Symmetry and Synchrony in Coupled Cell Networks 2: Group Networks

This paper continues the study of patterns of synchrony (equivalently, balanced colorings or flowinvariant subspaces) in symmetric coupled cell networks, and their relation to fixed-point spaces of subgroups of the symmetry group. Let Γ be a permutation group acting on the set of cells. We define the group network GΓ, whose architecture is entirely determined by the group orbits of Γ. We prove that if Γ has the “balanced extension property” then every balanced coloring of GΓ is a fixed-point coloring relative to the automorphism group of the group network. This theorem applies in particular when Γ is cyclic or dihedral, acting on cells as the symmetries of a regular polygon, and in these cases the automorphism group is Γ itself. In general, however, the automorphism group may be larger than Γ. Several examples of this phenomenon are discussed, including the finite simple group of order 168 in its permutation representation of degree 7. More dramatically, for some choices of Γ there exist balanced colorings of GΓ that are not fixed-point colorings. For example, there exists an exotic balanced 2-coloring when Γ is the symmetry group of the two-dimensional square lattice. This coloring is doubly periodic, and its reduction modulo 8 leads to a finite group with similar properties. Although these patterns do not arise from fixed-point spaces, we provide a group-theoretic explanation of their balance property in terms of a sublattice of index two.