**Math 506, Representation Theory, Fall 2016**

TuTh 9:30-10:50a.m., 141 Altgeld Hall

**My information****:**

Thomas Nevins

357 Altgeld Hall

217.265.6762

nevins AT illinois DOT edu

**Office hours****:** Tuesdays 4-4:50pm, Wednesdays 1-1:50pm.

**Problem session****:** Tuesdays, 4-4:50pm, 2 Illini Hall.

**Suggested Texts:** Alperin and Bell, Groups and Representations (Springer); Serre, Linear Representations of Finite Groups (also Springer). Both are downloadable via the UIUC Library.

**class summaries**

Representation theory is part of the algebraic study of symmetry. What sets it apart is its focus on the specific study of representations: usually, vector spaces on which symmetries act. Representations are ubiquitous in the study of symmetry, since symmetries of physical systems or geometric objects yield representations on spaces of states or fields (in physics) or functions (in geometry and analysis).

The course will constitute an introduction to representation theory.

Much of the course will be occupied with the study of representations of finite groups (in the “non-modular” situation, i.e. where the vector spaces live over fields of nonzero or at least sufficiently large characteristic to simplify life).

The representation theory of finite groups is easy enough to avoid most technical complications and have good answers to many fundamental questions, but subtle enough that it will allow us to illustrate many basic phenomena that pervade representation theory of all kinds.

Later in the course, we will choose, depending on the interests of the class and as time permits, a more advanced topic that allows us to appreciate some of the complexity and difficulty of representation theory. Possibilities might include the representation theory of compact Lie groups, or noncompact real groups; representations of finite groups of Lie type, induction and restriction, and the Harish-Chandra philosophy; the Deligne-Lusztig construction of characters of finite groups of Lie type; or the representation theory of complex semi simple Lie algebras and its relation to the geometry of flag varieties.

While grading of the course will be relaxed in attitude, there will be regular assigned homework to offer students an opportunity to test their mastery. [Students who wish to have more information before deciding whether to register are invited to contact Professor Nevins.]

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