Most primitive groups have messy invariants

Suppose G is a finite group of complex n × n matrices, and let R^G be the ring of invariants of G: i.e., those polynomials fixed by G. Many authors, from Klein to the present day, have described R^G by writing it as a direct sum Σ^δ_j=1 η_j$\text{C}$[θ₁ ,…, θ_n]. For example, if G is a unitary group generated by reflections, δ = 1. In this note we show that in general this approach is hopeless by proving that, for any ? > 0, the smallest possible δ is greater than | G |^n-1-? for almost all primitive groups. Since for any group we can choose δ ? | G |^n-1, this means that most primitive groups are about as bad as they can be. The upper bound on δ follows from Dade's theorem that the θ_i can be chosen to have degrees dividing | G |.