Combinatorics & algebra seminar-Webs and Grassmannian Cluster Algebras

Speaker: Kayla Wright, JHU
Date and time: Wednesday, November 5, 4–5 pm
Place: Phillips 108

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, planar diagrams known as webs. The rings of SL₃ and SL₄ tensor invariants possess rotation-invariant web bases, first introduced by Kuperberg for SL₃ and later extended to SL₄ 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 SL₃ and SL₄ 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.