Permissible symmetries of coupled cell networks

Abstract

AbstractWe consider coupled sets of identical cells and address the problem of which symmetries are permissible in such networks. For example, n linearly coupled cells with one independent variable in each cell cannot be constructed with the symmetry group An, the alternating group on n symbols. Using a graphical technique, we show that it is possible to construct cell networks with any desired finite group of symmetries. In particular, we show that any subgroup of Sn can be realized as the symmetries of a group of n cells. Special forms of coupling (especially low order polynomial coupling) are shown to restrict the possible symmetries. We give some upper and lower bounds for the degree of polynomial required to realize several classes of subgroups of Sn.