Tensor subalgebras and first fundamental theorems in invariant theory

Abstract

Let 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 = T G . 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.