Point Count of the Top-dimensional Open Positroid Variety
Calvin Yost-Wolff
TL;DR
The paper proves a cohomological explanation for the finite-field point count of the top-dimensional open positroid variety $\Pi_{k,n}^\circ$ in the coprime case $\gcd(k,n)=1$. By relating the split torus action on $\Pi_{k,n}^\circ$ to an anisotropic torus action on a twisted rational form, and showing that the cyclic rotation $\rho$ acts trivially on the $T$-equivariant compactly supported étale cohomology, the author deduces an isomorphism of cohomology under the Fr-modules for the untwisted and twisted forms. This cohomological triviality transfers to equality of $\mathbb{F}_q$-points counts for the two forms, yielding a point-count identity that factors the open-positroid count by the torus sizes and recovers the known Grassmannian count scaled by $|(\mathbb{F}_q^\times)^n|/|\mathbb{F}_{q^n}^\times|$. The approach combines cohomological purity results, Lang’s theorem for torsors, and a Moore determinant argument to connect the combinatorics of $\mathrm{Gr}(k,n)(\mathbb{F}_q)$ with torus-orbit quotients and rational $q$-Catalan numbers, providing a pathway to similar results in related settings.
Abstract
In [GL24], Galashin and Lam discovered that when $k$ and $n$ are coprime, the proportion of subspaces in $\mathrm{Gr}(k,n)(\mathbb{F}_q)$ that lie in the top-dimensional open positroid variety $Π_{k,n}^\circ(\mathbb{F}_q)$ is $|(\mathbb{F}_q^\times)^n|/|\mathbb{F}_{q^n}^\times|$. In this paper, I recover this point count identity by relating the split torus action on $(Π_{k,n}^\circ)_{\mathbb{F}_q}$ and an anisotropic torus action on a $\mathbb{F}_q$ rational form of $Π_{k,n}^\circ$. 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 $Π_{k,n}^\circ$ when $k$ and $n$ are coprime.