COORDINATES: Lectures meet Tues/Thur. 12:30--1:45 in MSB 215. The registrar calls this Sec 001, #12906
PREREQUISITES: Understanding of abstract linear algebra and the fundamentals of group theory and ring theory at the level of a first-year graduate algebra course. Some multilinear algebra (e.g., tensor products) will be useful, but will be reviewed quickly as needed.
TEXTS: (1) Bruce E. Sagan: The symmetric group: Representations, Combinatorial Algorithms, and Symmetric Functions, 2nd Ed. Graduate Texts in Mathematics, Springer, 2001, ISBN 0-387-95067-2.
(2) William Fulton and Joe Harris, Representation Theory. A First Course. Graduate Texts in Mathematics, 129, Springer-Verlag, 1991, ISBN 0-387-97495-4, 3-540-97495-4. Both of these texts are currently on sale (probably through 31 December 2006) and can both be ordered directly from Springer's website for $80 (including shipping).
WEB RESOURCES: The homepage for this course is http://www.math.uconn.edu/~troby/Math329S07/. It will include a copy of the syllabus and list of homework assignments. I will keep this updated throughout the quarter.
HOMEWORK: There will be at most a half-dozen homework assignments spread out through the term. To really learn the subject, I encourage you to do other exercises or work out examples on your own.
GRADING: Your grade will be based on homework, unless you do an optional presentation, which can then count in place of 1-2 homework assigments.
PRESENTATION: Presenting mathematics to others is a skill that the community as a whole still needs to improve, yet graduate students have very few opportunities to present mathematics at their level. I encourage anyone who wishes to do a presentation to come talk to me during the first third of the term. You may know of a topic or together we can find one that you can investigate on your own and present to the class. The benefits of doing this go far beyond the content that you'll undoubtedly learn well.
CONTENT: This course is an introduction to the representation theory of finite groups and Lie algebras. We will focus first on understanding the basic tools in the simplest context of finite groups, then branch out to Lie algebras. Specific topics will include: characters, induced representations, representations of the symmetric and general linear groups, symmetric functions, Schur-Weyl duality, representations of complex semi-simple Lie algebras, and the Weyl character formulae.
| MATH 329 LECTURE AND ASSIGNMENT SCHEDULE | |||
|---|---|---|---|
| Date | |
|
Refs/HW |
| 1/16 T | [S]v-viii, 1-10 [FH] v-viii, 1-2 | Introduction to Rep. Thy. | Links: (1) Wiki:GpRep; (2) Hist:AbsGp |
| 1/18 R | [S] 1.2-3 | Matrix reps. & G-modules | . |
| 1/23 T | [S] 1.4-6 | Reducibility, Mashke's Thm, Schur's Lemma | . |
| 1/25 R | [S] 1.7-8 | Commutant and Endomorphism Algs, Characters | HW (due 2/8/07): [S] 1.13: #1,2,4,5,6,9,12,15,16,17 |
| 1/30 T | [S] 1.8-10 | Character Inner Products; Gp. Alg. Decomp | . |
| 2/1 R | |||
| 2/6 T | |||
| 2/8 R | |||
| 2/13 T | |||
| 2/15 R | HW (due 3/1/07): [S] 2.12: #4,5,8,10; [FH] #1.3-4,10,11 | ||
| 2/20 T | |||
| 2/22 R | |||
| 2/27 T | |||
| 3/1 R | |||
| 3-11 MARCH: SPRING BREAK (NO CLASSES) | |||
| 3/13 T | |||
| 3/15 R | |||
| 3/20 T | |||
| 3/22 R | |||
| 3/27 T | Schur Functions | ||
| 3/29 R | RSK Correspondence | HW3 | |
| 4/3 T | |||
| 4/5 R | |||
| 4/10 T | |||
| 4/12 R | |||
| 4/17 T | HW4 | ||
| 4/19 R | |||
| 4/24 T | |||
| 4/26 R | |||
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