Tensor subalgebras and first fundamental theorems in invariant theory

AbstractLet V be an n-dimensional complex inner product space and let T:=T(V)⊗T(V∗) be the mixed tensor algebra over V. We characterize those subsets A of T for which there is a subgroup G of the unitary group U(n) such that A=TG. They are precisely the nondegenerate contraction-closed graded ∗-subalgebras of T. While the proof makes use of the First Fundamental Theorem for GL(n,C) (in the sense of Weyl), the characterization has as direct consequences First Fundamental Theorems for several subgroups of GL(n,C). Moreover, a Galois correspondence between linear algebraic ∗-subgroups of GL(n,C) and nondegenerate contraction-closed graded ∗-subalgebras of T is derived. We also consider some combinatorial applications, viz. to self-dual codes and to combinatorial parameters