Orthogonal Polynomial Stochastic Duality Functions for Multi-Species SEP$(2j)$ and Multi-Species IRW

TL;DR

This paper constructs orthogonal polynomial duality functions for multi-species symmetric exclusion and independent random walk processes on finite graphs, using Lie algebra representations, leading to explicit polynomial families as duality functions.

Contribution

It introduces a novel algebraic approach to derive orthogonal duality functions for multi-species processes via Lie algebra intertwiners.

Findings

Derived multivariate Krawtchouk polynomials as duality functions for multi-species SEP(2j).

Obtained Charlier polynomial products as duality functions for multi-species IRW.

Established a Lie algebraic framework connecting duality functions to representation theory.

Abstract

We obtain orthogonal polynomial self-duality functions for multi-species version of the symmetric exclusion process (SEP$(2j)$) and the independent random walker process (IRW) on a finite undirected graph. In each process, we have $n>1$ species of particles. In addition, we allow up to $2j$ particles to occupy each site in the multi-species SEP$(2j)$. The duality functions for the multi-species SEP$(2j)$ and the multi-species IRW come from unitary intertwiners between different $*$-representations of the special linear Lie algebra $\mathfrak{sl}_{n+1}$ and the Heisenberg Lie algebra $\mathfrak{h}_n$, respectively. The analysis leads to multivariate Krawtchouk polynomials as orthogonal duality functions for the multi-species SEP$(2j)$ and homogeneous products of Charlier polynomials as orthogonal duality functions for the multi-species IRW.