REGULAR PREHOMOGENEOUS VECTOR SPACES FOR VALUED DYNKIN QUIVERS

We introduce regular prehomogeneous vector spaces associated with an arbitrary valued Dynkin graph (Gamma, v) having a fixed oriented modulation (M, Omega) over the ground field K. Here K is of characteristic zero, but it may not be algebraically closed. We will construct a fundamental theory of such prehomogeneous vector spaces. Each generic point of a regular prehomogeneous vector space corresponds to a hom-orthogonal partial tilting Lambda-module, where Lambda is the tensor K-algebra of (M, Omega). We count the number of isomorphism classes of hom-orthogonal partial tilting L-modules of type B-n, C-n, F-4 and G(2). As a consequence of our theorem, we estimate lower and upper bounds for the number of basic relative invariants of regular prehomogeneous vector spaces for any valued Dynkin quiver.