Invariant theory of finite groups

Abstract

Let C[x] denote the ring of polynomials with complex coefficients in n variables x = (x ₁, x ₂,…, x _n). We are interested in studying polynomials which remain invariant under the action of a finite matrix group Г ⊂ GL(C ⁿ). The main result of this chapter is a collection of algorithms for finding a finite set {I ₁, I ₂,…, I _m} of fundamental invariants which generate the invariant subring C[x]^Г. These algorithms make use of the Molien series (Sect. 2.2) and the Cohen—Macaulay property (Sect. 2.3). In Sect. 2.4 we include a discussion of invariants of reflection groups, which is an important classical topic. Sections 2.6 and 2.7 are concerned with applications and special cases.