On the Generic Point Arrangements in Euclidean Space and Stratification of the Totally Nonzero Grassmannian

TL;DR

This paper studies the classification of generic point arrangements in Euclidean space using stratifications of the totally nonzero Grassmannian, providing enumerations in dimension two and highlighting open problems in higher dimensions.

Contribution

It introduces a stratification of the totally nonzero Grassmannian to parametrize and classify generic point arrangements, with explicit enumeration in 2D and open questions for higher dimensions.

Findings

Enumerates isomorphism classes in 2D using Euler-totient function.

Describes stratification of the totally nonzero Grassmannian.

Identifies open enumeration problems for dimensions ≥ 3.

Abstract

In this article, for positive integers $n\geq m\geq 1$, the parameter spaces for the isomorphism classes of the generic point arrangements of cardinality $n$, and the antipodal point arrangements of cardinality $2n$ in the Eulidean space $\mathbb{R}^m$ are described using the space of totally nonzero Grassmannian $Gr^{tnz}_{mn}(\mathbb{R})$. A stratification $\mathcal{S}^{tnz}_{mn}(\mathbb{R})$ of the totally nonzero Grassmannian $Gr^{tnz}_{mn}(\mathbb{R})$ is mentioned and the parameter spaces are respectively expressed as quotients of the space $\mathcal{S}^{tnz}_{mn}(\mathbb{R})$ of strata under suitable actions of the symmetric group $S_n$ and the semidirect product group $(\mathbb{R}^*)^n\rtimes S_n$. The cardinalities of the space $\mathcal{S}^{tnz}_{mn}(\mathbb{R})$ of strata and of the parameter spaces $S_n\backslash \mathcal{S}^{tnz}_{mn}(\mathbb{R}), ((\mathbb{R}^*)^n\rtimes S_n)\backslash \mathcal{S}^{tnz}_{mn}(\mathbb{R})$ are enumerated in dimension $m=2$. Interestingly enough, the enumerated value of the isomorphism classes of the generic point arrangements in the Euclidean plane is expressed in terms of the number theoretic Euler-totient function. The analogous enumeration questions are still open in higher dimensions for $m\geq 3$.