Syllabus
1) Linear algebraic groups, classical groups and their Lie algebras, real forms of complex algebraic groups.
2) Regular and locally regular representations.
3) Classification of irreducible representations of classical groups using highest weights, complete reducibility.
4) Fourier analysis on the group algebra of a finite group, classification of irreducible representations of the group of permutations.
5) Representations of associative algebras, double commutant theorm.
6) Invariants for GL, SO and Sp groups.
Annotation
Invariants of classical groups will be studied, mainly those induced by actions on vector spaces. Invariants are traces of endomorphisms or determinants.
Hereby we mean that these objects are invariants with respect to the adjoint action of the appropriate general linear group. In the application of the theory, we focus to the case of polynomial invariants of binary n-ics, i.e., invariant polynomials defined on the space of polynomials P^n(C^2).
The discriminant of a binary quadric is an example of such an invariant for n = 2.