Quantum algebra symmetry of the ASEP with second-class particles

TL;DR

This paper explores the quantum algebra symmetry of a two-component ASEP with second-class particles, constructing algebraic representations, proving reversibility, and explicitly deriving the reversible measure.

Contribution

It introduces the construction of $U_q[\mathfrak{gl}(3)]$ algebra representations for the ASEP with second-class particles and provides explicit reversible measures.

Findings

Constructed $U_q[\mathfrak{gl}(3)]$ representations commuting with the generator

Proved the process is reversible

Derived explicit reversible measure

Abstract

We consider a two-component asymmetric simple exclusion process (ASEP) on a finite lattice with reflecting boundary conditions. For this process, which is equivalent to the ASEP with second-class particles, we construct the representation matrices of the quantum algebra $U_q[\mathfrak{gl}(3)]$ that commute with the generator. As a byproduct we prove reversibility and obtain in explicit form the reversible measure. A review of the algebraic techniques used in the proofs is given.