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Graduate Texts in Mathematics #42
Linear Representations of Finite Groups
This book consists of three parts, rather different in level and purpose. The first part was originally written for quantum chemists. It describes the correspondence, due to Frobenius, between linear representations and characters. The second part is a course given in 1966 to second-year students of l’Ecole Normale. It completes in a certain sense the first part. The third part is an introduction to Brauer Theory.
- GenresMathematicsAlgebra
182 pages, Hardcover
First published September 1, 1977
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82 people want to read
About the author
Jean-Pierre Serre
70 books18 followersBorn in Bages, Pyrénées-Orientales, France, to pharmacist parents, Serre was educated at the Lycée de Nîmes and then from 1945 to 1948 at the École Normale Supérieure in Paris.
He was awarded his doctorate from the Sorbonne in 1951. From 1948 to 1954 he held positions at the Centre National de la Recherche Scientifique in Paris.
In 1956 he was elected professor at the Collège de France, a position he held until his retirement in 1994. His wife, Professor Josiane Heulot-Serre, was a chemist; she also was the director of the Ecole Normale Supérieure de Jeunes Filles. Their daughter is the historian and writer Claudine Monteil.
From a very young age he was an outstanding figure in the school of Henri Cartan, working on algebraic topology, several complex variables and then commutative algebra and algebraic geometry, where he introduced sheaf theory and homological algebra techniques. Serre's thesis concerned the Leray–Serre spectral sequence associated to a fibration. Together with Cartan, Serre established the technique of using Eilenberg–MacLane spaces for computing homotopy groups of spheres, which at that time was one of the major problems in topology.
In his speech at the Fields Medal award ceremony in 1954, Hermann Weyl gave high praise to Serre, and also made the point that the award was for the first time awarded to an algebraist. Serre subsequently changed his research focus. However, Weyl's perception that the central place of classical analysis had been challenged by abstract algebra
has subsequently been justified, as has his assessment of Serre's place in this change.
In the 1950s and 1960s, a fruitful collaboration between Serre and the two-years-younger Alexander Grothendieck led to important foundational work, much of it motivated by the Weil conjectures. Two major foundational papers by Serre were Faisceaux Algébriques Cohérents (FAC), on coherent cohomology, and Géometrie Algébrique et Géométrie
Analytique (GAGA).
In his paper FAC, Serre asked whether a finitely generated projective module over a polynomial ring is free. This question led to a great deal of activity in commutative algebra, and was finally answered in the affirmative by Daniel Quillen and Andrei Suslin independently in 1976. This result is now known as the Quillen-Suslin theorem.
Even at an early stage in his work Serre had perceived a need to construct more general and refined cohomology theories to tackle the Weil conjectures. The problem was that the cohomology of a coherent sheaf over a finite field couldn't capture as much topology as singular cohomology with integer coefficients. Amongst Serre's early candidate theories of 1954–55 was one based on Witt vector coefficients.
Around 1958 Serre suggested that isotrivial principal bundles on algebraic varieties — those that become trivial after pullback by a finite étale map — are important.
This acted as one important source of inspiration for Grothendieck to develop étale topology and the corresponding theory of étale cohomology. These tools, developed in full by Grothendieck and collaborators in Séminaire de géométrie algébrique (SGA) 4 and SGA 5, provided the tools for the eventual proof of the Weil conjectures.
From 1959 onward Serre's interests turned towards group theory, number theory, in particular Galois representations and modular forms.
Amongst his most original contributions were: his "Conjecture II" (still open) on Galois cohomology; his use of group actions on Trees (with H. Bass); the Borel-Serre compactification; results on the number of points of curves over finite fields; Galois representations in ℓ-adic cohomology and the proof that these representations have often a "large" image; the concept of p-adic modular form; and the Serre conjecture (now a theorem) on mod-p representations that made Fermat's last theorem a connected part of mainstream arithmetic geometry.
Serre, at twenty-seven in 1954, is the youngest ever to be awarded
He was awarded his doctorate from the Sorbonne in 1951. From 1948 to 1954 he held positions at the Centre National de la Recherche Scientifique in Paris.
In 1956 he was elected professor at the Collège de France, a position he held until his retirement in 1994. His wife, Professor Josiane Heulot-Serre, was a chemist; she also was the director of the Ecole Normale Supérieure de Jeunes Filles. Their daughter is the historian and writer Claudine Monteil.
From a very young age he was an outstanding figure in the school of Henri Cartan, working on algebraic topology, several complex variables and then commutative algebra and algebraic geometry, where he introduced sheaf theory and homological algebra techniques. Serre's thesis concerned the Leray–Serre spectral sequence associated to a fibration. Together with Cartan, Serre established the technique of using Eilenberg–MacLane spaces for computing homotopy groups of spheres, which at that time was one of the major problems in topology.
In his speech at the Fields Medal award ceremony in 1954, Hermann Weyl gave high praise to Serre, and also made the point that the award was for the first time awarded to an algebraist. Serre subsequently changed his research focus. However, Weyl's perception that the central place of classical analysis had been challenged by abstract algebra
has subsequently been justified, as has his assessment of Serre's place in this change.
In the 1950s and 1960s, a fruitful collaboration between Serre and the two-years-younger Alexander Grothendieck led to important foundational work, much of it motivated by the Weil conjectures. Two major foundational papers by Serre were Faisceaux Algébriques Cohérents (FAC), on coherent cohomology, and Géometrie Algébrique et Géométrie
Analytique (GAGA).
In his paper FAC, Serre asked whether a finitely generated projective module over a polynomial ring is free. This question led to a great deal of activity in commutative algebra, and was finally answered in the affirmative by Daniel Quillen and Andrei Suslin independently in 1976. This result is now known as the Quillen-Suslin theorem.
Even at an early stage in his work Serre had perceived a need to construct more general and refined cohomology theories to tackle the Weil conjectures. The problem was that the cohomology of a coherent sheaf over a finite field couldn't capture as much topology as singular cohomology with integer coefficients. Amongst Serre's early candidate theories of 1954–55 was one based on Witt vector coefficients.
Around 1958 Serre suggested that isotrivial principal bundles on algebraic varieties — those that become trivial after pullback by a finite étale map — are important.
This acted as one important source of inspiration for Grothendieck to develop étale topology and the corresponding theory of étale cohomology. These tools, developed in full by Grothendieck and collaborators in Séminaire de géométrie algébrique (SGA) 4 and SGA 5, provided the tools for the eventual proof of the Weil conjectures.
From 1959 onward Serre's interests turned towards group theory, number theory, in particular Galois representations and modular forms.
Amongst his most original contributions were: his "Conjecture II" (still open) on Galois cohomology; his use of group actions on Trees (with H. Bass); the Borel-Serre compactification; results on the number of points of curves over finite fields; Galois representations in ℓ-adic cohomology and the proof that these representations have often a "large" image; the concept of p-adic modular form; and the Serre conjecture (now a theorem) on mod-p representations that made Fermat's last theorem a connected part of mainstream arithmetic geometry.
Serre, at twenty-seven in 1954, is the youngest ever to be awarded
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Displaying 1 - 4 of 4 reviews
May 13, 2016
Extremely elegant proofs and logical structure. Lacks intuition a bit. Exercises aren't all that challenging.
June 9, 2019
An absolutely essential book for representation theory. Well translated etc, the only thing I would say is a fair few of the proofs are needlessly complicated. For instance, I remember early on a proof where we have to prove something for all finite dimensional vector spaces, which one would naturally do by induction starting from a line and working upwards. However, Serre insists on this very obtuse proof in strong induction, starting from the 0 vector space? Very weird.
July 12, 2007
It's in the usual style of Serre. But it's quite good once you are comfortable with some basic operations of algebra like tensors, semisimple theory etc. It might be a bit scary to start with but becomes easy later on.
April 2, 2025
I read the first section for a directed study/research thing for a EX credit. I did all the problems (sorta)
Serre is an excellent writer. He is quite terse but his style is pleasantly readable. This is especially true in his proofs. They are always clear but are so austere that you must read them quite carefully.
The problems are not too difficult, although their statement can be a bit awkward.
The examples section is pretty boring, but it was made for chemists so that is not surprising.
I wish the compact group section was more detailed, but fiddling with those ideas on my own was quite fun. (Much closer to analysis so that was fun)
After I do more algebra stuff I wanna return and finish the second section!
Serre is an excellent writer. He is quite terse but his style is pleasantly readable. This is especially true in his proofs. They are always clear but are so austere that you must read them quite carefully.
The problems are not too difficult, although their statement can be a bit awkward.
The examples section is pretty boring, but it was made for chemists so that is not surprising.
I wish the compact group section was more detailed, but fiddling with those ideas on my own was quite fun. (Much closer to analysis so that was fun)
After I do more algebra stuff I wanna return and finish the second section!
Displaying 1 - 4 of 4 reviews