Relatively free invariant algebras of finite reflection groups

Abstract

Let G G be a finite subgroup of G l n ( K ) Gl_{n}(K) ( K (K is a field of characteristic 0 0 and n ≥ 2 ) n\geq 2) acting by linear substitution on a relatively free algebra K ⟨ x 1 , … , x n ⟩ / I K\langle x_{1},\dots ,x_{n}\rangle /I of a variety of unitary associative algebras. The algebra of invariants is relatively free if and only if G G is a pseudo-reflection group and I I contains the polynomial [ [ x 2 , x 1 ] , x 1 ] . [[x_{2},x_{1}],x_{1}].