Quivers, Geometric Invariant Theory, and Moduli of Linear Dynamical Systems.pdfVIP

QUIVERS, GEOMETRIC INVARIANT THEORY, AND MODULI OF LINEAR DYNAMICAL SYSTEMS MARKUS BADER Abstract. We use geometric invariant theory and the language of quivers to study compactifications of moduli spaces of linear dynamical systems. A general approach to this problem is presented and applied to two well known cases: We show how both Lomadze’s and Helmke’s compactification arises naturally as a geometric invariant theory quotient. Both moduli spaces are proven to be smooth projective manifolds. Furthermore, a description of Lomadze’s compactification as a Quot scheme is 7 given, whereas Helmke’s compactification is shown to be an algebraic Grassmann bundle over a Quot 0 scheme. This gives an algebro-geometric description of both compactifications. As an application, we 0 determine the cohomology ring of Helmke’s compactification and prove that the two compactifications 2 are not isomorphic when the number of outputs is positive. c e D 0. Introduction 4 In this article, we study actions of products of general linear groups on spaces of matrices. We ] present general techniques from algebraic geometry that we apply to two concrete examples, namely to G two different compactifications of the moduli space of controllable linear dynamical systems. The first A section introduces geometric invariant theory and representation theory of quivers in a tutorial way. h.