Representation theory

A.Y. 2016/2017
9
Max ECTS
69
Overall hours
SSD
MAT/02
Language
Italian
Learning objectives
The aim of the course is to present the basic Ideas of Representation Theory for finite groups (in the 6-credits part) and of Lie algebras (In the advanced 3-credits part).
Expected learning outcomes
Knowledge of the basic ideas of Representation Theory for finite groups (in the 6-credits part) and of Lie algebras (In the advanced 3-credits part).
Course syllabus and organization

Single session

Lesson period
First semester
Teoria della rappresentazione (prima parte)
Course syllabus
In the course (6 credits) we present basic ideas of representation theory for finite groups.

1. Definitions and examples. Irreducible, reducible and completely reducible representations of finite groups.
2. Representations and modules. Simple and semisimple modules: characterizations.
3. Applications to the group algebra. Maschke's Theorem.
4. Characters of finite groups Basic definitions and properties, irreducible characters, orthogonality relations, linear characters.
5. Character tables. Examples.
6. Applications of Character Theory. Solubility criteria, Burnside's Theorem, existence of normal subgroups and how to determine them.
7. Product of representations.
8. Induced representations and characters. Frobenius' Theorem.
9. Representations of symmetric groups. Partitions and Young tableaux, degrees of the irreducible representations of
Teoria della rappresentazione mod/2
Course syllabus
Introduction to Lie algebras and their representations.
Teoria della rappresentazione (prima parte)
MAT/02 - ALGEBRA - University credits: 6
Practicals: 20 hours
Lessons: 28 hours
Professors: Bianchi Mariagrazia, Pacifici Emanuele
Teoria della rappresentazione mod/2
MAT/02 - ALGEBRA - University credits: 3
Lessons: 21 hours
Professors: Bianchi Mariagrazia, Pacifici Emanuele