This is inevitably an imprecise question, but there are already several questions like this on the site so I thought i'd try anyway.

If I understand correctly, for any reductive algebraic group $G$ the points of $G$ over the field with one element should be $G(\mathbb{F}_1) = W$ where $W$ is the Weyl group of $G$. My question, stated briefly (and generally), is the following:

For a reductive group there are several notions which make sense over any base field, what are the analogs of these for $W$:

- Parabolic subgroups
- Levi subgroups
- Borel subgroups
- Bruhat decomposition

I will be satisfied with an answer for the case $G=GL_n$ so let's consider this case in a little more detail:

In this case: $GL_n(\mathbb{F}_1)=W=\Sigma_n$. It seems to me like the only reasonable notion of a **Levi subgroup** of $\Sigma_n$ would be a **Young subgroup** $\Sigma_{\lambda} := \Sigma_{\lambda_1} \times \dots \times \Sigma_{\lambda_l}$ for some partition $\lambda = (\lambda_1,...,\lambda_l)$. **Is this correct?** If so what are the corresponding parabolic subgroups?

Following this line of thought we get that the maximal torus should be trivial (which is somehow reasonable). **Should the Borel be the trivial subgroup then?** If so this is somehow disappointing...