Reflection Quotients in Riemannian Geometry. a Geometric Converse to Chevalley’s Theorem

Chevalley’s theorem and it’s converse, the Sheppard-Todd theorem, assert that finite reflection groups are distinguished by the fact that the ring of invariant polynomials is freely generated. We show that in the Euclidean case, a weaker condition suffices to characterize finite reflection groups, namely that a freely-generated polynomial subring is closed with respect to the gradient product. From the standpoint of invariant theory, finite reflection groups are distinguished by the property that the corresponding algebra of invariants is freely generated. The existence of free generators of the invariant algebra is known as Chevalley’s theorem [2]. The converse, i.e. the statement that a finite group with a freely generated invariant algebra is necessarily generated by reflections, is known as the Sheppard-Todd theorem [6]. The customary proofs of these results use algebraic methods. The purpose of this note is to propose an alternate characterization of finite reflection groups over the reals, and to prove the result using the language and ideas of Riemannian geometry. Riemannian theory is relevant because a finite reflection group over R determines a Euclidean structure (this can be seem by a simple averaging argument), and therefore, without loss of generality, the elements of the reflection group can be assumed to be Euclidean automorphisms. Now the structure of Riemannian geometry is specified by a fundamental covariant: the gradient operation. Axiomatically then, the gradient operation is preserved by all Riemannian automorphisms. Specializing to Euclidean space, if P (x, . . . , x), Q(x, . . . , x) are two polynomials that are invariant with respect to some Euclidean reflections, then the corresponding gradient product