Two-parameter Poisson-Dirichlet measures and reversible exchangeable fragmentation-coalescence processes

We show that for 0 < alpha < 1 and 0 > -alpha, the Poisson-Dirichlet distribution with parameter (alpha, theta) is the unique reversible distribution of a rather natural fragmentation-coalescence process. This completes earlier results in the literature for certain split-and-merge transformations and the parameter alpha = 0.