Math 298: Representation Theory, Group study
Winter 2004

If you signed up for Math 299, crn 53613 (299-068 (53613)) , please DROP this and add Math 298, crn 40072.

Meeting time: M 2:10-3:00pm, Kerr 593   crn: 40072
Instructor: Eric Rains, Kerr Hall 677, phone: 754-8160
Monica Vazirani, Kerr Hall 579, phone: 752-2218,
Office hours: Eric's in his office (677) quite often, and to quote him: when his door is open, it means his door is open.
You're always welcomed to come by Monica's MAT-21A office hours if no other students are there, and otherwise by appointment. Monday and Wednesday between noon and 3 are especially good.
Text: Serre, Jean-Pierre Linear representations of finite groups. Translated from the second French edition by Leonard L. Scott. Graduate Texts in Mathematics, Vol. 42. Springer-Verlag, New York-Heidelberg, 1977. x+170 pp. ISBN 0-387-90190-6

Until you get copies, chapter 1 is in my box. If you get the last copy, don't TAKE it, but make a copy and leave the last there for the next student. Chapter's 1-8 are being copied and available for $6.60 at Navin's copy place, on 3rd and University (between A and B, by Roma).

Fulton & Harris Representations Theory, a first course .
If people would like to have a peek at the first few pages of this (and table of contents) I can also make some copies. I don't mind if you decide to read out of this book instead.
In class presentations: Depending on how many units you take this for, you will be asked to present a chapter, or a section of a chapter, (or even just explain a concept, construction, definition, or present a homework solution) in class. Contact us soon about which chapter-- earlier chapters should be easier to do, although they come up sooner!
Below we'll list scheduled presentations:

Ch 2 --- Connie Wilmarth
Ch 3 --- Alice Stevens
Ch 4 --- Joseph Crayne, with Kei Nakamura talking about connections to Fourier theory
Ch 5 --- everyone taking turns presenting examples and hw questions. also on Thursday, February 19th from 3:10-4 in 693 Kerr we'll have a makeup meeting.
There is a talk in that room at 4:10...just to give you a heads up.
Ch 6 --- Rick
Ch 7 --- Yvonne (March 1st probably)
Ch 8 ---

Problem Sets: Ch 1-- pay special attention to Thm 1, and its proof. Try this exercise: Let V, V' be representations of G. Let f: V -- > V' be a G-map (that is f ° rho(g) = rho'(g) ° f (or more simply f°g=g°f) for all g in G). Show (1) ker f is a subrepresentation of V (i.e. is G-invariant) and (2) Im f is a subrepr of V'.

Ch 2. Try exercises 2.2, 2.3, 2.4, maybe 2.5, 2.6a,b, 2.7.
And pay special attention to Schur's Lemma in Ch2.

Ch 3. Try exercises 3.1, 3.2, 3.3, 3.4, and make sure to understand Example 2, page 29.

Ch 5. Exercises 5.1, 5.4 (on pg 42, not the one pg43), 5.5

Ch 6. Exercises 6.1 (Maschke's Thm), 6.2, 6.4, 6.5, 6.6, 6.9 if you like number theory and things about algebraic integers
Also try this: let G = {1,a,a^2} be the cyclic group of order 3. (a) Give an explicit ring isomorphism CG = C x C x C. (b) Give an explicit ring isomorphism RG = R x C.

Ch 7. Exercises 7.2, 7.3, 7.4

Ch 8. 8.2, 8.7, 8.9
Grading: Now that this is a 298, group study, we must assign letter grades. Hope that's ok. Sorry for the extra paperwork. You can sign up online. Technically I'm supposed to sign this form outside of Celia's office. Anyone who is taking this for more than 1 unit should run it by me and/or bring me this form to sign.
If you signed up for Math 299, crn 53613 (299-068 (53613)) , please DROP this and add Math 298, crn 40072.
Prerequisites: linear algebra, algebra
Participants, emails: lesphilli@math; lesyoung@math

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