Point Count of the Top-dimensional Open Positroid Variety

TL;DR

This paper derives a point count formula for the top-dimensional open positroid variety over finite fields by analyzing torus actions and cohomology, confirming a prior result for coprime parameters.

Contribution

It provides a new proof of the point count identity using torus actions and cohomology, revealing the trivial action of cyclic rotation when parameters are coprime.

Findings

Confirmed the point count formula for open positroid varieties over finite fields.

Showed cyclic rotation acts trivially on torus-equivariant cohomology under coprimality.

Connected torus actions with point counting in algebraic geometry.

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 $\Pi_{k,n}^\circ(\mathbb{F}_q)$ 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 $(\Pi_{k,n}^\circ)_{\mathbb{F}_q}$ and an anisotropic torus action on a $\mathbb{F}_q$ rational form of $\Pi_{k,n}^\circ$. 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 $\Pi_{k,n}^\circ$ when $k$ and $n$ are coprime.