Representation theory studies maps from groups into the general linear
group of a finite-dimensional vector space. For finite groups the theory
comes in two distinct flavours. In the 'semisimple case' (for example
over the field of complex numbers) one can use character theory to
completely understand the representations. This by far is not sufficient
when the characteristic of the field divides the order of the group.
Modular Representation Theory of finite Groups comprises this second
situation. Many additional tools are needed for this case. To mention
some, there is the systematic use of Grothendieck groups leading to the
Cartan matrix and the decomposition matrix of the group as well as
Green's direct analysis of indecomposable representations. There is also
the strategy of writing the category of all representations as the
direct product of certain subcategories, the so-called 'blocks' of the
group. Brauer's work then establishes correspondences between the blocks
of the original group and blocks of certain subgroups the philosophy
being that one is thereby reduced to a simpler situation. In particular,
one can measure how nonsemisimple a category a block is by the size and
structure of its so-called 'defect group'. All these concepts are made
explicit for the example of the special linear group of two-by-two
matrices over a finite prime field.
Although the presentation is strongly biased towards the module
theoretic point of view an attempt is made to strike a certain balance
by also showing the reader the group theoretic approach. In particular,
in the case of defect groups a detailed proof of the equivalence of the
two approaches is given.
This book aims to familiarize students at the masters level with the
basic results, tools, and techniques of a beautiful and important
algebraic theory. Some basic algebra together with the semisimple case
are assumed to be known, although all facts to be used are restated
(without proofs) in the text. Otherwise the book is entirely
self-contained.
