Recent work in mathematics and physics linked reaction-diffusion processes with integrable quantum spin chains. Specifically, the generator of the asymmetric simple exclusion process is the sum of operators that satisfy Hecke algebra relations. We explore whether Hamiltonians defined on Hecke algebras and their deformations can lead to generators of solid-on-solid models, models that as compared to traditional reaction-diffusion processes have an extra conservation law: the total number of particles is unchanged.