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On classical invariant theory and binary cubics

Gerald W. Schwarz — 1987

Annales de l'institut Fourier

Let G be a reductive complex algebraic group, and let C [ m V ] G denote the algebra of invariant polynomial functions on the direct sum of m copies of the representations space V of G . There is a smallest integer n = n ( V ) such that generators and relations of C [ m V ] G can be obtained from those of C [ n V ] G by polarization and restitution for all m > n . We bound and the degrees of generators and relations of C [ n V ] G , extending results of Vust. We apply our techniques to compute the invariant theory of binary cubics.

Linear maps preserving orbits

Gerald W. Schwarz — 2012

Annales de l’institut Fourier

Let H GL ( V ) be a connected complex reductive group where V is a finite-dimensional complex vector space. Let v V and let G = { g GL ( V ) g H v = H v } . Following Raïs we say that the orbit H v is if the identity component of G is H . If H is semisimple, we say that H v is for H if the identity component of G is an extension of H by a torus. We classify the H -orbits which are not (semi)-characteristic in many cases.

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