course roughly a half-dozen times at Princeton or Caltech, and each time additional polish was added. I am grateful to all the students in those courses for the feedback and insight they provided. Parts of the actual manuscript were written during stays at the ETH-Zurich, Hebrew University, and the Technion. I appreciate their hospitality and, in par- ticular, the courtesy shown to me by CM . Graf, W. Hunziker, M. Ben-Artzi, and J. Avron. The preparation of the manuscript, which involved taming both TeX and my handwriting, was well handled by C. Galvez, to whom I'm grateful. I benefited from a careful reading and comments from S. Miller. I hope you will enjoy this book. I can't think of any other course of mathematics with so much innate beauty so close to the surface.
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