Basic Representation Theory

Representation Theory studies how a given group may act on vector spaces. It is a fundamental tool to study groups using linear algebra.

Representation Theory plays an important role in many recent developments of mathematics and theoretical physics. The course aims to introduce basic concepts and results of the classical theory of complex representations of finite groups and simplest examples of representations of Lie groups and Lie algebras.

Prerequisites: Linear Algebra, Elementary Finite Groups Theory.


Curriculum:

  1. Linear representations of groups. Definitions and examples. Irreducible representations. Schur’s Lemma. Complete reducibility.
  2. Characters of representations. Number of irreducible characters. Character tables and orthogonality relations. Group algebra.
  3. First examples: abelian groups, dihedral group D_n , groups S_3 , S_4 , A_4.
  4. Representations of symmetric groups, Young diagrams.
  5. Examples of Lie groups and Lie algebras. Covering of SO(3,R) by SU(2).
  6. Compact groups and their representations. Peter-Weyl theorem.
  7. Representations of Lie algebra sl(2,C). Clebsch-Gordan decomposition.
  8. Connection between representations of Lie groups and Lie algebras.

Textbooks

  • W. Fulton, J. Harris, Representation theory. A first course., Berlin: Springer, 1991. (Grad. Texts Math., v. 129)
  • E.B. Vinberg, Linear Representations of Groups, 1989.