Invariants of finite cyclic groups acting on generic matrices

Abstract

A classical theorem of E. Noether asserts that if R is a commutative ring, finitely generated over a field k, and G is any finite group of k- automorphisms of R, then the fixed ring (or ring of invariants) R” is also finitely generated. The question naturally arises as to what extent Noether’s theorem can be generalized to the noncommutative case. If R is also Noetherian and IG/ ~ ’ E k, all is well: RG is finitely generated, a result of Montgomery and Smail [6]. However, it is false in general, even for Pf rings [6]. Moreover, a recent result in Dicks and Formanek [l] (and, somewhat later, Kharchenko [4]), shows that almost the opposite of Noether’s theorem holds in the free aIgebra. That is, they prove that if G acts linearly on the free algebra F= k(xj,..., x,>, then P is finitely generated if and only if G acts by scalar matrices. In the present paper we consider the analogous problem for an algebra of generic matrices. That is, let U = k(X i,..., X,f be the generic matrix algebra generated over a field k by the m x m (m 2 2) generic matrices X1,..., X, (da 2). Let G act linearly on U; that is, for each gE G, Xp = c, cliiXi, for uij E k. Thus g corresponds to the dxd matrix A = (a,). If G consists of scalar matrices and \G/ -’ E k, then UC is always finitely generated. For, consider the free algebra F= k(x, ,..., xd> with the same action; since I/= ir’, a homomorphic image of F, it follows that v =