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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.
MSC:
16G20 Representations of quivers and partially ordered sets
11S90 Prehomogeneous vector spaces
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