A matrix theory for finite group actions

We introduce a new matrix theory to investigate finite group actions on spaces. Given a finite group action, we associate it with a family of orbit matrices. The spectral radius of an action is also introduced. It is shown that the spectral raduis is bounded below by a constant depending only on some geometric invariants of the underlying Riemannian manifolds. The relation between the eigenspaces of orbit matrices and regular representations of finite groups are also investigated. In particular, we obtain that the eigenvalues of orbit matrices reveal some structures of the groups.