# O.K.Sheinman

## Basic Representation Theory

Group symmetry is one of the most fundamental
geometrical properties of the real world. It suffices to say
that three of the four fundamental types of physical
interactions (namely, electromagnetic, weak, and strong)
are controlled by symmetries of the first three
special unitary groups: *U(1)*, *SU(2)*, and *SU(3)*,
respectively. The representation theory of these groups
and of some related structures will be considered in the
proposed course.

### Syllabus

1. Representations of finite groups:

- Schur lemma
- complete reducibility
- 1-st and 2-d Burnside theorems
- characters and equivalence

2.
Representations of the special unitary group:

- Peter-Weyl theorem
- Weyl formula for characters
- representations of
*SU(2)*
- the hydrogen atom
- quantum numbers and the periodic system of elements

3.
Representations of the Lie algebra *sl(2,C)*:

- finite-dimensional irreducible modules
- Casimir operator

4.
Additional subjectmatter:
- Representations of the simplest Kac-Moody
algebra $\widehat{sl}^(2,{\bf C})$
and combinatorial
analysis.

### Books:

1. W. Fulton, J. Harris. *Representation Theory. A first course*,
Springer-Verlag, 1991