Linear Symmetries of the Unsquared Measurement Variety

2020/07/24 by Ioannis Gkioulekas, Steven J. Gortler, Gkioulekas, Ioannis +5
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Metric Geometry (math.MG) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2007.12649

openalex publication_date 2020/07/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a new family of algebraic varieties, L_d,n, which we call the unsquared measurement varieties. This family is parameterized by a number of points n and a dimension d. These varieties arise naturally from problems in rigidity theory and distance geometry. In those applications, it can be useful to understand the group of linear automorphisms of L_d,n. Notably, a result of Regge implies that L_2,4 has an unexpected linear automorphism. In this paper, we give a complete characterization of the linear automorphisms of L_d,n for all n and d. We show, that apart from L_2,4 the unsquared measurement varieties have no unexpected automorphisms. Moreover, for L_2,4 we characterize the full automorphism group.

Citations

• Point Configurations and Cayley-Menger Varieties
• 23040 symmetries of hyperbolic tetrahedra
• On graphs and rigidity of plane skeletal structures
• Determining Generic Point Configurations From Unlabeled Path or Loop Lengths

Related

• Determining Generic Point Configurations From Unlabeled Path or Loop Lengths
• Automorphism Groups of Configuration Spaces and Discriminant Varieties
• Automorphisms of classical geometries in the sense of Klein
• On linearity of automorphism groups of algebras and groups
• Géométrie, points rationnels et itérés des automorphismes de l'espace affine