Symmetric Semialgebraic Sets and Orbit Spaces

Abstract

The aim of this project is to explore the link between real-valued polynomials which are invariant under the action of a Lie group, and the orbit space of a real representation of that Lie group. This is an application of real semialgebraic geometry - the study of regions in Euclidean space de ned by polynomial equations and inequalities. Given a real representation space V of a nite group or compact Lie group G there is an orbit space V=G. In the mid 1980s, Claudio Processi and Gerald Schwarz [14][15] discovered a method for explicitly characterising the orbit space V=G as a semialgebraic set in real Euclidean space. These methods involve the application of realvalued polynomials on V which are invariant under the action of G on V . The key theorem of Procesi and Schwarz will be introduced and proven. Their methods will then be applied to S22 , the space of pairs of real symmetric 2 2 matrices, to obtain a semialgebraic description of S22=O(2), where O(2) is the orthogonal group of order 2 acting on S22 . Two di erent ways will be shown for describing the unit ball in R2 as a spectrahedron, a solution of a linear matrix inequality. Each of these spectrahedra will be shown to be O(2) invariant, and therefore to possess an orbit space with respect to O(2). It will also be shown that the two spectrahedra are no longer the same when de ned over S22 instead of R2. Instead of a unit ball they describe more complicated shapes. The methods of Procesi and Schwarz will then be used to identify the orbit spaces of these two spectrahedra. Some observations will be made about their geometry and the way in which they are related. It will then be shown under what conditions the two spectrahedra coincide, and how the unit ball in R2 may be recovered by a suitable embedding of R2 into S22 .