Selected Title s i n This Serie s 15 Charle s W. Curtis , Pioneer s of representation theory: Frobenius, Burnside, Schur, and Brauer, 199 9 14 Vladimi r Maz'ya and Tatyana Shaposhnikova, Jacque s Hadamard, a universal mathematician, 199 8 13 Lar s Garding, Mathematic s and mathematicians: Mathematics in Sweden before 1950, 1998 12 Walte r Rudin, Th e way I remember it, 1997 11 Jun e Barrow-Green, Poincar e and the three body problem, 1997 10 Joh n Stillwell, Source s of hyperbolic geometry, 1996 9 Bruc e C. Berndt an d Robert A . Rankin, Ramanujan : Letters and commentary, 1995 8 Kare n Hunger Parshall and David E. Rowe, Th e emergence of the American mathematical research community, 1876-1900: J. J. Sylvester, Felix Klein, and E. H. Moore, 1994 7 Hen k J. M. Bos, Lecture s in the history of mathematics, 1993 6 Smilk a Zdravkovska and Peter L. Duren, Editors, Golde n years of Moscow mathematics, 199 3 5 Georg e W. Mackey , Th e scope and history of commutative and noncommutativ e harmonic analysis, 1992 4 Charle s W. M c Arthur, Operation s analysis in the U.S. Army Eighth Air Force in World War II, 1990 3 Pete r L. Duren et al., Editors, A century of mathematics in America, part III, 1989 2 Pete r L. Duren et al., Editors, A century of mathematics in America, part II, 1989 1 Pete r L. Duren et al., Editors, A century of mathematics in America, part I, 1988 org. This page intentionally left blank org. PIONEERS OF REPRESENTATION THEORY: Frobenius, Burnside, Schur, and Brauer org. This page intentionally left blank org. History of Mathematics Volum e 15 PIOKBtS OF M U T A T I ON THEORY: Frobenius, Burnside, Schur, and Brauer Charles W. Curtis America n Mathematica l Societ y Londo n Mathematica l Societ y org. Editorial Boar d American Mathematica l Societ y Londo n Mathematical Societ y George E. Andrews Davi d Fowler, Chair Bruce Chandler Jerem y J. Gray Karen Parshall, Chair S . J. Patterso n George B. Seligman 1991 Mathematics Subject Classification. Primar y 01A55, 01A60, 20C15, 20C20; Secondary 01A70, 16G10, 20G05. A list of photograph credits and archival sources is included at the beginning of this volume. Library o f Congress Cataloging-in-Publication Dat a Curtis, Charles W. Pioneers o f representation theor y : Probenius , Burnside , Schur , an d Brauer / Charle s W. Curtis. p. cm. — (History of mathematics, ISSN 0899-2428 ; v. 15) Includes bibliographical references and index. ISBN 0-8218-9002-6 (alk. paper) 1. Representations of groups—History. 2 . Finite groups—History. I . Title. II . Series. QA176.C87 199 9 512/.2—dc21 99-1498 3 CIP Copying an d reprinting. Individua l reader s of this publication, an d nonprofit librarie s acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permissio n is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted onl y under licens e from the American Mathematica l Society . Request s for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Request s can also be made by e-mail to [email protected]. © 199 9 by the American Mathematical Society. All rights reserved. Printed in the United States of America. The American Mathematical Society retains all rights except those granted to the United States Government. @ Th e paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. The London Mathematical Society is incorporated under Royal Charter and is registered with the Charity Commissioners. Visit the AMS home page at URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 1 6 15 14 13 12 11 org. To the memory of Irving Reiner org. This page intentionally left blank org. Contents Preface x i Photo Credits x v Archival Sources xv i Chapter I. Som e 19th-century Algebra and Number Theory 1 1. Introductio n 2 2. Rule r and Compass Constructions and Cyclotomic Fields 6 3. Equation s Denning Subfields of the Cyclotomic Field 1 0 4. Th e Quadratic Subfield of the Cyclotomic Field 1 2 5. Lagrang e Resolvents and Gauss Sums 1 5 6. Quadrati c Reciprocity: a proof based on Gauss sums 1 9 7. Dirichlet' s L-series and Characters mod k 2 0 8. Character s associated with quadratic forms 2 7 Chapter II. Frobeniu s and the Invention of Character Theory 3 5 1. Frobeniu s in Berlin and Zurich: 1850-190 0 3 5 2. Researc h on Finite Groups and Number Theory: 1880-189 6 4 0 3. Character s of Finite Groups 5 0 4. Grou p Representations and Characters 6 3 5. Th e Characters of the Symmetric Group 7 3 6. A n Application of Character Theory: Frobenius Groups 8 1 Chapter III. Burnside : Representations and Structure of Finite Groups 8 7 1. Burnsid e at Cambridge and Greenwich 8 7 2. Burnside' s Early Research on Finite Groups: 1890-190 0 9 7 3. Fro m Lie Groups to Representations of Finite Groups: 1898-190 0 10 2 4. Foundation s of Representation Theory: 1900-190 5 11 1 5. Th e pa<76-Theorem and Other Applications of Character Theory 12 0 Chapter IV. Schur : A New Beginning 12 9 1. Issa i Schur in Berlin: 1894-193 9 12 9 2. Schur' s New Foundations of Character Theory 13 8 3. Join t Work of Frobenius and Schur: 1905-190 6 14 9 4. Th e Schur Index 15 7 5. Projectiv e Representations of Finite Groups 16 6 6. Permutatio n Groups and Centralizer Rings 17 2 Chapter V. Polynomia l Representations of GL (C) 111 n 1. Invarian t Theory and Representations of GL : Jacque s Deruyts 17 7 n 2. Polynomia l Representations of GL : Issa i Schur 18 2 n 3. Schur' s Character Formula 19 3 org.