Young diagrammatic methods in non-commutative invariant theory

In this paper we will study some aspects of non-commutative invariant theory. Let V be a finite-dimensional vector space over a field K of characteristic zero and let K[V] = K⊕V⊕S 2(V)⊕…, and K′V› = K⊕V⊕⊕2(V)⊕⊕3 V⊕& be respectively the symmetric algebra and the tensor algebra over V. Let G be a subgroup of GL(V). Then G acts on K[V] and K′V›. Much of this paper is devoted to the study of the (non-commutative) invariant ring K′V› G of G acting on K′V›. In the first part of this paper, we shall study the invariant ring in the following situation.