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Linear Representations of Finite Groups

Jean-Pierre Serre
Publisher: 
Springer Verlag
Publication Date: 
1977
Number of Pages: 
170
Format: 
Hardcover
Series: 
Graduate Texts in Mathematics 42
Price: 
74.95
ISBN: 
0387901906
Category: 
Textbook
BLL Rating: 

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Michael Berg
, on
03/8/2020
]
Serre’s Linear Representations of Finite Groups, translated from the French original, Représentations Lineares des Groups Finis, is another gem by the author, widely acknowledged as one of the very greatest expositors of mathematical prose.  As is the case with so many books by Serre, this book is compact, as is the prose and presentation.  It is also exceptionally lucid.
 
I first had occasion to study the book in graduate school in the 1980s, in connection with my doctoral thesis, and it made a great deal of difference, as well as a huge and lasting impact.  At that stage, I had already come across a number of works by Serre.  These included his stunning but (in the words of one of my earlier professors, the late Basil Gordon) “austere” Course of Arithmetic, as well as a fabulous short article of his on a number-theoretic theme that was used in a seminar I had attended some years earlier.  In my encounter with the book under review, my main objective was to come to terms with induced representations and Serre did not disappoint.  Other sources that I had looked at were, not surprisingly, heavily oriented toward calculating induced characters and the according relations, but I had found these approaches cumbersome and somewhat unmotivated.  I was looking for something more holistic, so to speak.  In Serre’s treatment, it all centered around linear algebra, and I was immediately roped in.  Everything got much clearer --- fast.
 
The present book is noteworthy not just because of its elegance and concision, however.  It is also rather unusual in the sense that Serre directs the first third of it at quantum chemists: his late wife Josiane  Heulot-Serre was a chemist, and this part of the book is a reproduction of what Serre wrote for, among others, her students.  The focus of this part is, accordingly, on character theory, but done very smoothly: see my earlier remark.
 
After this, the second part is focused on expanding on the themes of the first part, appropriate to Serre’s own courses at the École Normale Supérieure.  Says Serre in connection herewith:  “The methods used are those of linear algebra (in a wider sense than in the first part): group algebras, modules, noncommutative tensor products, semisimple algebras.”  Indeed: very effective and very pretty.
 
Finally, the book’s third and last part is devoted to Brauer theory, and Serre first presented some of this material in the Séminaire de Géométrie Algébrique, I.H.E.S., 1965/66.  
 
Nothing more needs to be said: this book is another treasure by a grandmaster.  I think it is the definitive source from which to learn group representation theory.

 

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.
 

Part I: Representations and Characters; 1. Generalities on Linear Representation; 2. Character Theory; 3. Subgroups, products, induced representations; 4. Compact Groups; 5. Examples; Bibliography Part I; Part II: Representation in Characteristic Zero; 6. The Group Algebra; 7. Induced Representations- Mackey's Criterion; 8. Examples of Induced Representations; 9. Artin's Theorem; 10. A Theorem of Brauer; 11. Applications of Brauer's Theorem; 12. Rationality Questions; 13. Rationality Questions: Examples; Bibliography Part II; Part III: Introduction to Brauer Theory; 14. The Groups Rk(G), Rk(G) and Pk(G); 15. The cde Triangle; 16. Theorems; 17. Proofs; 18. Modular Characters; 19. Application to Artin Representations; Appendix; Bibliography part III; Index of Notation; Index of Terminology.

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