Math 8246 Group Theory Spring Semester 2004
Instructor
Peter Webb, 350 Vincent Hall, 625 3491, webb@math.umn.edu, http://www.math.umn.edu/~webb
Office Hours
2:00-3:00 MWF or by appointment.
Text
There will be no text to purchase. Lecture notes on representation theory are available from http://www.math.umn.edu/~webb
Other useful texts
DJS Robinson, A course in the theory of groups, 2nd edition, Springer-Verlag, ISBN 0387944613.
J.L. Alperin, Local representation theory, Cambridge University Press 1993, ISBN 052144926X.
D.J. Benson, Representations and cohomology, vols I and II, Cambridge University Press 1991
K.W. Gruenberg, Cohomological Topics in Group Theory, Lecture Notes in Math. 143, Springer 1970, now available in an electronic version.
J.-P. Serre, Linear representations of finite groups, Springer Graduate Texts in Mathematics
Course Content
We will continue with representation theory from where we left off last semester. The first thing will be to finish off what we were doing with projective modules for group algebras. After that we will study splitting fields, Brauer's theorem on the number of simple modules in positive characteristic, representations over rings of algebraic integers and the decomposition map. Next will come Brauer characters, and after that the theory of vertices and sources and Green correspondence. We will then study blocks and prove Brauer's first main theorem.
At this point it may be that we know enough modular representation theory to consider ourselves well-educated in that direction, and I propose to say something about group cohomology. We will start with the definitions, talk about resolutions and the restriction, corestriction and inflation mappings, and identify the low-dimensional cohomology groups. This will involve a discussion of extension theory and the Schur multiplier. As an application of group cohomology I may then talk about crystallographic groups. This is a rather pretty subject in which pictures such as some of those produced by Escher may appear, and at this advanced level the classification of crystal structures comes down to a question of group theory and a group cohomology calculation. At this point it will be a question of seeing how long this syllabus has taken....
Instead of some of the above, I could teach the representation theory of symmetric groups in arbitrary characteristic, and its relation to the representation theory of the general linear groups via 'Schur-Weyl duality'.
Course Assessment
I will assign a set of homework problems roughly every 2 weeks, giving a total of at most six homework assignments altogether. If you make a genuine attempt at 50% or more of the questions you will get an A for the course. You do not have to obtain correct solutions to these questions, only make genuine attempts (in my opinion). I am well aware that the existence of homework in advanced courses is considered by students quite negatively. I believe myself that it is extremely difficult to obtain a sound and permanently-lasting command of the material presented without doing some work which actively involves the student. If people wish to propose other forms of active involvement, I will be extremely willing to discuss these. As it is, it should be possible for everyone who wishes to obtain an A on this course.
Expectations of written work
Most of the time in the conventional homework problems, to satisfy my criterion of making a genuine attempt you will need to write down explanations for the calculations and arguments you make. Where explanations need to be given, these should be written out in sentences i.e. with verbs, capital letters at the beginning, periods at the end, etc. and not in an abbreviated form.
I encourage you to form study groups. However everything to be handed in must be written up in your own words. If two students hand in identical assignments, they will both receive no credit.
Incompletes
These will only be given in exceptional circumstances. A student must have satisfactorily completed all but a small portion of the work in the course, have a compelling reason for the incomplete, and must make prior arrangements with me for how the incomplete will be removed, well before the end of the quarter.
Proof-reading the representation theory notes
The notes which I hand out to you may one day be a book. I will be very appreciative of any comments you have about what I write. These may be lists of typographical errors, mathematical errors, or comments that perhaps more explanation would be in order in such and such a place, more background should be given as motivation etc. etc. In advance I give you my thanks.
Date of this version of the schedule: 1/20/2004