E 23

This talk has two parts. In the first part, we describe the menagerie of indecomposable representations over a field of prime characteristic, with a special focus on orthogonal representations of algebraic groups, i.e., those quadratic forms that are preserved by the group. In the second part, we focus more generally on homogenous polynomials that are preserved by the group. The classical ``linear preserver problem`` starts with a polynomial and asks: what is the group of linear transformations that preserve it? This problem has been solved for many interesting polynomials, usually by means that are special to the particular polynomial under consideration. We turn this problem on its head by starting with a polynomial that is preserved by a simple algebraic group and observe that the full preserver can be described by a general theorem. The results are new even in the case where the field is the complex numbers, and as an application we shed some light on a 125+ year old problem. This is joint work with Bob Guralnick.

Prof. Dr. Skip Garibaldi

Emory University, Atlanta, USA