Separating invariants of classical groups

Abstract

The proposed project is dedicated to some problems in Invariant Theory. Assume that the base field F is infinite of an arbitrary characteristic and G is a linear classical group, i.e. it belongs to the following list of groups: GL(n), O(n), Sp(n), SL(n), SO(n), where the characteristic of F is different from two in case G is O(n) or SO(n). Consider the action of of this group over the direct sum H of several copies of the space of matrices n x n. The set of all polynomial functions between H and F, which are constants over the orbits of the action of G over H, is called the algebra of G-invariants of matrices. Generators of this algebra are known for all groups with the exception of G=O(n) or SO(n) in case of the field of characteristic two. The notion of separating invariants was introduced by Harm Derksen e Gregor Kemper in 2002 as a simplification of the notion of generators of the algebra of invariants. Working o the proposed project, we will obtain the minimal exact degree of elements of the separating sets of invariants of matrices for groups GL(n), O(n), Sp(n) and of invariants of bilinear forms in case n is less than 5. We will construct some indecomposable invariants of the representations of the generalized quiver and investigate its properties. We will describe the generalized quivers with polynomial algebras of invariants.