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Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer
Charles W. Curtis, University of Oregon, Eugene, OR
A co-publication of the AMS and the London Mathematical Society.

History of Mathematics
1999; 292 pp; softcover
Volume: 15
Reprint/Revision History:
reprinted 2000
ISBN-10: 0-8218-2677-8
ISBN-13: 978-0-8218-2677-5
List Price: US$48
Member Price: US$38.40
Order Code: HMATH/15.S
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The year 1897 was marked by two important mathematical events: the publication of the first paper on representations of finite groups by Ferdinand Georg Frobenius (1849-1917) and the appearance of the first treatise in English on the theory of finite groups by William Burnside (1852-1927). Burnside soon developed his own approach to representations of finite groups. In the next few years, working independently, Frobenius and Burnside explored the new subject and its applications to finite group theory.

They were soon joined in this enterprise by Issai Schur (1875-1941) and some years later, by Richard Brauer (1901-1977). These mathematicians' pioneering research is the subject of this book. It presents an account of the early history of representation theory through an analysis of the published work of the principals and others with whom the principals' work was interwoven. Also included are biographical sketches and enough mathematics to enable readers to follow the development of the subject. An introductory chapter contains some of the results involving characters of finite abelian groups by Lagrange, Gauss, and Dirichlet, which were part of the mathematical tradition from which Frobenius drew his inspiration.

This book presents the early history of an active branch of mathematics. It includes enough detail to enable readers to learn the mathematics along with the history. The volume would be a suitable text for a course on representations of finite groups, particularly one emphasizing an historical point of view.


Graduate students and research mathematicians; mathematical historians.


"Any historian interested in dealing with such matters, however, will derive considerable benefit and mathematical insight from a careful reading of Curtis's mathematically rich and well-documented book. I see it as a fine example of how mathematicians and historians can cooperate productively in the challenging task of piecing together the history of mathematics."

-- Historia Mathematica

"Parts of the book will appeal to any reader with an interest in the history of mathematics. For instance, there are biographies of each of the major contributors at the beginnings of the chapters, and numerous smaller biographical sketches are scattered throughout the text. There are also revealing glimpses into the mathematical culture of the early twentieth century found in the numerous citations from the correspondence between these men. Curtis' accounts of the major upheavals in the lives of German mathematicians caused by the anti-semitism of the Nazi regime are also quite engaging. Graduate students just starting out on a research program in algebra or representation theory would benefit immensely from it. Curtis has produced a rare but necessary sort of book that fills this need for a more formal introduction to the historical mathematical background of the twentieth-century mathematics usually seen first in graduate school."

-- MAA Online

"This book is likely to be of interest to any mathematician who has had occasion in any of his/her own work to use group representation theory in any of its many contemporary guises. This is a beautiful and carefully written book, which succeeds at many levels. The mathematics discussed is powerful, and influences many areas of modern mathematics (and other sciences). The story of its evolution and its various diversifications has its own fascination, and serves to remind us how a single mathematical question can lead to the creation of vital new areas. The mathematical contribution of the main characters inspires admiration, while we also gain some insight into their lives at a human level. In short, the book fascinates both as mathematics and as history."

-- The LMS Newsletter

"This is a masterly account, from a master of the subject, of the history of the representation theory of (mostly finite) groups. It features the four great pillars of the classical theory, but goes back much further to the roots of the theory, and includes descriptions of the contributions of the very many other mathematicians involved in building the splendid edifice it has become today."

-- Zentralblatt MATH

"One could probably use [the book] as the basic text for an exciting graduate course on characters and representations of finite groups ... a clear and modern exposition of the most influential papers in the development of character theory and representation theory of finite groups up to about 1950 ... stands as an achievement which should turn out to be as much a beginning as an end."

-- Mathematical Reviews

Table of Contents

  • Some 19th-century algebra and number theory
  • Frobenius and the invention of character theory
  • Burnside: Representations and structure of finite groups
  • Schur: A new beginning
  • Polynomial representations of \(GL_n(\mathbb{C})\)
  • Richard Brauer and Emmy Noether: 1926-1933
  • Modular representation theory
  • Bibliography
  • Index
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