Instructor
Time and Location
MWF Per. 7, Little 237.
Office Hours:
(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.
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.
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Attendance and Late Policy
Attendance is recommended.
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Special accommodations
Students requesting classroom accommodation must first register with the Dean of Student Office. The Dean of Students Office will provide documentation to the student who must then provide this documentation to the instructor when requesting accommodation.
Honor Code
UF students are bound by The Honor Pledge which states, “We, the members of the University of Florida community, pledge to hold ourselves and our peers to the highest standards of honor and integrity by abiding by the Honor Code.
On all work submitted for credit by students at the University of Florida, the following pledge is either
required or implied: “On my honor, I have neither given nor received unauthorized aid in doing this
assignment.”
The Honor Code (http://www.dso.ufl.edu/sccr/process/student-conduct-honor-code/)
specifies a number of behaviors that are in violation of this code and the possible sanctions.
Furthermore, you are obliged to report any condition that facilitates academic misconduct to appropriate
personnel. If you have any questions or concerns, please consult with the instructor of this class.
Course Evaluations
Students are expected to provide feedback on the quality of instruction in this course based on 10 criteria. These evaluations are conducted online at https://evaluations.ufl.edu. Evaluations are typically open during the last two or three weeks of the semester, but students will be given specific times when they are open. Summary results of these assessments are available to students at https://evaluations.ufl.edu/results/.