Orthogonal polynomial duality of a two-species asymmetric exclusion process

TL;DR

This paper studies a two-species asymmetric exclusion process, proving its self-duality using orthogonal polynomials and clarifying earlier constructions related to quantum groups.

Contribution

It establishes the self-duality of a specific two-species ASEP with respect to q-Krawtchouk polynomials and refines the construction of the process from quantum group representations.

Findings

Proves self-duality of type D ASEP for certain cases

Connects the process to orthogonal polynomials (q-Krawtchouk)

Provides a more efficient argument for non-negativity conditions

Abstract

We examine type D ASEP, a two--species interacting particle system which generalizes the usual asymmetric simple exclusion process. For certain cases of type D ASEP, the process does not give priority for one species over another, even though there is nontrivial interaction between the two species. For those specific cases, we prove that the type D ASEP is self--dual with respect to an independent product of $q$--Krawtchouk polynomials. The type D ASEP was originally constructed in arXiv:2011.13473, using the type D quantum groups $\mathcal{U}_q(\mathfrak{so}_6)$ and $\mathcal{U}_q(\mathfrak{so}_8)$. That paper claimed that certain states needed to be "discarded'' in order to ensure non--negativity. Here, we also provide a more efficient argument for the same claim.