Tensor subalgebras and First Fundamental Theorems in invariant theory

2006/04/11 by Schrijver, Alexander · 1 citation
#15A72 #20Gxx #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.math/0604240

Abstract

Let V=\oCⁿ 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 \UU(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,\oC) (in the sense of Weyl), the characterization has as direct consequences First Fundamental Theorems for several subgroups of \GL(n,\oC). Moreover, a Galois connection between linear algebraic *-subgroups of \GL(n,\oC) and nondegenerate contraction-closed *-subalgebras of T is derived.

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