Representations of Finite Groups: A Hundred Years, Part II

Recapitulation The origin of the representation theory of finite groups can be traced back to a correspondence between R. Dedekind and F. G. Frobenius that took place in April of 1896. The present article is based on several lectures given by the author in 1996 in commemoration of the centennial of this occasion. In Part I of this article we recounted the story of how Dedekind proposed to Frobenius the problem of factoring a certain homogeneous polynomial arising from a determinant (called the “group determinant”) associated with a finite group G . In the case when G is abelian, Dedekind was able to factor the group determinant into linear factors using the characters of G (namely, homomorphisms of G into the group of nonzero complex numbers). In a stroke of genius, Frobenius invented a general character theory for arbitrary finite groups, and used it to give a complete solution to Dedekind’s group determinant problem. Interestingly, Frobenius’s first definition of (nonabelian) characters was given in a rather ad hoc fashion, via the eigenvalues of a certain set of commuting matrices. This work led Frobenius to formulate, in 1897, the modern definition of a (matrix) representation of a group G as a homomorphism D : G → GLn(C) (for some n). With this definition in place, the character χD : G → C of the representation is simply defined by χD(g) = trace(D(g)) (for every g ∈ G). The idea of studying a group through its various representations opened the door to a whole new direction of research in group theory and its applications. Having surveyed Frobenius’s invention of character theory and his subsequent monumental contributions to representation theory in Part I of this article, we now move on to tell the story of another giant of the subject, the English group theorist W. Burnside. This comprises Part II of the article, which can be read largely independently of Part I. For the reader’s convenience, the few bibliographical references needed from Part I are reproduced here, with the same letter codes for the sake of consistency. As in Part I, [F: (53)] refers to paper (53) in Frobenius’s collected works [F]. Burnside’s papers are referred to by the year of publication, from the master list compiled by Wagner and Mosenthal in [B]. Consultation of the original papers is, however, not necessary for following the general exposition in this article.