Shephard–Todd theorem

In mathematics, the Chevalley–Shephard–Todd theorem in invariant theory of finite groups states that the ring of invariants of a finite group acting on a complex vector space is a polynomial ring if and only if the group is generated by pseudoreflections. In the case of subgroups of the complex general linear group the theorem was first proved by G. C. Shephard and J. A. Todd (1954) who gave a case-by-case proof. Claude Chevalley (1955) soon afterwards gave a uniform proof. It has been extended to finite linear groups over an arbitrary field in the non-modular case by Jean-Pierre Serre.

Let V be a finite-dimensional vector space over a field K and let G be a finite subgroup of the general linear group GL(V). An element s of GL(V) is called a pseudoreflection if it fixes a codimension 1 subspace of V and is not the identity transformation I, or equivalently, if the kernel Ker (s − I) has codimension one in V. Assume that the order of G is relatively prime to the characteristic of K (the so-called non-modular case). Then the following properties are equivalent:

In the case when the field K is the field C of complex numbers, the first condition is usually stated as "G is a complex reflection group". Shephard and Todd derived a full classification of such groups.

Broer (2007) gave an extension of the Chevalley–Shephard–Todd theorem to positive characteristic.

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