6510.001, Fall 2024
Representation Theory
Conley

The course will meet Mondays and Wednesdays, 12:00-1:20, in GAB 473. It will count towards the algebra breadth requirement. The format will be mainly lectures. Grading will be based on attendance and some combination of in-class presentations and maybe a small number of homework problems. It is necessary for me to note that use of screens in class will not permitted for any purpose other than note-taking.

We will loosely follow “Representation Theory: A First Course”, by Fulton and Harris. My tentative plan is to begin with Chapters 1-3, which introduce representations of finite groups, including the orthogonality relations and induction. We will then jump to Part II, on representations of Lie groups and Lie algebras. Rather than develop the abstract theory via nilpotent and solvable Lie algebras, the Killing form, and the Cartan criterion, we will go immediately to Chapters 11-13, which introduce finite dimensional complex semisimple Lie algebras by looking in detail at sl(2) [aka o(3)] and sl(3). We may also look at Section 16.2, on sp(4) [aka o(5)]. In the context of these examples, we will discuss the following topics:

• Cartan subalgebras and roots

• The Weyl group

• Weights and highest weights

• The classification of finite dimensional irreducible representations

Time permitting, we may also discuss the following topics:

• Semisimplicity of finite dimensional representations

• The Weyl character formula

• The Kostant multiplicity formula

• Special cases of the Littlewood-Richardson rule

The agenda is flexible; let me know if you have preferences. Here are some additional possibilities:

• Verma modules

• Infinitesimal characters

• A taste of injectives and projectives via sl(2)

• A taste of quantum groups via U_q[sl(2)]

• A taste of Kac-Moody algebras via affine sl(2)

• A taste of Lie superalgebras via osp(1|2)

• A taste of finite fields via SL(2, q)

• A taste of compact groups via SU(2): orthogonality relations again, the Weyl character formula again, and the Peter-Weyl theorem