Representation Theory
A.Y. 2018/2019
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).
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
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 S-n .
10. Introduction to Lie algebras and their representations.
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 S-n .
10. Introduction to Lie algebras and their representations.
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
Lessons: 28 hours
Professors:
Bianchi Mariagrazia, Pacifici Emanuele
Teoria della rappresentazione mod/2
MAT/02 - ALGEBRA - University credits: 3
Lessons: 21 hours
Professor:
Pacifici Emanuele