Class groups of open Richardson varieties in the Grassmannian are trivial

Abstract

We prove that the divisor class group of any open Richardson variety in the Grassmannian is trivial. Our proof uses Nagata's criterion, localizing the coordinate ring at a suitable set of Pl\"ucker coordinates. We prove that these Pl\"ucker coordinates are prime elements by showing that the subscheme they define is an open subscheme of a positroid variety. Our results hold over any field and over the integers.