Credits: 4 (3 lectures and 1 tutorial weekly)
Prerequisites: MAT 634 (Differential Geometry) and MAT 640 Graduate Algebra I
Overview: Representations of groups realize the group elements as linear transformations on vector spaces, or even more concretely, as matrices. This enables the use of linear algebra to study algebra, and connects group theory with other areas such as geometry, harmonic analysis and number theory. In this course, we will first study the representation theory of finite groups, and then that of compact groups.
Part 1: Representations of Finite Groups
Review of group actions, representations, unitarizability and unitary equivalence of finite dimensional representations of finite groups, complete reducibility, group algebra as a *-algebra, regular representations, matrix coefficients, Schur’s lemmas, tensor products of representations, orthogonality of matrix coefficients, orthogonality of characters, direct sum decompositions, projection formulas, dimension theorem, character tables, Frobenius-Schur theorem on real and quaternionic representations, Fourier analysis on finite groups, subgroups of index 2, induced representations, Frobenius character formula, Frobenius reciprocity, Mackey irreducibility criterion.
Part 2: Representations of Compact Groups
Review of manifolds and Lie groups, the classical compact groups, topological properties of G and G/H, invariant forms and integration, Haar measure, examples of Haar measure for matrix groups, matrix coefficients, characters, Schur orthogonality, review of spectral theory, Schur’s lemma, regular representations, Frobenius reciprocity, Peter-Weyl theorem, representations and harmonic analysis of SU(2), Fourier theory.
- Representation Theory of Finite Groups by Benjamin Steinberg, Springer.
- Representations of Finite and Compact Groups by Barry Simon, Graduate Studies in Mathematics, American Mathematical Society.
- Representation Theory – A First Course by William Fulton and Joe Harris, Springer.
- A First Course on Representation Theory and Linear Lie Groups by S C Bagchi, S Madan, A Sitaram, and U B Tewari, Universities Press.
- Compact Lie Groups by Mark R Sepanski, Springer.
- Short Courses in Mathematics by S Kumaresan, Universities Press.