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On the Popov-Pommerening conjecture for groups of type \(A_ n\). (English) Zbl 0682.14005

The Popov-Pommerening conjecture claims that, given a reductive algebraic group G over an algebraically closed field k and a closed subgroup H of G which is normalized by a maximal torus of G, then for any affine algebraic G-variety X the algebra \(k[X]^ H\) of H-invariant regular functions on X is finitely generated. It is proved in the paper under review that the Popov-Pommerening conjecture is true if G is a reductive group of type \(A_ n\), \(n\leq 4\). It is also shown that there exists a Grosshans subgroup H of \(GL_ 5(k)\) such that the invariant subalgebra \(A^ H\) (where A is a “letter place algebra”) is not spanned by the invariant standard bitableaux.
Reviewer: V.L.Popov

MSC:

14L24 Geometric invariant theory
14L30 Group actions on varieties or schemes (quotients)
14L35 Classical groups (algebro-geometric aspects)
15A72 Vector and tensor algebra, theory of invariants
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