Pioneers of Representation Theory: Frobenius, Burnside, Schur, and BrauerThe 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. Members of the LMS may order directly from the AMS at the AMS member price. The LMS is registered with the Charity Commissioners. |
Contents
Some 19thCentury Algebra and Number Theory | 1 |
Frobenius and the Invention of Character Theory | 35 |
Burnside Representations and Structure of Finite Groups | 87 |
Schur A New Beginning | 125 |
Polynomial Representations of G LnC | 173 |
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Common terms and phrases
A-module abelian group Berlin Burnside Burnside's Burnside's Theorem central simple algebra Chapter character theory coefficients commutative completely reducible completing the proof complex numbers conjugacy classes conjugate contains corresponding Dedekind defined degree denote direct sum eigenvalues elements of G equations equivalent factor set finite group finite order follows formula Frobenius Frobenius's functions Galois group Gauss given group algebra group determinant group G group matrix group of linear group theory groups of finite invariant irreducible characters irreducible polynomial irreducible representation isomorphic left ideals left o-module linear substitutions Math mathematical modular representation multiplication nonzero number theory obtained orthogonality relations p-block paper permutation groups polynomial representations preceding theorem proved quadratic forms rational regular representation representation of G representation theory representations of finite result Richard Brauer ring roots of unity satisfying Schur index semisimple simple groups splitting field subfield subgroup H symmetric group variables vector space