Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer (History of Mathematics)

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. Co-published with the London Mathematical Society beginning with Volume 4. Members of the LMS may order directly from the AMS at the AMS member price. The LMS is registered with the Charity Commissioners. Front Matter Cover Title page Dedication Contents Preface Photo Credits Archival Sources Chapter I. Some 19th-century Algebra and Number Theory 1. Introduction 2. Ruler and Compass Constructions and Cyclotomic Fields 3. Equations Defining Subfields of the Cyclotomic Field 4. The Quadratic Subfield of the Cyclotomic Field 5. Lagrange Resolvents and Gauss Sums 6. Quadratic Reciprocity: a proof based on Gauss sums 7. Dirichlet's L-series and Characters mod k 8. Characters associated with quadratic forms Chapter II. Frobenius and the Invention of Character Theory 1. Frobenius in Berlin and Zürich: 1850-1900 2. Research on Finite Groups and Number Theory: 1880-1896 3. Characters of Finite Groups 4. Group Representations and Characters 5. The Characters of the Symmetric Group 6. An Application of Character Theory: Frobenius Groups Chapter III. Burnside: Representations and Structure of Finite Groups 1. Burnside at Cambridge and Greenwich 2. Burnside's Early Research on Finite Groups: 1890-1900 3. From Lie Groups to Representations of Finite Groups: 1898-1900 4. Foundations of Representation Theory: 1900-1905 5. The p^a q^b -Theorem and Other Applications of Character Theory Chapter IV. Schur: A New Beginning 1. Issai Schur in Berlin: 1894-1939 2. Schur's New Foundations of Character Theory 3. Joint Work of Frobenius and Schur: 1905-1906 4. The Schur Index 5. Projective Representations of Finite Groups 6. Permutation Groups and Centralizer Rings Chapter V. Polynomial Representations of GL_n(C) 1. Invariant Theory and Representations of GL_n: Jacques Deruyts 2. Polynomial Representations of GL_n: Issai Schur 3. Schur's Character Formula Chapter VI. Richard Brauer and Emmy Noether: 1926-1933 1. Richard Brauer in Berlin, Königsberg, and North America 2. Emmy Noether and her Impact on Representation Theory 3. Simple Algebras and the Brauer Group: 1926-1933 Chapter VII. Modular Representation Theory 1. Brauer and Nesbitt: Opening Moves 2. Nonsemisimple Algebras: Brauer, Nakayama, and Nesbitt 3. Blocks and the Classification of Finite Simple Groups 4. The Brauer Induction Theorem and its Applications Bibliography Index Errata Back Cover "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. The volume would be a suitable text for a course on representations of finite groups, particularly one emphasizing an historical point of view."--Jacket