Learning objectives
The students will learn the basic definitions, problems, and techniques in Representation Theory. At the end of the course, they will be able to solve basic exercises in the context finite groups and finite-dimensional algebras.
Prerequisites
Linear Algebra: canonical Jordan form, diagonalization and triangularization, bilinear, sesquilinear, symmetric, Hermitian, and quadratic forms.
Course unit content
The course is an introduction to Representation Theory. In the first part, we will discuss the basics of the theory and introduce the fundamental problems. In the second part, we will study in details the representation theory of finite groups, in particular symmetric groups, with their character theory. In the third part, we will discuss the representation theory of quivers and of finite–dimensional algebras, relying on basic tools from homological algebra.
Full programme
Language: categories and functors.
Basic notions in representation theory. Associative algebras and representations.
Basic notions in Lie algebras. Representation theory of sl(2).
Representations of matrix algebras.
Representations of finite groups and Maschke’s Theorem.
Representations of quivers and Gabriel’s Theorem.
Introduction to homological algebras.
Representations of finite-dimensional algebras.
Bibliography
[E] P. Etingof et al., Introduction to representation theory.
[FH] W. Fulton, J. Harris, Representation Theory. A first course.
Both books are freely available online.
Teaching methods
The topics of the course will be discussed during the lectures, together with examples, applications, and exercises.
Assessment methods and criteria
Every forth-night one lecture will be focused on exercises, both computational and theoretical. At the end of the course there will be a written exam. A passing grade will give access to a subsequent oral exam, consisting in an interview at the board, during which the student will be asked to discuss, explain, and prove the main results of the course.
Other information
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