Let V be a finite-dimensional vector space over a fieldK and let G be a finite subgroup of the general linear groupGL. An element s of GL is called a pseudoreflection if it fixes a codimension 1 subspace of V and is not the identity transformationI, or equivalently, if the kernel Ker has codimension one in V. Assume that the order of G is relatively prime to the characteristic of K. Then the following properties are equivalent:
In the case when the fieldK 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.
Examples
Let V be one-dimensional. Then any finite group faithfully acting on V is a subgroup of the multiplicative group of the field K, and hence a cyclic group. It follows that G consists of roots of unity of order dividing n, where n is its order, so G is generated by pseudoreflections. In this case, K = K is the polynomial ring in one variable and the algebra of invariants of G is the subalgebra generated by xn, hence it is a polynomial algebra.
Let V = Kn be the standard n-dimensional vector space and G be the symmetric groupSn acting by permutations of the elements of the standard basis. The symmetric group is generated by transpositions, which act by reflections on V. On the other hand, by the main theorem of symmetric functions, the algebra of invariants is the polynomial algebra generated by the elementary symmetric functions e1,... en.
Let V = K2 and G be the cyclic group of order 2 acting by ±I. In this case, G is not generated by pseudoreflections, since the nonidentity element s of G acts without fixed points, so that dim Ker = 0. On the other hand, the algebra of invariants is the subalgebra of K = K generated by the homogeneous elements x2, xy, and y2 of degree 2. This subalgebra is not a polynomial algebra because of the relation x2y2 = 2.
Generalizations
gave an extension of the Chevalley–Shephard–Todd theorem to positive characteristic. There has been much work on the question of when a reductive algebraic group acting on a vector space has a polynomial ring of invariants. In the case when the algebraic group is simple all cases when the invariant ring is polynomial have been classified by In general, the ring of invariants of a finite group acting linearly on a complex vector space is Cohen-Macaulay, so it is a finite rank free module over a polynomial subring.
