MATH3371/4241 Representation Theory III/IV
The central topic in the course are group representations, which are, quite literally, representations of groups as groups of matrices. Representation theory is one of the central areas in mathematics with many applications within pure mathematics, say in number theory, but also in physics (e.g. quantum mechanics) and chemistry (e.g. crystallography).
In the first term, we will cover the representation theory of finite groups. This topic is particularly neat and enjoys a very satisfying calculus of so-called group characters. This provides a striking solution to the problem of determining all the representations of a finite group.
The other main topic of investigation is the representation theory of some classical matrix groups (linear Lie groups) such as SL₂(ℝ), the 2 by 2 matrices of determinant 1, and SO(3), the group of rotations in 3 dimensions. To study those we will associate to these groups their so-called Lie algebras. These are vector spaces and their representation theory can be studied solely by algebraic means. Via the exponential map one can then obtain representations for the matrix groups.
Outline of Course
Aim: To develop and illustrate the representation theory and that of complex characters of finite groups and classical Lie groups and Lie algebras via highest weight theory.
- Representation theory of finite groups: Basic notions, Schur's Lemma, unitarizable representations, Maschke's theorem, tensor products.
- Character theory and tables: Revise conjugacy classes. Character of a representation and its properties, irreducible characters. Orthogonality of irreducible characters. Number and degree of irreducible characters. Construction of the character table of a finite group, orthogonality properties. Plancherel formula, Fourier inversion.
- Induced representations: Construction and basic properties. Frobenius reciprocity. Basics of the Mackey machine.
- Modules over the group algebra.
- Representations of Abelian groups, dihedral groups, quaternion group, S₃, S₄, S₅, A₄, A₅, finite Heisenberg group (throughout the term).
- Linear Lie groups and their Lie algebras, exponential map.
- Lie group and Lie algebra representations. (Finite)-dimensional representations of ℝ and S¹. Relationship with Fourier series.
- Finite-dimensional representations of SL₂(ℂ) and its Lie algebra 𝔰𝔩₂(ℂ) via highest weight theory. Representations of GL₂(ℂ).
- Representations of SU(2) and SO(3). Spherical harmonics.
- Finite-dimensional representations of SL₃(ℂ) and its Lie algebra 𝔰𝔩₃(ℂ). Outlook to SLₙ(ℂ) and 𝔰𝔩ₙ(ℂ).
- Time permitting: Representation theory of the Heisenberg group and Lie algebra.