Representations of Groups
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INSTRUCTOR: Charles Conley, GAB 419, (940) 565-3326
OFFICE HOURS: MW 11:00-12:30, T 11-12
CLASS MEETS: MW 12:30-1:50, GAB 438
GRADING: Your grade will be based on weekly problem sets; there will be no exams.
PREREQUISITES: The most important prerequisites are linear algebra (at the level of 4450) and abstract group theory (at the level of 5520). It will help if you know some manifold theory (5450), but vector calculus (3740) will be sufficient.
TEXT: The text is Representation Theory, a First Course, by Fulton and Harris. Another good reference is Representations of Compact Lie Groups, by Brocker and tom Dieck.
TOPICS: The first part of the course will be an introduction to representations of finite groups. We will begin with representations of the finite cyclic groups, including a discussion of finite Fourier transforms. Then we will consider representations of various small finite groups, such as the Klein four group, S_n, A_n, and D_n for n = 3, 4, 5, and the quaternionic group. In this setting we will meet unitary representations, irreducible representations, Schur's lemma, characters, orthogonality relations, dual representations, and tensor products. One of the central results of this part of the course is that the irreducible representations and the conjugacy classes of finite groups are in bijection (but not in any natural way), and the sum of the squares of the dimensions of the irreducible representations is the order of the group. We may also treat induction and Tanaka-Krein duality, a way to reconstruct the group from its representations.
The second part of the course will be an introduction to Lie groups and their representations. We will work with groups of matrices, focusing on low dimensional examples such as the circle, the torus, SO_3, SU_2, and maybe O_3, U_2, and SU_3. In this setting we will see again the ingredients of the first part of the course, along with some new ones: the exponential map, invariant integration, compactness, Lie algebras, and the Peter-Weyl theorem. If time permits we will cover the Clebsch-Gordan rules and the Weyl character formula.