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

Abstract

In this article, for positive integers $n\ge m\ge 1$, 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 $G{r}_{mn}^{tnz}(\mathbb{R})$. A stratification ${\mathcal{S}}_{mn}^{tnz}(\mathbb{R})$ of the totally nonzero Grassmannian $G{r}_{mn}^{tnz}(\mathbb{R})$ is mentioned and the parameter spaces are respectively expressed as quotients of the space ${\mathcal{S}}_{mn}^{tnz}(\mathbb{R})$ of strata under suitable actions of the symmetric group $S_n$ and the semidirect product group $({\mathbb{R}}^{\ast }{)}^n⋊S_n$. The cardinalities of the space ${\mathcal{S}}_{mn}^{tnz}(\mathbb{R})$ of strata and of the parameter spaces $S_n\backslash {\mathcal{S}}_{mn}^{tnz}(\mathbb{R}),(({\mathbb{R}}^{\ast }{)}^n⋊S_n)\backslash {\mathcal{S}}_{mn}^{tnz}(\mathbb{R})$ 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 $m\ge 3$.

15 pages

References in corpus (1)

• On Infinity Type Hyperplane Arrangements and Convex Positive Bijections