ABSTRACT: We prove that if the symmetric algebra of a finite dimensional representation of a reductive group over a field of positive characteristic admits good filtrations (dual Weyl module filtrations), then the ring of invariants with respect to this group action is F-regular, in particular, Cohen - Macaulay. Utilizing a theorem of Andersen - Jantzen, we get the following.For any finite quiver, the coordinate ring of the representation space of a given dimension vector satisfies the assumption of the theorem above, where the reductive group in consideration is the direct product of appropriate GL, SL, Sp, or the trivial groups, corresponding to the vertices. If the characteristic is not two, then SO is also allowed. Determinantal rings are typical examples of the rings of invariants obtained in such a way.
 
 

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