[Paper Review] New Examples of Potential Theory on Bratelli Diagrams

Jason Fulman|ArXiv.org|Dec 17, 1999

Random Matrices and Applications11 references3 citations

TL;DR

This paper introduces new examples of potential theory on Bratelli diagrams derived from Macdonald polynomials, with a focus on Hall-Littlewood and Schur functions. It constructs harmonic functions using unipotent upper triangular matrices over finite fields and provides sampling algorithms for associated probability measures, revealing deep connections between combinatorics, representation theory, and stochastic processes on graded graphs.

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.

Motivation & Objective

• To extend potential theory to Bratelli diagrams arising from Macdonald polynomials, particularly Hall-Littlewood and Schur functions.
• To establish connections between harmonic functions on these diagrams and the Jordan forms of unipotent upper triangular matrices over finite fields.
• To construct explicit families of harmonic functions on combinatorial graphs derived from symmetric functions.
• To develop algorithms for sampling from the probability measures associated with these harmonic functions.
• To generalize the Plancherel measure and Young lattice via natural deformations induced by symmetric functions.

Proposed method

• Utilizes the combinatorial structure of Bratelli diagrams where vertices are integer partitions and edges represent branching rules.
• Applies Green's polynomials as combinatorial dimensions to define the multiplicity of paths in the diagram.
• Leverages the distribution of Jordan forms of random unipotent upper triangular matrices over finite fields to define harmonic functions.
• Employs Macdonald polynomials as generating functions for the harmonic functions on the diagram.
• Derives sampling algorithms based on the stochastic interpretation of the harmonic functions via matrix conjugacy classes.
• Uses representation-theoretic and symmetric function theory to construct and verify the harmonic functions.

Experimental results

Research questions

• RQ1 How can potential theory be systematically developed on Bratelli diagrams associated with Macdonald polynomials?
• RQ2 What is the probabilistic interpretation of harmonic functions on these diagrams in terms of unipotent matrices over finite fields?
• RQ3 How do Hall-Littlewood polynomials give rise to harmonic functions on partition lattices with specific branching multiplicities?
• RQ4 What deformations of the Young lattice and Plancherel measure arise from Schur functions in this framework?
• RQ5 What efficient sampling algorithms can be constructed for the probability measures underlying these harmonic functions?

Key findings

• Harmonic functions on Bratelli diagrams arising from Hall-Littlewood polynomials are constructed using the Jordan form distribution of random unipotent upper triangular matrices over finite fields.
• The branching multiplicities in the diagram are integers, and the combinatorial dimensions correspond to Green's polynomials.
• The Schur function case yields natural deformations of the Young lattice and a generalization of the Plancherel measure.
• Explicit sampling algorithms are provided for the probability measures associated with the harmonic functions on these diagrams.
• The framework unifies combinatorial representation theory with potential theory on graded graphs through symmetric functions.
• The results demonstrate a deep interplay between algebraic combinatorics, finite group representations, and stochastic processes on discrete structures.