Symbol functions for symmetric frameworks

TL;DR

This paper develops a unified framework using symbol functions to analyze rigidity properties of symmetric bar-joint frameworks, extending classical results and enabling new constructions of rigid modes.

Contribution

It introduces a symbol function approach for symmetric frameworks, generalizing rigidity analysis and constructing rigid modes in novel contexts.

Findings

Unified symbol function formulation for frameworks with abelian symmetry.

Extension of rigidity theory to broader classes of frameworks.

Construction methods for generalized rigid unit modes.

Abstract

We prove a variant of the well-known result that intertwiners for the bilateral shift on `$\ell^2(Z)$ are unitarily equivalent to multiplication operators on $L^2(T)$. This enables us to unify and extend fundamental aspects of rigidity theory for bar-joint frameworks with an abelian symmetry group. In particular, we formulate the symbol function for a wide class of frameworks and show how to construct generalised rigid unit modes in a variety of new contexts.