Representation theory

A.Y. 2020/2021
9
Max ECTS
73
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
- Both lectures and exercise classes will be delivered through the application Zoom, and there will be the possibility to recover them at any time because they will be recorded and posted on ARIEL.

- The program and the material for the course will not change.

- The oral part of the exam will take place through the Zoom application until it will be necessary to do so (whereas the written part requires no modification, because it consists of a homework); there are no substantial changes on the examination methods, only logistical issues that will be communicated in due course via the ARIEL platform.
Prerequisites for admission
Basics of Algebra, in particular of Group Theory.
Assessment methods and Criteria
The exam is divided into a 6-credit part (related to group representations) and a 3-credit part (optional, related to Lie algebras). There is no temporal or order restriction between the performance of the two parts.

1) For the 6-credit part, the exam consists of a written test and an oral test.

- In the written test, some open-answer exercises will be assigned, aimed at verifying the ability to solve problems related to the course topics. The written test is in the form of a homework: the student receives a text containing three or four exercises, to be carried out within about seven days and then returned to the teachers. The result of the written test will be communicated at the beginning of the oral test.

- During the oral exam the student's written work will be discussed, and the student will be asked to illustrate some results of the program of the course, in order to evaluate the knowledge and understanding of the topics covered, as well as the ability to know how to apply them.

The exam is passed if the written test and the oral test are passed. The result will be communicated immediately at the end of the oral exam.

2) For the 3-credit part, the exam consists in the preparation and presentation (in front of anyone interested) of a seminar on a topic agreed with the teachers, which starts from the program carried out during the course but reaches a deeper level of detail.

The result will be communicated immediately at the end of the seminar.


For those who intend to take both parts of the exam, the final result is communicated immediately after the last part, and is determined by the weighted average of the results achieved in the two parts.
Teoria della rappresentazione (prima parte)
Course syllabus
1. Definitions and examples. Irreducible and completely reducible representations of finite groups.
2. Representations and modules. Simple and semisimple modules.
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.
Teaching methods
Traditional lectures.
Teaching Resources
C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras", Interscience Publ. New York (1962).

I.M. Isaacs, "Character Theory of finite groups", Academic Press (1976).

M.P. Malliavin, "Les groupes finis et leurs représentations complexes", Volume 1. Masson, 1981
Teoria della rappresentazione mod/2
Course syllabus
Smooth manifolds, tangent space and tangent bundle of a smooth manifold, smooth vector fields and associated Lie algebra. Lie groups and associated Lie algebras: left invariant smooth vector fields. Functor between the category of Lie groups and that of Lie algebras.
Examples of Lie algebras. Adjoint representation. Ideals. Solvable, nilpotent and semisimple algebras. Theorems by Engel and Lie.
Killing form and characterization of semisimple algebras. Modules for Lie algebras and Weyl's Theorem on complete reducibility of modules for a semisimple Lie algebra. Modules of sl(2,C). Toral subalgebras and Cartan decomposition; roots systems.
Teaching methods
Traditional lectures.
Teaching Resources
J.E. Humphreys, "Introduction to Lie Algebras and Representation Theory", Springer (1972).

W. Fulton, J. Harris, "Representation Theory: A First Course", Springer (1991).

J.M. Lee, "Introduction to Smooth Manifolds", Springer (2012).
Teoria della rappresentazione (prima parte)
MAT/02 - ALGEBRA - University credits: 6
Practicals: 24 hours
Lessons: 28 hours
Professors: Bianchi Mariagrazia, Pacifici Emanuele
Teoria della rappresentazione mod/2
MAT/02 - ALGEBRA - University credits: 3
Lessons: 21 hours