Preface
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 charac-
ters.
This
is a fundamental result,
of
constant use in mathematics as well as in
quantum chemistry
or
physics. I have tried to give proofs as elementary as
possible, using only the definition
of
a group and the rudiments
of
linear algebra.
The
examples (Chapter 5) have been chosen from those useful to chemists.
The second part is a course given in 1966 to second-year students
of
I'Ecole
Nonnale.
It
completes the first on the following points:
(a) degrees
of
representations and integrality properties
of
characters (Chapter 6);
(b) induced representations, theorems
of
Artin and Brauer, and applications
(Chapters
7-1
J);
(c) rationality questions
(Chapters]
2 and ) 3).
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.
The
third part is an introduction to Brauer theory: passage from characteristic 0
to characteristic p (and conversely). I have freely used the language
of
abelian
categories (projective modules, Grothendieck groups), which is well suited to
this sort
of
question.
The
principal results are:
(a)
The
fact that the decomposition homomorphism is surjective: all irreducible
representations in characteristic
p can be lifted "virtually"
(i.e.,
in a suitable
Grothendieck group) to characteristic
o.
(b)
The
Fong-Swan
theorem, which allows suppression
of
the
word'
'virtually"
in the preceding statement, provided that the
group
under consideration is
p-solvable.
I have also given several applications to the Artin representations.
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