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A classification of multiplicity free actions. (English) Zbl 0869.14021

This paper is concerned with the classification of multiplicity-free linear actions of reductive complex algebraic groups on vector spaces. The action of a complex reductive algebraic group \(G\) on an affine variety \(V\) is called multiplicity-free if each irreducible representation of \(G\) occurs at most once in the ring of regular functions \(C[V]\). A complete classification of multiplicity-free actions where \(V\) is an irreducible \(G\)-module was given by V. Kac [J. Algebra 64, 190-213 (1980; Zbl 0431.17007)]. In this article, the authors give a classification where \(V\) is a \(G\)-module, but not irreducible. They find many examples where the action does not decompose into a direct product of irreducible actions.

MSC:

14L30 Group actions on varieties or schemes (quotients)

Citations:

Zbl 0431.17007
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