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Representations of Hecke algebras and Markov dualities for interacting particle systems

Alexander PovolotskyPavel PyatovRoger TribeBruce WestburyOleg Zaboronski
Dec 2022
摘要
Many continuous reaction-diffusion models on $\mathbb{Z}$ (annihilating orcoalescing random walks, exclusion processes, voter models) admit a rich set ofMarkov duality functions which determine the single time distribution. A commonfeature of these models is that their generators are given by sums of two-siteidempotent operators. In this paper, we classify all continuous time Markovprocesses on $\{0,1\}^{\mathbb{Z}}$ whose generators have this property,although to simplify the calculations we only consider models with equal leftand right jumping rates. The classification leads to six familiar models andthree exceptional models. The generators of all but the exceptional models turnout to belong to an infinite dimensional Hecke algebra, and the dualityfunctions appear as spanning vectors for small-dimensional irreduciblerepresentations of this Hecke algebra. A second classification exploresgenerators built from two site operators satisfying the Hecke algebrarelations. The duality functions are intertwiners between configuration andco-ordinate representations of Hecke algebras, which results in a novelco-ordinate representations of the Hecke algebra. The standard Baxterisationprocedure leads to new solutions of the Young-Baxter equation corresponding toparticle systems which do not preserve the number of particles.
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