Tensor subalgebras and First Fundamental Theorems in invariant theory

Details Tensor subalgebras and First Fundamental Theorems in invariant theory Tensor subalgebras and First Fundamental Theorems in invariant theory

2006-04-11

arxiv.org/abs/math/0604240v1

Mathematics

Representation Theory

Scientific paper

Let $V=\oC^n$ and let $T:=T(V)\otimes T(V^*)$ 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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