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Representation Theory of Finite Groups (Summer Semester 2015)

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.

The time and date of the excercise class is negotiable.

Schedule
Lecture: Wednesday 11:30-13:00 SR 3.60 Begin: 15.4.2015
Problem class: Friday 14:00-15:30 SR 2.59 Begin: 24.4.2015

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.