Higher Dimers, SL_3 and SL_4 Webs and Grassmannian Cluster Algebras.
Presenter
December 12, 2025
Abstract
Two classically studied rings in algebraic geometry and representation theory are the homogeneous coordinate ring of the Grassmannian of k-planes in n-space and the ring of SL_r tensor invariants. Both admit rich combinatorial models—in particular, the focus of this workshop: webs. The rings of SL_3 and SL_4 tensor invariants possess rotation-invariant web bases, first introduced by Kuperberg for SL_3 and later extended to SL_4 by Gaetz–Pechenik–Pfannerer–Striker–Swanson. We use these web bases to explore the cluster algebra structure of the homogeneous coordinate ring of the Grassmannian. Specifically, we study the generators of our cluster algebras via SL_3 and SL_4 webs as well as higher dimer covers on certain planar bicolored graphs. By connecting these two web bases, we derive combinatorial formulas for the generators using a phenomenon called web duality, first observed in small cases by Fraser–Lam–Le. We further show that, for large families of and webs, this web duality can be realized through the transpose of standard Young tableaux, which index the basis webs. In the talk, we will present this web duality using explicit examples and focus on the cluster algebraic applications of this phenomenon.