Embedding Q-Processes Into Stochastic Processes On Permutation Groups

Embedding Q-processes into stochastic processes on permutation groups is studied in this paper.Namely,given a Markov process X=(X_t)_{t∈[0,∞})(X₀=z) on a finite set E,can it be represented as((?)_tz)_{t∈[0,∞})? where z∈E and ((?)_t)_{t∈[0,∞})((?)₀ is the identity map)is a Q-process on some transitive permutation groupsΓof E.Let (Q(x,y))_x,y∈E be the Q-matrix of X=(X_t)_{t∈[0,∞}).Then there exists embedding(?)_t)_{t∈[0,∞})if and only ifWhere (?) is the collection of irreducible representations ofΓ,(?)₊ is the set of irreducible representationsρ∈Γthat appear with positive multiplicityν_ρ>0 inthe decomposition of the permutation representation,(?)₀ is the family of irreduciblerepresentations that do not appear,χ_ρis the character ofρand d_ρis the dimensionofρ.Furthermore,some properties of the Markov chains ((?)_t)_{t∈[0,∞}) are also studied.The obtained results extend those of S.N.Evans([12]).