Multiplicity free spaces with a one-dimensional quotient
Let \(V\) be a finite dimensional complex vector space and let \(\mathbb{C} \left[ V\right] \) be the algebra of polynomials on \(V.\) Let \(\left( G,V\right) \) be a finite dimensional rational representation of a connected reductive group \(G.\) Then \(G\) acts on \(\mathbb{C}\left[ V\right] \) by \(g.\varphi\left( x\right) =\varphi\left( g^{-1}x\right) ,\;g\in G,\,\varphi\in\mathbb{C}\left[ V\right] .\) If each irreducible representation of \(G\) occurs at most once in the representation of \(\left( G,\mathbb{C}\left[ V\right] \right) \) then \(\left( G,V\right) \) is said to be multiplicity free. Denoting the semisimple part of \(G\) by \(G^{\prime},\) a multiplicity free space \(\left( G,V\right) \) is said to have a one dimensional quotient if there is a homogeneous polynomial which is not in the ring of invariants \(\mathbb{C}\left[ V\right] ^{G}\) but generates \(\mathbb{C}\left[ V\right] ^{G^{\prime}}\). The term ``one dimensional quotient'' refers to the fact that the categorical quotient \(V/\!/G^{\prime}\) has dimension 1.NEWLINENEWLINEThis paper is a complete list of multiplicity free spaces with a one dimensional quotient, up to geometric equivalence, which are indecomposable and saturated. Here, \(\left( G,V\right) \) is ``saturated'' if the dimension of the center of the representation is equal to the number of irreducible summands of \(V;\) unsaturated representations become saturated by adding a torus. Starting the the classification of all multiplicity free spaces in [\textit{C. Benson} and \textit{G. Ratliff}, J. Algebra 181, No. 1, 152--186 (1996; Zbl 0869.14021)], a list of 21 different types of spaces with one dimensional quotient is given, along with the weighted Dynkin diagram (when applicable) and the homogeneous polynomial which generates \(\mathbb{C}\left[ V\right] ^{G^{\prime}}.\)NEWLINENEWLINEThis classification shows that, with one exception (namely \(\text{SL} \left( 3\right) \times \mathrm{Sp}\left( n\right) \times\mathbb{C}^{\ast}\)), an irreducible multiplicity-free space \(\left( G,V\right) \) will not have a one dimensional quotient if and only if \(\mathbb{C}\left[ V\right] ^{G^{\prime} }=\mathbb{C}\).
scientific article; zbMATH DE number 6199361