Multiplicity Free Actions, Birth and Death Processes on Partitions, and Biane’s Quantum Ornstein–Uhlenbeck Semigroups

We identify all classical Markov processes obtained by restricting Biane’s quantum Ornstein–Uhlenbeck semigroups to commutative \(\text {C}^*\) ${\text{C}}^{\ast }$ -algebras related to Gelfand pairs built on Heisenberg groups. It turns out that they form a family of (mostly) multivariate continuous-time pure birth and pure death chains, with birth and death rates defined in terms of the generalized binomial coefficients for multiplicity free actions. The state spaces for some of those processes are some sets of partitions (equivalently, Young diagrams).