Time and Location
MWF Per. 7, Little 235.
(Tentative) MTW Per. 6, Little 432, and by appointment.
Description and Goals
Group representation theory is the study of group
actions on vector spaces where each group element acts as a linear transformation.
We will concentrate the case of finite groups, but we will consider
vector spaces both of characteristic 0 (“ordinary” representation theory)
and of characteristic p (“modular” representation theory). The aim will be
to alternate between developing the theory and examining examples.
First semester (Fall 2014)
Part I. Generalities.
We shall begin with general background on modules that apply in all cases and work out
some small examples. Topics include:
Group algebras; modules and representations; direct sums and indecomposable modules;
Hom spaces,tensor products; canonical isomorphisms; induction and restriction,
permutation modules; the trace map, tensor identity, Nakayama relations (Frobius reciprocity); Mackey formula.
Part II. Character Theory.
In the ordinary theory all modules are semisimple and the theory can be further developed
in the form of character theory. Highlights will include Burnside’s p^aq^b Theorem and
Frobenius’s Theorem, and Brauer’s characterization of characters. These
are spectacular applications of character theory to answer questions about groups.
Part III. Modular representation theory.
The modular theory is more subtle and homological algebra comes into play.
We shall consider projective modules, extensions, and filtrations.
The ordinary and modular theories are tied together by the study of modules
over a local principal ideal domain whose quotient field is of characteristic zero
and whose residue field is of characteristic p. This leads to Brauer’s block theory, whereby ordinary and modular representations are grouped into “blocks”.
Block theory leads to deeper arithmetical results on characters through
the study of modular representations, and also has many fascinating
problems of its own. We hope to end by discussing a famous
conjecture of Alperin which is now one of the main focal points in this subject.
Second semester (Spring 2015)
In the second semester we shall explore the theory of blocks further.
To each block there is associated a conjugacy class of p-subgroups called the defect groups.
Brauer’s Main Theorems establish the relationship of the blocks of a group
to the blocks of subgroups closely related to their defect groups, such as their
normalizers and centralizers. These results yield useful information
about the values of characters on p-singular elements.
After developing these general results we will
look at some important special cases and applications, including the theory
of blocks of defect 1 and Glauberman’s Z*-theorem.
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Attendance and Late Policy
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