Twists, Higher Dimer Covers, and Web Duality for Grassmannian Cluster Algebras

TL;DR

This paper introduces a twisted boundary measurement map for Grassmannian cluster algebras, extending web duality and providing Laurent expansions for twisted web immanants, with implications for cluster variable classification.

Contribution

It develops a new twisted boundary measurement map using face weights, extends web duality to larger cases, and connects these to cluster variable formulas and classifications.

Findings

Web duality holds for many SL3 and SL4 webs.

The new map recovers and extends formulas for cluster variable twists.

Evidence supports conjectures on cluster variable classification in Grassmannians.

Abstract

We study a twisted version of Fraser, Lam, and Le's higher boundary measurement map, using face weights instead of edge weights, thereby providing Laurent polynomial expansions, in Pl\"ucker coordinates, for twisted web immanants for Grassmannians. In some small cases, Fraser, Lam, and Le observe a phenomenon they call "web duality'', where web immanants coincide with web invariants, and they conjecture that this duality corresponds to transposing the standard Young tableaux that index basis webs. We show that this duality continues to hold for a large set of $\text{SL}_3$ and $\text{SL}_4$ webs. Combining this with our twisted higher boundary measurement map, we recover and extend formulas of Elkin-Musiker-Wright for twists of certain cluster variables. We also provide evidence supporting conjectures of Fomin-Pylyavskyy as well as one by Cheung-Dechant-He-Heyes-Hirst-Li concerning classification of cluster variables of low Pl\"ucker degree in $\mathbb{C}[\text{Gr}(3,n)]$.