# Invariant synchrony subspaces of sets of matrices

## TL;DR

This paper investigates invariant synchrony subspaces of matrices, exploring their properties, applications in graph and network theory, and introduces algorithms for their computation, with generalizations to non-square matrices and connections to design theory.

## Contribution

It introduces a comprehensive study of invariant synchrony subspaces, including properties, applications, and a novel algorithm for their identification, extending to non-square matrices and tactical decompositions.

## Key findings

- Invariant subspaces form a lattice structure.
- Applications in graph partitions and network analysis.
- Introduction of the split and cir algorithm.

## Abstract

A synchrony subspace of R^n is defined by setting certain components of the vectors equal according to an equivalence relation. Synchrony subspaces invariant under a given set of square matrices form a lattice. Applications of these invariant synchrony subspaces include equitable and almost equitable partitions of the vertices of a graph used in many areas of graph theory, balanced and exo-balanced partitions of coupled cell networks, and coset partitions of Cayley graphs. We study the basic properties of invariant synchrony subspaces and provide many examples of the applications. We also present what we call the split and cir algorithm for finding the lattice of invariant synchrony subspaces. Our theory and algorithm is further generalized for non-square matrices. This leads to the notion of tactical decompositions studied for its application in design theory.