Some prehomogeneous representations defined by cubic forms
There are four fundamental irreducible prehomogeneous vector spaces with relative invariants of degree 4 in the classification table of Sato-Kimura of reduced prehomogeneous vector spaces. In this paper, the author constructs ``\(k\)-forms'' of these prehomogeneous vector spaces and determines the generic stabilizer where \(k\) is an arbitrary commutative ring. Instead of considering in the framework of classical theory of prehomogeneous vector space, he starts with the axiomatic construction of the linear representation of an algebraic group \(G(k)\) defined over the commutative ring \(k\). Then he determines the stabilizer subgroup of a generic point belonging to an open subscheme defined by a quartic form. Under some conditions, by ``extending'' to the algebraic closed field \(K\) from the commutative ring \(k\), it is proved that the group \(G(K)\) acts on the open subscheme transitively.
scientific article; zbMATH DE number 1268107