Simultaneous similarity of matrices

The main aim of the representation theory is the description of modules over a given algebra. The representation of an algebra is uniquely determined by the images of the generators of the algebra, therefore it may be identified with a tuple of matrices satisfying certain properties. Two representations are equivalent if and only if the corresponding tuples of matrices are simultaneously similar. This transfers the original representation-theoretic problem into a linear-algebraic one. The purpose of the proposed project is the investigation of simultaneous similarity on tuples of matrices. We are going to investigate two aspects of this group action, connected to invariant theory and to algebraic geometry. On the invariant-theoretic side we are going to provide invariants that fully characterize orbits of the action under investigation. This will significantly improve the existing results on invariants, which can be used only for verification if the closures of two orbits intersect. On the algebro-geometric side we are going to characterize the irreducible varieties of some varieties of modules, in particular od those that are the largest in some sense. We are also going to provide new classes of algebras with irreducible varieties of modules, which includes answering some explicit open problems. Our research will significantly contribute to the fields of linear algebra, representation theory and invariant theory. It is also expected to have an impact on some neighbouring areas, such as algebraic geometry, multilinear algebra or functional analysis.