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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.
16G20 Representations of quivers and partially ordered sets
11S90 Prehomogeneous vector spaces
Full Text: DOI Euclid