# Multiplicity free actions, birth and death processes on partitions, and Biane's quantum Ornstein-Uhlenbeck semigroups

## TL;DR

This paper introduces new multivariate birth and death processes on partitions, connecting classical Markov processes with Biane's quantum Ornstein-Uhlenbeck semigroups through multiplicity free actions.

## Contribution

It develops a family of multivariate pure birth and death chains linked to multiplicity free actions, extending the understanding of quantum Ornstein-Uhlenbeck semigroups in a classical setting.

## Key findings

- Processes are defined on partitions and Young diagrams.
- Chains are classical Markov processes derived from quantum semigroups.
- Connections established between quantum and classical stochastic processes.

## Abstract

We introduce a family of 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 the introduced processes are some sets of partitions (equivalently, Young diagrams). The chains turn out to be the classical Markov processes obtained by restricting Biane's quantum Ornstein-Uhlenbeck semigroups to commutative C*-algebras related to Gelfand pairs built on Heisenberg groups.