manuscripta math. 82, 161 - 170 (1994) manuscri~ta mathemat~ca Springer-Verlag 1994 When is a Ring of Torus Invariants a Polynomial Ring? David L. Wehlau 1 Let p : T --* GL(V) be a finite dimensional rational representation of a torus over an algebraically closed field k. We give necessary and sufficient conditions on the arrangement of the weights of V within the character lattice of T for the ring of invariants, k[V] T, to have a homogeneous system of parameters con- sisting of monomials (Theorem 4.1). Using this we give two simple constructive criteria each of which gives necessary and sufficient conditions for k[V] T to be a polynomial ring (Theorem 5.8 and Theorem 5.10). 1. Introduction Let p : G --~ G L ( V ) be a finite dimensional rational representation ofa reductive algebraic group defined over an algebraically closed field k. Denote by k[V] the algebra of polynomial functions on V. The action of G on V given by p induces an action of G on k[V]. An element of k[V] fixed by this action is called an invariant and the set of invariants forms a finitely generated subalgebra called the ring of invariants, denoted k[V] a. The fundamental problem in invariant theory is to describe the ring of in- variants. A first step toward understanding k[V] a is to determine for which representations p is k[V] a a polynomial ring (see [11]). G.C. Shephard and J.A. Todd showed in 1954 [13] that the ring ofinvariants of a finite group over a field of characteristic zero is a polynomial ring if and only if the action of G on V is generated by pseudo-reflections. Their proof of the "if" half of this theorem was by classification. In 1955 C. Chevalley [3] gave an elegant algebraic proof that, if the action of a finite group G on V is generated by reflections (still assuming char(k) = 0), then k[V] a is a polynomial ring. Soon thereafter, J.P. Serre observed that Chevalley's proof was also valid for actions generated by pseudo-reflections and that it therefore provided an algebraic proof of the "if" half of the Shephard-Todd theorem. A method for classifying those reductive groups having a polynomial ring of invariants was suggested in 1976 by V. Kac, V. Popov and E. Vinberg [5]. Us- a Research supported in part by NSERC Grant OGP 137522