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

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 $m\geq 3$.