Web Diagrams of Cluster Variables for Grassmannian Gr(4,8)

TL;DR

This paper computes web diagrams for quadratic and cubic cluster variables in the Grassmannian Gr(4,8) using hourglass plabic graphs and Lam's method, advancing understanding of cluster algebra invariants.

Contribution

It applies two existing methods to explicitly compute web diagrams and dual webs for cluster variables in Gr(4,8), extending previous work to higher complexity.

Findings

Computed web diagrams for Gr(4,8) cluster variables.

Extended methods to quadratic and cubic cases.

Provided explicit dual webs for these variables.

Abstract

Gaetz, Pechenik, Pfannerer, Striker, and Swanson introduced the concept of hourglass plabic graphs and provided a method for computing web diagrams and invariants corresponding to $4\times n$ Young tableaux, while Elkin, Musiker, and Wright applied Lam's method to explicitly compute the webs compatible with cluster variables in Gr(3,n) and their twists, namely, the preimages of the immanant map introduced by Fraser, Lam, and Le. In this paper, we use these two methods to compute both the web diagrams and the dual webs corresponding to quadratic and cubic cluster variables in the Grassmannian cluster algebra C[Gr(4,8)].