# Class groups of open Richardson varieties in the Grassmannian are trivial

## TL;DR

This paper proves that the divisor class group of any open Richardson variety in the Grassmannian is trivial, using Nagata's criterion and properties of Plücker coordinates, applicable over any field or integers.

## Contribution

It establishes the triviality of the divisor class group for open Richardson varieties in the Grassmannian, a result previously unknown.

## Key findings

- Divisor class group of open Richardson varieties is trivial
- Plücker coordinates are prime elements in this context
- Results hold over any field and over the integers

## 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.