Point Count of the Top-dimensional Open Positroid Variety

In [GL24], Galashin and Lam discovered that when $k$ and $n$ are coprime, the proportion of subspaces in $\mathrm{G}\mathrm{r}(k,n)({\mathbb{F}}_q)$ that lie in the top-dimensional open positroid variety ${\mathrm{\Pi }}_{k,n}^{\circ }({\mathbb{F}}_q)$ is $\mathrm{\mid }({\mathbb{F}}_q^{\times }{)}^n\mathrm{\mid }\mathrm{/}\mathrm{\mid }{\mathbb{F}}_{q^n}^{\times }\mathrm{\mid }$. In this paper, I recover this point count identity by relating the split torus action on $({\mathrm{\Pi }}_{k,n}^{\circ }{)}_{{\mathbb{F}}_q}$ and an anisotropic torus action on a ${\mathbb{F}}_q$ rational form of ${\mathrm{\Pi }}_{k,n}^{\circ }$. The main step in the point count argument and the main technical result in this paper is that cyclic rotation acts trivially on the torus-equivariant cohomology of ${\mathrm{\Pi }}_{k,n}^{\circ }$ when $k$ and $n$ are coprime.