Point Count of the Top-dimensional Open Positroid Variety
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866910024506277888 |
|---|---|
| author | Yost-Wolff, Calvin |
| author_facet | Yost-Wolff, Calvin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_15316 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Point Count of the Top-dimensional Open Positroid Variety Yost-Wolff, Calvin Combinatorics Algebraic Geometry Representation Theory 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. |
| title | Point Count of the Top-dimensional Open Positroid Variety |
| topic | Combinatorics Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2602.15316 |