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

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 Plücker 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 Plücker degree in $\mathbb{C}[\text{Gr}(3,n)]$.

💡 Research Summary

The paper “Twists, Higher Dimer Covers, and Web Duality for Grassmannian Cluster Algebras” develops a twisted version of the higher boundary measurement map introduced by Fraser, Lam, and Lee (FLL). The authors replace edge weights with face weights on plabic graphs, which allows them to express twisted web immanants as Laurent polynomial expansions in Plücker coordinates. Their main technical achievement is Theorem 3.4, which shows that for any element f in the coordinate ring C