# Course Identification

## Lecturers and Teaching Assistants

## Course Schedule and Location

**Tutorials**

Tuesday, 10:15 - 12:00, Goldsmith, Rm 208

## Field of Study, Course Type and Credit Points

## Comments

## Prerequisites

Linear algebra. Other topics that can help are: representation theory of finite groups, basic functional analyses, basic topology and differential topology, Lie algebras. Each of these topics will be briefly reviewed if majority of the audience knows it, or taught slower otherwise.

## Restrictions

## Language of Instruction

## Attendance and participation

## Grade Type

## Grade Breakdown (in %)

## Evaluation Type

**Final assignment**

## Scheduled date 1

## Estimated Weekly Independent Workload (in hours)

## Syllabus

(1) Representations of a finite group G

(a) Groups, actions of groups on sets, natural constructions (depending on the audience).

(b) Basic definitions: representation of a group G, morphisms of representations.

(c) Irreducible representations. Schurs lemmas.

(d) Natural constructions with representations.

(e) Complete reducibility.

(f) Intertwining numbers and their properties.

(g) Decomposition of the regular representation.

(h) Group algebra and its structure.

(i) Burnside theorem and its corollaries.

(j) Characters, Orthogonality relations. Character rings.

(n) Representations of finite abelian groups. Fourier transform.

(k) Brauers theorem (optional)

(l)Equivariant sheaves

(l) Restriction and induction

(m) Mackey theory

(2) Some results about representations of topological groups.

(a) Representations of commutative groups and Fourier transform.

(b) Basic results about representations of the compact group G = SO(3).

(3) Representations of general compact groups

(a) Basic definitions and properties

(b) Peter-Weyl theorem

(4) Lie groups and Lie algebras (if time permits)

(5) Representations of compact Lie groups via representations of Lie algebras (if time permits)

(6) Representations of general Lie groups (if time permits)

(a) Lie groups and Lie algebras

(b) The space of smooth vectors, Garding theorem on density, Dixmier-Malliavin theorem, the

action of the Lie algebra

(c) Cocompact subgroups, smooth induction

(d) Representations of SL(2,C)

## Learning Outcomes

Upon successful completion of this course students should be able to:

- Use linear symmetries in order to reduce problems to simpler ones, and in order to introduce additional structure to infinite-dimensional vector spaces.
- Get a taste of what representation theory is about and will be able to read advanced textbooks and survey papers on the subject.

## Reading List

1. J.P. Serre: Linear representations of finite groups

2. Barry Simon: Representations of finite and compact groups

3. A.A. Kirillov: Elements of representation theory