I am vaguely aware of the following facts:
Let $V$ be a finite-dimensional real vector space with a positive-definite inner product $g$. Let $g_{\otimes n}$ denote the natural extension of $g$ to $V^{\otimes n}=V\otimes\cdots\otimes V.$ Let $\rho:\text{O}(V,g)\to\text{O}(V^{\otimes n},g_{\otimes n})$ be the standard group homomorphism. Then
- The $g_{\otimes n}$-invariant linear maps $V^{\otimes n}\to\mathbb{R}$ exist only when $n$ is even, and they are spanned by the complete contractions $$x_{\sigma(1)}\otimes\cdots\otimes x_{\sigma(n)}\mapsto g(x_{\tau(1)},x_{\tau(2)})\cdots g(x_{\tau(n-1)},x_{\tau(n)})$$ where $\sigma$ and $\tau$ range over all permutations of $\{1,\ldots,n\}$.
- If $W\subset V^{\otimes n}$ is a linear subspace such that $(\rho(T))(W)\subset W$ for all $T\in\text{O}(V,g)$ and such that the dimension of the vector space $$\Big\{\text{bilinear symmetric }S:W\times W\to\mathbb{R}\text{ s.t. }(\rho(T))^\ast S=S\quad\forall T\in\text{O}(V,g)\Big\}$$ is one, then there is no nontrivial proper linear subspace $U\subset V$ with $(\rho(T))(U)\subset U$ for all $T\in\text{O}(V,g).$
My reference is chapter 9 in Besse's book on four-dimensional geometry. My questions are the following:
- Does the above remain true if $g$ is only non-degenerate? It seems like the argument for #2 may break down since it relies on orthogonal projection. I couldn't make a guess on #1 since I'm not familiar with Weyl's book on the classical groups.
- Where can I find a reference with such a statement? Weyl's book doesn't seem to cover the indefinite case.