Invariant synchrony subspaces of sets of matrices

John M. Neuberger, Nandor Sieben, James W. Swift

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


A synchrony subspace of Rn 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 ordered by inclusion 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 nonsquare matrices. This leads to the notion of tactical decompositions studied for its application in design theory.

Original languageEnglish (US)
Pages (from-to)964-993
Number of pages30
JournalSIAM Journal on Applied Dynamical Systems
Issue number2
StatePublished - 2020


  • Almost Equitable Partitions
  • Balanced Partitions
  • Coupled Cell Network
  • Equitable Partition
  • Exo-Balanced Partitons
  • Synchrony

ASJC Scopus subject areas

  • Analysis
  • Modeling and Simulation


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