New Examples of Potential Theory on Bratelli Diagrams

TL;DR

This paper explores potential theory on Bratteli diagrams linked to Macdonald polynomials, focusing on Hall-Littlewood and Schur functions, constructing harmonic functions, and providing sampling algorithms for related probability measures.

Contribution

It introduces new potential-theoretic models on Bratteli diagrams derived from Macdonald polynomials, including explicit harmonic functions and sampling algorithms.

Findings

Harmonic functions are constructed for diagrams from Hall-Littlewood and Schur functions.

Algorithms for sampling from the associated probability measures are provided.

Connections between harmonic functions and combinatorial structures like Green's polynomials are established.

Abstract

We consider potential theory on Bratteli diagrams arising from Macdonald polynomials. The case of Hall-Littlewood polynomials is particularly interesting; the elements of the diagram are partitions, the branching multiplicities are integers, the combinatorial dimensions are Green's polynomials, and the Jordan form of a randomly chosen unipotent upper triangular matrix over a finite field gives rise to a harmonic function. The case of Schur functions yields natural deformations of the Young lattice and Plancharel measure. Many harmonic functions are constructed and algorithms for sampling from the underlying probability measures are given.