# zbMATH — the first resource for mathematics

Regular prehomogeneous vector spaces for valued Dynkin quivers. (English) Zbl 1454.16017
Summary: We introduce regular prehomogeneous vector spaces associated with an arbitrary valued Dynkin graph $$(\Gamma, \boldsymbol{v})$$ having a fixed oriented modulation $$( \mathfrak{M}, \Omega)$$ 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 $$( \mathfrak{M}, \Omega)$$. We count the number of isomorphism classes of hom-orthogonal partial tilting $$\Lambda$$-modules of type $$\mathbf{B}_n, \mathbf{C}_n, \mathbf{F}_4$$ and $$\mathbf{G}_2$$. 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.
##### MSC:
 16G20 Representations of quivers and partially ordered sets 11S90 Prehomogeneous vector spaces
Full Text: