# Geometry and Topology of Symmetric Point Arrangements

## TL;DR

This paper explores the geometric and topological properties of symmetric point arrangements in Euclidean space, focusing on their arrangement spaces and how symmetries influence their deformability and mirror transformations.

## Contribution

It provides a characterization of symmetric arrangements through their arrangement spaces and analyzes the conditions affecting their continuous deformations and mirror symmetries.

## Key findings

- Arrangement space characterizations for symmetry properties
- Deformation conditions depend on representation decomposition
- Mirror symmetry deformation depends on dimension and representation

## Abstract

We investigate point arrangements $v_i\in\mathbb R^d,i\in \{1,...,n \}$ with certain prescribed symmetries. The arrangement space of $v$ is the column span of the matrix in which the $v_i$ are the rows. We characterize properties of $v$ in terms of the arrangement space, e.g. we characterize whether an arrangement possesses certain symmetries or whether it can be continuously deformed into another arrangement while preserving symmetry in the process. We show that whether a symmetric arrangement can be continuously deformed into its mirror image depends non-trivially on several factors, e.g. the decomposition of its representation into irreducible constituents, and whether we are in even or odd dimensions.