Multiple partition structures and harmonic functions on branching graphs. (English) Zbl 1531.92066
Summary: We introduce and study multiple partition structures which are sequences of probability measures on families of Young diagrams subjected to a consistency condition. The multiple partition structures are generalizations of Kingman’s partition structures, and are motivated by a problem of population genetics. They are related to harmonic functions and coherent systems of probability measures on a certain branching graph. The vertices of this graph are multiple Young diagrams (or multiple partitions), and the edges depend on the Jack parameter. If the value of the Jack parameter is equal to one the branching graph under considerations reflects the branching rule for the irreducible representations of the wreath product of a finite group with the symmetric group. If the value of the Jack parameter is zero then the coherent systems of probability measures are precisely the multiple partition structures. We establish a bijective correspondence between the set of harmonic functions on the graph and probability measures on the generalized Thoma set. The correspondence is determined by a canonical integral representation of harmonic functions. As a consequence we obtain a representation theorem for multiple partition structures.
We give an example of a multiple partition structure which is expected to be relevant for a model of population genetics for the genetic variation of a sample of gametes from a large population. Namely, we construct a probability measure on the wreath product of a finite group with the symmetric group. If the finite group contains the identity element only then it coincides with the well-known Ewens probability measure on the symmetric group. The constructed probability measure defines a multiple partition structure which is a generalization of the Ewens partition structure studied by Kingman. We show that this multiple partition structure can be represented in terms of a multiple analogue of the Poisson-Dirichlet distribution called the multiple Poisson-Dirichlet distribution in the paper.
We give an example of a multiple partition structure which is expected to be relevant for a model of population genetics for the genetic variation of a sample of gametes from a large population. Namely, we construct a probability measure on the wreath product of a finite group with the symmetric group. If the finite group contains the identity element only then it coincides with the well-known Ewens probability measure on the symmetric group. The constructed probability measure defines a multiple partition structure which is a generalization of the Ewens partition structure studied by Kingman. We show that this multiple partition structure can be represented in terms of a multiple analogue of the Poisson-Dirichlet distribution called the multiple Poisson-Dirichlet distribution in the paper.
MSC:
| 92D20 | Protein sequences, DNA sequences |
| 92D10 | Genetics and epigenetics |
| 05E05 | Symmetric functions and generalizations |
| 65P10 | Numerical methods for Hamiltonian systems including symplectic integrators |
| 31C20 | Discrete potential theory |
| 28A99 | Classical measure theory |
Keywords:
partition structures; harmonic functions on branching graphs; symmetric functions; wreath products; probability measures on finite groups; Ewens sampling formula; Poisson-Dirichlet distribution and its generalizationsReferences:
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