Point Count of the Top-dimensional Open Positroid Variety

Abstract: In [GL24], Galashin and Lam discovered that when $k$ and $n$ are coprime, the proportion of subspaces in $\mathrm{Gr}(k,n)(\mathbb{F}<em>q)$ that lie in the top-dimensional open positroid variety $Π</em>{k,n}<sup>\circ(\mathbb{F}_q)$ is $$|(\mathbb{F}<em>q^{\times)^{n|/|\mathbb{F}}}</em>{q<sup>n}<sup>\times|$$. In this paper, I recover this point count identity by relating the split torus action on $(Π<em>{k,n}^\circ)</em>{\mathbb{F}<em>q}$ and an anisotropic torus action on a $\mathbb{F}_q$ rational form of $Π</em>{k,n}<sup>\circ$. 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 $Π_{k,n}<sup>\circ$ when $k$ and $n$ are coprime.