Comparative Analyses of the Type D ASEP: Stochastic Fusion and Crystal Bases

TL;DR

This paper compares probabilistic and algebraic approaches to the Type D ASEP, revealing that different methods produce distinct processes and highlighting limitations of previous generalizations to Lie algebras.

Contribution

It introduces stochastic fusion for Type D ASEP and constructs a fused system using crystal bases and Casimir elements, showing these approaches yield different processes.

Findings

Stochastic fusion affects generator matrices and stationary distributions.

Algebraic construction using $U_q(so_6)$ produces a different process.

Previous ASEP analysis does not extend to all finite-dimensional simple Lie algebras.

Abstract

The Type D asymmetric simple exclusion process (ASEP) is a particle system involving two classes of particles that can be viewed from both a probabilistic and an algebraic perspective (arXiv:2011.13473). From a probabilistic perspective, we perform stochastic fusion on the Type D ASEP and analyze the outcome on generator matrices, limits of drift speed, stationary distributions, and Markov self-duality. From an algebraic perspective, we construct a fused Type D ASEP system from a Casimir element of $U_q(so_6)$, using crystal bases to analyze and manipulate various representations of $U_q(so_6)$. We conclude that both approaches produce different processes and therefore the previous method of arXiv:1908.02359, which analyzed the usual ASEP, does not generalize to all finite-dimensional simple Lie algebras.