KIT - Department of Mathematics - Representation Theory of Finite Groups (Winter Semester 2016/17)

$Webrelaunch 2020$

Department of Mathematics

Home Page

Institutes

Institute of Algebra and Geometry

Research Group Number Theory and Algebraic Geometry

Darstellungstheorie

Darstellungstheorie

Representation Theory of Finite Groups (Winter Semester 2016/17)

Lecturer: PD Dr. Fabian Januszewski
Classes: Lecture (0153700), Problem class (0153710)
Weekly hours: 2+1

The (German) announcement of the lecture is available here. The class may be held in English upon request. For an English description of the contents, see below.

Schedule
Lecture:	Thursday 11:30-13:00	SR 2.58
Problem class:	Monday 14:00-15:30 (every 2nd week)	SR 2.58

Lecturers
Lecturer, Problem classes	PD Dr. Fabian Januszewski
	Office hours: jederzeit wenn ich da bin.
	Room Kollegiengebäude Mathematik (20.30)
	Email: fabian.januszewski@math.uni-paderborn.de

This lecture will be an introduction to the representation theory of finite groups.

A representation of a group $G$ is a group homomorphism
$\rho:\quad G\to{\rm{Aut}}_k(V)$
into the automorphism group of a $k$ -vector space $V$ . In order words a representation is a $k$ -linear action of $G$ on $V$ .

We may think of $\rho$ as a linearization of $G$ . From this perspective a representation allows us to use linear algebra on $V$ to understand $\rho(g)$ and thus the structure of $G$ . Representation theory makes arguments of linear algebra available in the context of group theory. On the other hand the structure of $G$ tells us a lot about $\rho$ , which has important applications in parctice.

Representation theory has many applications in almost all fields of mathematics and even in quantum physik.

For example the Fourier transform is a special case of the representation theory of locally compact abelian groups. Therefore representation theory may be seen as a non-abelian generalization of Fourier transform to arbitrary groups.

In the lecture we will introduce the representation theory of finite groups. We will learn how to classify all representations of such a group under suitable hypotheses on the pair $(G,k)$ (if $k$ is of characteristic $0$ this condition is satisfied for all finite groups $G$ ).

Afterwards we will makes this classification explicit in the case of the symmetric group $S_n$ . We will learn what Young Diagrams are, and study the Hook Length Formula to determine the dimensions of irreducibles.