Question
Let $H$ denote the subgroup of permutation matrices of $\operatorname{GL}(n,\mathbb{R})$. Is $H$ a reductive group?
My Work
I know that $H$ is a linear algebraic group being a finite subgroup of $\operatorname{GL}(n,\mathbb{R})$. And, I have found various facts saying that the permutation matrices as an algebraic subgroup of $\operatorname{GL}(n,\mathbb{C})$ is reductive, e.g.,
Symmetric subgroups of $\operatorname{GL}(n,\mathbb{C})$ are linearly reductive and being linearly reductive is equivalent to being reductive over algebraically closed fields of zero characteristic.
I have tried to apply the following facts to this problem but have not had success:
- A linear algebraic group over a field of characteristic $0$ is reductive if and only if its Lie algebra is a reductive Lie algebra.
- Let $G$ be a reductive group and $H$ a closed subgroup, then the $H$ is reductive if and only if the quotient space $G/H$ is affine.
I have primarily tried applying fact (2) above, since $\operatorname{GL}(n,\mathbb{R})$ is a reductive group and the subgroup of permutation matrices is a closed subgroup. But, I am unable to determine whether $G/H$ is affine.
Motivation
I am curious about this question because I have a real algebraic variety with a group action on it by real permutation matrices, and I am curious about applying techniques of Geometric Invariant Theory to this problem. I do not know a lot about GIT, but from I have read one gets nice statements statements when the group is algebraic and reductive, hence my question.