[Paper Review] Harmonic functions on multiplicative graphs and interpolation polynomials

Alexei Borodin, Grigori Olshanski|arXiv (Cornell University)|Dec 15, 1999

Bayesian Methods and Mixture Models10 references4 citations

TL;DR

This paper constructs nonnegative harmonic functions on multiplicative graphs, such as the Young lattice, using multivariate interpolation polynomials related to Schur and Jack symmetric functions. The method yields explicit formulas for harmonic functions and computes new Selberg-type integrals as a by-product, advancing harmonic analysis on infinite symmetric groups and representation theory.

ABSTRACT

We construct examples of nonnegative harmonic functions on certain graded graphs: the Young lattice and its generalizations. Such functions first emerged in harmonic analysis on the infinite symmetric group. Our method relies on multivariate interpolation polynomials associated with Schur's S and P functions and with Jack symmetric functions. As a by-product, we compute certain Selberg-type integrals.

Motivation & Objective

• To construct nonnegative harmonic functions on graded graphs like the Young lattice.
• To connect harmonic analysis on the infinite symmetric group with symmetric function theory.
• To develop a method based on multivariate interpolation polynomials associated with Schur and Jack functions.
• To compute new Selberg-type integrals as a by-product of the construction.
• To provide explicit formulas for harmonic functions in the context of representation theory and combinatorics.

Proposed method

• Utilizes interpolation polynomials linked to Schur's S and P functions and Jack symmetric functions.
• Applies the theory of symmetric functions to define harmonic functions on multiplicative graphs.
• Employs a graded graph structure where vertices are indexed by integer partitions.
• Derives explicit expressions for harmonic functions using symmetric function identities.
• Relies on the algebraic properties of interpolation polynomials to ensure nonnegativity and harmonicity.
• Computes Selberg-type integrals through the evaluation of certain symmetric function integrals arising in the construction.

Experimental results

Research questions

• RQ1 How can nonnegative harmonic functions be systematically constructed on multiplicative graphs such as the Young lattice?
• RQ2 What is the relationship between interpolation polynomials and harmonic functions in the context of symmetric functions?
• RQ3 How do Schur and Jack symmetric functions contribute to the construction of harmonic functions on graded graphs?
• RQ4 What new Selberg-type integrals emerge from this construction?
• RQ5 What is the role of the infinite symmetric group in the harmonic analysis of such graphs?

Key findings

• The authors construct explicit families of nonnegative harmonic functions on the Young lattice and its generalizations.
• The harmonic functions are derived from interpolation polynomials associated with Schur and Jack symmetric functions.
• The method yields new evaluations of Selberg-type integrals through symmetric function theory.
• The construction provides a direct link between harmonic analysis on the infinite symmetric group and symmetric function identities.
• The interpolation polynomials used are shown to be closed under certain duality and limit operations, ensuring consistency in the harmonic function construction.
• The results extend previous work in representation theory and provide a new algebraic framework for studying harmonic functions on graded graphs.