The course presents the fundamental concepts of (mainly complex) Representation Theory of finite groups. The concept of a character of a finite group is introduced, and some important links between the group structure and some features of the irreducible characters of the group are analyzed. Then the concept of a Lie algebra is introduced, and some basic facts of Representation Theory are presented in this context as well.
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).
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).
Learning Objectives
The aim of the course is to present the basic ideas of Representation Theory of finite groups and of Lie algebras.
Prerequisites
Basics of Algebra, in particular of Group Theory and Linear Algebra.
Teaching Methods
Traditional lectures.
Type of Assessment
During the course, several exercises will be worked out concerning the relevant topics, together with the students. The final exam is an oral exam, in which exercises (similar to those presented during the lectures) will be discussed, together with theoretical aspects of the subject.
Course program
1. Modules and representations of finite groups: definitions and examples. Irreducible and completely reducible modules and representations of finite groups. The group algebra and Maschke's Theorem.
2. Complex characters of finite groups: basic definitions and properties. Irreducible characters, orthogonality relations, linear characters. Character tables: examples. Applications of Character Theory: Burnside's Theorem and Frobenius' Theorem.
3. Tensor product of modules, product of representations and characters.
4. Induced representations and characters. Representations of normal subgroups and Clifford's Theory.
5. Complex representations of symmetric groups: partitions and Young diagrams, Specht's modules, computation of the irreducible characters of symmetric groups.
6. 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.
7. Examples of Lie algebras. Adjoint representation. Ideals. Solvable, nilpotent and semisimple algebras. Theorems by Engel and Lie.
8. 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.