Let $\Bbb R^n_{++} = \{x\in \Bbb R^n\mid x_1,\ldots,x_n> 0\}$ be the positive orthant.
What are group actions which leaves $\Bbb R^n_{++}$ invariant?
I think I have already found these ones:
- The group of permutation matrices $S_n$ and its subgroups.
- The group of diagonal matrices with positive diagonal elements.
- The group of component-wise powers, i.e. $\{x\mapsto (x_1^{\alpha},\ldots,x_n^{\alpha})\mid \alpha\in \Bbb R, \alpha \neq 0\}$ with the composition. Perhaps, this can be further extended to $\{x\mapsto (x_1^{\alpha_1},\ldots,x_n^{\alpha_n})\mid \alpha_1,\ldots,\alpha_n\neq 0 \}$.
- A combination of the above actions, e.g. $\{x\mapsto P\,(x_{1}^{\alpha},\ldots,x_n^{\alpha})\mid \alpha\in \Bbb R, \alpha \neq 0, P \in S_n\}$.
Is it all? Also, these actions are heavily using the fact that $\Bbb R^n_{++}$ is the interior of a polyhedral cone (the nonnegative orthant), I guess (hope) that there are actions which could be defined for general convex pointed cones in finite dimension.