On configurations of subspaces of a Hilbert space with fixed angles between them

We investigate the set of irreducible configurations of subspaces of a Hilbert space for which the angle between every two subspaces is fixed. This is the problem of *-representations of certain algebras generated by idempotents and depending on parameters (on the set of angles). We separate the class of problems of finite and tame representation type. For these problems, we indicate conditions on angles under which the configurations of subspaces exist and describe all irreducible representations.