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- Homepage: http://www.math.uconn.edu/~troby
- Offices: MBS M404, CUE 123: phone: 860-486-8385

- Office hours: Thursday 11-12 in MSB 404 and by appointment. Feel free to catch me after class. I'm happy to answer questions anytime by email, which I check frequently.

* 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.

* 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.

- The fine Electronic Journal of Combinatorics also has has a directory of webpages of combinatorialists, survey articles, and more.
- A similar effort which includes the journal Annals of Combinatorics is run out of Nakai University by Bill Chen.

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