A Classification of Multiplicity Free Actions Original Research Article

The action of a complex algebraic groupGon an affine varietyVis said to be multiplicity free if each irreducible representation ofGoccurs at most once in the ring image[V] of regular functions onV. This paper concerns the classical setting whereVis a vector space andGacts linearly onV. The irreducible linear multiplicity free actions have been classified by V. Kac. The focus here is on nonirreducible linear multiplicity free actions. We exhibit many such actions which do not decompose as direct products of irreducible actions. Our main results provide a classification for all linear multiplicity free actions on a vector space.