Purity of Weak Separation on Surfaces

Weak separation is a combinatorial condition describing cluster compatibility in certain important cluster algebras, notably in the coordinate ring of the Grassmannian. Oh, Speyer, and Postnikov gave a beautiful proof of the purity of the weak separation complex in this case. I will explain how one can formulate a version of weak separation that lives on a cylinder rather than on a disk. Conjecturally, the corresponding simplicial complexes are also pure.