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 exampleover 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."
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