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

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.

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