Representations of Finite Groups
The theory of group representations occurs in many contexts. First, it is developed for its own sake: determine all irreducible representations of a given group. See for instance Curtis-Reiner’s Methods of Representation Theory (Wiley-Interscience, 1981).
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XVIII
Representations of Finite Groups
The theory of group representations occurs in many contexts. Fir st , it is developed for its own sake: determine all irreducible representation s of a given gro up. See for inst ance Curti s-Reiner ' s Methods ofRepresentation Theory (WileyInterscien ce , 1981). It is also used in classifying finite simple groups. But already in this book we have seen applications of repr esent ations to Galoi s theory and the determ ination of the Galoi s group ove r the rationals. In addition , there is an analogous theory for topological groups. In this case , the closest analogy is with compact groups, and the reader will find a self-contained treatment of the compact case entirely similar to §5 of this chapter in my book SL 2(R ) (Springer Verlag) , Chapter 11 , §2 . Essenti ally , finite sums are replaced by integrals , other wise the formalis m is the same . The analysis co mes onl y in two places. One of them is to show that eve ry irreducible representation of a comp act group is finite dimensional; the other is Schur' s lemm a. The detail s of these extra considerations are carried out completely in the above -mentioned reference . 1 was care ful to wr ite up §5 with the analo gy in mind . Sim ilarly , readers will find analogous material on induced repre sentations in SL 2(R), Chapter III, §2 (wh ich is also self-contained). Examples of the general theory come in various shapes . Theorem 8.4 may be viewed as an example , showing how a certain repre sent ation can be expressed as a direct sum of induced repre sentations from I-dimensional repre sentations. Examples of repre sent ation s of S3 and S4 are given in the exercises. The entire last section works out completely the simple characte rs for the group GL 2 (F ) when F is a finite field , and shows how these characters essentially co me from indu ced characters . For other exampl es also leading into Lie groups, see W. Fulton and J. Harr is , Representation Theory , Spr inger Verlag 199 1.
663 S. Lang, Algebra © Springer Science+Business Media LLC 2002
664
§1.
REPRESENTATIONS OF FINITE GROUPS
XVIII , §1
REPRESENTATIONS AND SEMISIMPLICITY
Let R be a commutative ring and G a group . We form the group algebra R[G) . As explained in Chapter II , §3 it consists of all formal linear comb inations
with coefficients natural way ,
au E
R, almost all of which are O. The product is taken in the
Let E be an R-module . Every algebra-homomorphism R[G)
~
EndR(E)
induces a group-homomorphism G ~ AutR(E) ,
and thus a representation of the ring R[G) in E gives rise to a representation of the group. Given such representations , we also say that R[GJ, or G, operate on E. We note that the representation makes E into a module over the ring R[G) . Conversely , given a repre sentation of the group, say p : G ~ AutR(E) , we can extend p to a representation of R[G) as follows . Let Q' = 2: auO" and x E E. We define p( a)x =
L a"p«(J)x .
It is immediately verified that p has been extended to a ring -homomorphism of R[G] into EndR(E).
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