## Welcome to Tom Roby's Math 329 homepage! (Spring 2007)

(last updated: 21 February 2007)

### Class Information

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.

(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.

GRADING: Your grade will be based on a takehome midterm exam, homework, and a project. The breakdown of points is:

Homework Midterm Project
50% 25% 25%

CONTENT: This course is an introduction to the representation theory of finite groups and Lie algebras. Representation theory studies the way in which a group may act on a vector space, which means that each element of the group is represented as a matrix. These matrix representations can be built up from irreducible ones, which form the building blocks for the construction of all representations of a given group. Understanding the representations of a group gives us new insight into the group itself, as well as objects on which the group acts.

Representation theory has applications all over mathematics, from algebra and number theory, to analysis, geometry, combinatorics, and mathematical physics. 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.

LEARNING GOALS: Besides becoming fluent in some basics of this area and conversant with some deeper aspects, I expect that students will learn or improve their ability:

• to read mathematics independently from different books and research papers;
• to navigate the research literature using such tools as MathSciNet and the ArXiv;
• to write mathematics clearly; and
• to present mathematics to others lucidly.

DISABILITIES If you have a documented disability and wish to discuss academic accommodations, or if you would need assistance in the event of an emergency, please contact me as soon as possible.

HOMEWORK: Homework will be assigned occasionally. I will grade it on exposition as well as correctness.