Two Dualities: Markov and Schur–Weyl

Abstract

Abstract We show that quantum Schur–Weyl duality leads to Markov duality for a variety of asymmetric interacting particle systems. In particular, we consider the following three cases: (1) Using a Schur–Weyl duality between a two-parameter quantum group and a two-parameter Hecke algebra from [6], we recover the Markov self-duality of multi-species ASEP previously discovered in [23] and [3]. (2) From a Schur–Weyl duality between a co-ideal subalgebra of a quantum group and a Hecke algebra of type B [2], we find a Markov duality for a multi-species open ASEP on the semi-infinite line. The duality functional has not previously appeared in the literature. (3) A “fused” Hecke algebra from [15] leads to a new process, which we call braided ASEP. In braided ASEP, up to $m$ particles may occupy a site and up to $m$ particles may jump at a time. The Schur–Weyl duality between this Hecke algebra and a quantum group lead to a Markov duality. The duality function had previously appeared as the duality function of the multi-species ASEP$(q,m/2)$ [23] and the stochastic multi-species higher spin vertex model [24].