Weak Lumpability of Finite Markov Chains and Positive Invariance of Cones

We consider weak lumpability of general finite homogeneous Markov chains evolving in discrete time, that is when a lumped Markov chain with respect to a partition of the initial state space is also a homogeneous Markov chain. We show that weak lumpability is equivalent to the existence of a decomposable polyhedral cone which is positively invariant by the transition probability matrix of the original chain. It allows us, in a unified way, to derive new results on lumpability of reducible Markov chains and to obtain spectral properties associated with lumpability