Dimers, webs, and local systems

TL;DR

This paper explores the relationship between determinants of Kasteleyn matrices and multiwebs in planar bipartite graphs with SL_n-local systems, providing new insights into their enumeration and properties.

Contribution

It establishes a novel connection between Kasteleyn matrix determinants and n-multiwebs, extending web enumeration to graphs with SL_n-local systems.

Findings

Determinant of Kasteleyn matrix counts n-multiwebs with web-trace weights

Provides methods to study random n-multiwebs on simple surfaces

Extends web enumeration theory to SL_n-local systems

Abstract

For a planar bipartite graph $\mathcal G$ equipped with a $\mathrm{SL}_n$-local system, we show that the determinant of the associated Kasteleyn matrix counts "$n$-multiwebs" (generalizations of $n$-webs) in $\mathcal G$, weighted by their web-traces. We use this fact to study random $n$-multiwebs in graphs on some simple surfaces.