ABSTRACT
Chapter 7 Groups Generated by Reflections Many of the examples of rings of invariants that we have computed have turned out to be polynomial algebras, and we have seen several necessary conditions for the ring of invariants to be polynomial, such as 5.5.5 and 4.4.3. At present no necessary and sufficient conditions are known which assure that the ring of invariants of a finite group is a polynomial algebra. In the nonmodular case, however, i.e. when the group order is prime to the characteristic of the field, e.g. if the field has characteristic zero, the groups with polynomial algebras of invariants are precisely those generated by pseudoreflections. This was originally observed by Shephard-Todd based on computing all examples with the aid of their classification theorem. See sections 7.3 for a discussion of the classification and section 7.4 for a proof of the theorem of Shephard-Todd without the use of the classification. This chapter is devoted to the nonmodular invariant theory of finite pseudoreflection groups. It represents only a small part of the known, unknown and interesting phenomena connected with these groups. See for example the many books, survey articles and research papers listed in the reference list.
