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

Abstract

We obtain orthogonal polynomial self-duality functions for the 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 𝔰𝔩ₙ₊₁ and the Heisenberg Lie algebra 𝔥ₙ, 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.