After successful completion of the course, students are able to

• define the invariant ring of a representation of a group, and compute it in simple cases
• show that the invariant ring is finitely generated, provided that the representation is semisimple
• prove the first fundamental theorem of invariant theory for the general linear group
• show the equivalence of the multilinear formulation of the first fundamental theorem with the polynomial formulation using polarisation and restitution
• prove Schur-Weyl duality between the general linear group and the symmetric group

Let W be a (finite dimensional, complex) vector space.  A map f from W to the complex numbers is polynomial, if it is a polynomial in the coordinates with respect to any basis of W.

Let G be a group, eg. the special linear group SL(n) of n x n matrices with determinant 1, and let W be a G-module.  For example, G might act on the n-dimensional complex vector space by ordinary matrix multiplication.  A polynomial function f is invariant, if we have f(g w) = f(w) for all elements of the group g and all vectors w.

Following Hermann Weyl we will thus determine the invariant functions for a few classical groups.

Whiteboard presentation, interspersed with short exercises.

exam

Lineare Algebra