Characterization of bilinear spaces with unimodular isometry group

We study finite-dimensional bilinear spaces and their isometry groups. To each bilinear space we associate two canonical filtrations, which yield structural results on . Prominent among these is an explicit formula for the number of indecomposable and degenerate blocks of of a given dimension. Equipped with this material, we proceed to characterize those bilinear spaces whose isometry group is contained in the special linear group. This characterization can easily be implemented in practice by means of an algorithm. As an application, we determine the real -by- matrices whose congruence class is disconnected.