Relatively free invariant algebras of finite reflection groups

M\'aty\'as Domokos

Trans. Amer. Math. Soc. 348 (1996), pp. 2217-2234 

Abstract. Let G be a finite subgroup of GL_n(K) (K is a field of
characteristic 0 and n \ge 2) acting by linear substitution on a 
relatively free algebra K<x_1,...,x_n> of a variety of unitary 
associative algebras. The algebra of invariants is relatively free 
if and only if G is a pseudo-reflection group and I contains the 
polynomial [[x_2,x_1],x_1].