The Rigid Unit Mode spectrum for symmetric frameworks
Eleftherios Kastis, Derek Kitson
TL;DR
This work develops a unified operator-theoretic framework to analyze the RUM spectrum of symmetric frameworks with a discrete abelian symmetry group. By formulating a Gamma-gain framework and defining orbit matrices on the dual group, it connects symmetry-generated joint spectra to the RUM spectrum and to translational degrees of freedom in the covering framework. The main results show that the RUM spectrum $\Omega({\mathcal{G}})$ contains a nonempty subset $\Omega_{js}(\tau)$ determined by joint eigenvalues of symmetry generators, and that $\Omega({\mathcal{G}})$ can be expressed as the union of Bohr-Fourier spectra of twisted almost periodic flexes; it also characterizes when every almost periodic flex reduces to a translation. These insights provide a versatile toolkit for analyzing phase-periodic and almost-periodic flexing in symmetric networks and for constructing flexes with prescribed symmetry properties.
Abstract
We establish several fundamental properties of the Rigid Unit Mode (RUM) spectrum for symmetric frameworks with a discrete abelian symmetry group and arbitrary linear constraints. In particular, we identify a nonempty subset of the RUM spectrum which derives from the joint eigenvalues of generators for the linear part of the symmetry group. These joint eigenvalues give rise to $χ$-symmetric flexes which span the space of translations for the framework. We show that the RUM spectrum is a union of Bohr-Fourier spectra arising from twisted almost-periodic flexes of the framework. We also characterise frameworks for which every almost periodic flex is a translation.