Linear representations of finite groups
Michael Aschbacher · Cambridge University Press eBooks · 2000
Chapter 12 considers FG -representations where G is a finite group, F is a splitting field for G , and the characteristic of F does not divide the order of G . Under these hypotheses, FG -representation theory goes particularly smoothly. For example Maschke's Theorem says each FG -representation is the sum of irreducibles, while, as F is a splitting field for G , each irreducible FG -representation is absolutely irreducible. Section 34 begins the analysis of the characters of such representations. We find that, if m is the number of conjugacy classes of G , then G has exactly m irreducible characters (χ i : 1 ≤ i < m ), and that these characters form a basis for the space of class functions from G into F . A result of Brauer (which is beyond the scope of this book) shows that, under the hypothesis of the first paragraph, the representation theory of G over F is equivalent to the theory of G over ℂ, so section 35 specializes to the case F = ℂ. The character table of G over ℂ is defined; this is the m by m complex matrix (χ i ( g j )), where ( g j 1 ≤ j ≤ m ) is a set of representatives for the conjugacy classes of G . Various numerical relations on the character table are established; among these the orthogonality relations of lemma 35.5 are most fundamental. The concepts of induced representations and induced characters are also discussed.