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. 1. Introduction. In this note we shall discuss the relation between pairwise distances of orbit points of finite groups acting effectively and isometrically on compact Riemannian manifolds and their group structures. Our approach is to consider corresponding matrices of these pairwise distances and then relate the eigenspaces of these matrices to the regular representation of the groups. The motivation of these ideas traces back to the papers (Cl, 2), (R), (W2, 3). To get a sense of how the pairwise distances of the orbit points can reveal the structure of the group itself, let us consider the following two simple examples.