Comparative Analyses of the Type D ASEP: Stochastic Fusion and Crystal Bases
- 1Cornell University, USA
- 2Princeton University, USA
- 3University of Michigan, USA
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 𝒰q(so6), using crystal bases to analyze and manipulate various representations of 𝒰q(so6). 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 𝒰q(sl2) 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.
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BibTeX
@article{brodsky2026comparative,
title = {{Comparative Analyses of the Type D ASEP: Stochastic Fusion and Crystal Bases}},
author = {Erik Brodsky and Eva Engel and Connor Panish and Lillian Stolberg},
journal = {Symmetry, Integrability and Geometry: Methods and Applications},
year = {2026},
month = oct,
doi = {10.3842/sigma.2026.095},
url = {https://sigma-journal.com/2026/095/},
note = {CC BY-SA 4.0},
}
RIS
TY - JOUR
TI - Comparative Analyses of the Type D ASEP: Stochastic Fusion and Crystal Bases
AU - Erik Brodsky
AU - Eva Engel
AU - Connor Panish
AU - Lillian Stolberg
PY - 2026
DA - 2026/10/02
JO - Symmetry, Integrability and Geometry: Methods and Applications
DO - 10.3842/sigma.2026.095
UR - https://sigma-journal.com/2026/095/
AB - 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 𝒰_q(so_6), using crystal bases to analyze and manipulate various representations of 𝒰_q(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 𝒰_q(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.
SN - 1815-0659
ER -