SIGMA 22 (2026), 095, 52 pages      arXiv:2407.21015      https://doi.org/10.3842/SIGMA.2026.095

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

Erik Brodsky ^a, Eva Engel ^b, Connor Panish ^c and Lillian Stolberg ^d
a) Department of Mathematics, Cornell University, Ithaca, NY, USA
^b) Department of Mathematics, Princeton University, Princeton, NJ, USA
^c) Department of Statistics, Cornell University, Ithaca, NY, USA
^d) Department of Mathematics, University of Michigan, Ann Arbor, MI, USA

Received February 11, 2025, in final form September 02, 2026; Published online October 02, 2026

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.

Key words: asymmetric simple exclusion process; quantum groups; stochastic fusion; Markov self-duality; crystal bases; Young tableaux.

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