Invariants of finite cyclic groups acting on generic matrices (Q1073890)

Let R be an affine algebra over some field k and let G be a finite group of k-algebra automorphisms of R. If R is commutative, then the fixed subalgebra \(R^ G\) is also affine, by a classical theorem of E. Noether, whereas for R free, \(R^ G\) is known to be almost never affine. In the present article, the authors consider the case where \(R=k\{X_ 1,...,X_ d\}\) is the algebra generated by d generic \(m\times m\)- matrices \(X_ i\) (m,d\(\geq 2)\) and \(G=<g>\) is a finite cyclic group with \(| G| =n\) nonzero in k which acts on R by acting linearly on the k-subspace V of R generated by the \(X_ i's\). The main result asserts that if the matrix A representing the action of g on V is not scalar, then \(R^ G\) is not affine whenever \(m\geq n-[\sqrt{n}]+1\). Moreover, if A has a characteristic root \(\alpha\) such that \(\alpha^ q=1\) for some q with \(0<q<n\), then \(R^ G\) is not affine whenever \(m\geq 2\).