# Seminar/Proseminar: Representation Theory of Finite Groups

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## Generalities

Organizer: Lars Kindler
Date: Thursdays, 4-6pm, Place: SR 210/A3, Arnimallee 3

Official website in course registry

or come to office 109, Arnimallee 3

Prerequisites: Linear Algebra and some familiarity with basic notions of algebra, in particular group theory and some ring theory.

Guidelines: This is a seminar, which means the participants give the talks. Please write to me if you would like to reserve one of the talks for you, or if you have any questions.

First meeting: Thursday, April 20

## Description

A representation of a group $G$ is, informally, a collection of invertible linear transformations of some vector space over a field (or, more generally, of some module over a ring), which multiply according to the same multiplication table as $G$. In other words, it is the choice a set of symmetries of a vector space, which reflect the group structure of $G$.

As symmetry is one of the fundamental concepts of mathematics, representations of groups arise in many different areas of mathematics: number theory, topology, combinatorics, differential geometry, algebraic geometry, ....

In this seminar we will learn the foundations of the theory of representations of finite groups. After defining the basic notions, we will first study representations of groups on vector spaces over fields of characteristic $0$. This leads to a beautiful and mostly complete theory, which is already very useful.

## Talks

DateTopicSpeaker
20.04.2017Algebraic foundationsLars
27.04.2017Definitions and first propertiesJakub
04.05.2017Semi-simple representations and semi-simple algebrasLukas
11.05.2017The structure of semi-simple algebrasJoaquim
18.05.2017The character table of a finite groupCrystal
01.06.2017Orthogonality relations of charactersLouis
08.06.2017Consequences of the orthogonality relationsSimon
15.06.2017Algebraic integers and the degree of simple representationsElisabeth
22.06.2017Burnside's $p^aq^b$-theoremLars
29.06.2017Characters of cyclic groups and direct productsYusuf
06.07.2017Inflation, induction and restriction
13.07.2017Frobenius reciprocity
20.07.2017t.b.a.