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

Abstract

The Type D asymmetric simple exclusion process (ASEP), a particle system involving two classes of particles, was studied from both probabilistic and algebraic perspectives by Kuan et al. (2022). From a probabilistic perspective, we perform stochastic fusion on the Type D ASEP introduced by Kuan (2019) and analyze the outcome on the generator matrix, 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 $\mathcal{U}_q(\mathfrak{so}_6)$, using crystal bases to analyze and manipulate various representations of $\mathcal{U}_q(\mathfrak{so}_6)$. We know that the same generators for the normal ASEP are produced by stochastic fusion and by a suitable ground state transformation on a central element of $\mathcal{U}_q(\mathfrak{sl}_2)$ in a symmetric tensor product representation. However, in the case of the Type D ASEP, we find that the probabilistic and algebraic generator matrices are not the same and thus represent different processes. We conclude that the relationship between stochastic fusion and a ground state transformation (specifically, what we term the Type A ground state transformation) established by Kuan (2019) does not generalize to all finite-dimensional simple Lie algebras.