Transition Support Matrices and Local–Global Dynamics in Non-Abelian Group Cellular Automata

Abstract

In this paper, we study matrix-type structures associated with group cellular automata over finite non-abelian groups. We introduce a transition support matrix that encodes locally admissible transitions and provides a finite combinatorial description of local transition behavior. We show that the irreducibility of this matrix reflects local propagation properties and, under an additional realizability assumption, can imply topological transitivity. We also construct an explicit example over the symmetric group S3, showing that matrix irreducibility alone is not sufficient to guarantee transitivity in the non-abelian setting. These results illustrate the distinction between local transition structure and global dynamical behavior in non-abelian group cellular automata, and indicate some limitations of purely matrix-based approaches beyond the abelian case.