State symmetries in matrices and vectors on finite state spaces

Abstract

State symmetries are defined as permutations which act on vector spaces of column vectors and square matrices, resulting in isotropy groups for specific vector spaces. A large number of properties for such objects is shown, to provide a rigorous basis for future applications. The main statement characterises the state symmetry of vector sequences $(v^{(i)})$ which are generated by powers of a generator matrix $M$: $v^{(i)}= M^i v^{(0)}$. A section of examples illustrates some applications of the theory.