Variations on a theme of Kasteleyn, with application to the totally nonnegative Grassmannian

TL;DR

This paper offers a concise proof of a classical Kasteleyn result and its variants, which are crucial for parametrizing positroid varieties in the study of the totally nonnegative Grassmannian.

Contribution

It presents a new, simplified proof of Kasteleyn's theorem and its variants, enhancing understanding of positroid parametrization.

Findings

Provided a short proof of Kasteleyn's classical result

Proved several variants of Kasteleyn's theorem

Applied results to the parametrization of positroid varieties

Abstract

We provide a short proof of a classical result of Kasteleyn, and prove several variants thereof. One of these results has become key in the parametrization of positroid varieties, and thus deserves the short direct proof which we provide.