Block-diagonalized rigidity matrices of symmetric frameworks and applications

TL;DR

This paper proves that the rigidity matrix of symmetric frameworks can be block-diagonalized using group representation theory, enabling new insights into their rigidity and flexibility properties.

Contribution

It provides a complete proof that the rigidity matrix can be block-diagonalized for symmetric frameworks, extending Maxwell's rule to all dimensions and realization types.

Findings

Block-diagonalization of rigidity matrices using group representation theory.

Generalization of Fowler-Guest symmetry extension of Maxwell's rule.

Applicable to both injective and non-injective realizations across all dimensions.

Abstract

In this paper, we give a complete self-contained proof that the rigidity matrix of a symmetric bar and joint framework (as well as its transpose) can be transformed into a block-diagonalized form using techniques from group representation theory. This theorem is basic to a number of useful and interesting results concerning the rigidity and flexibility of symmetric frameworks. As an example, we use this theorem to prove a generalization of the Fowler-Guest symmetry extension of Maxwell's rule which can be applied to both injective and non-injective realizations in all dimensions.