Combinatorics of braid variety cluster structures and its associated geometry

In this talk, I will talk about two separate joint works related to combinatorics of braid variety cluster structures and associated geometry. 

On the first day, I will talk about two combinatorial models that explain cluster structures on braid varieties, independently suggested by Casals–Gorsky–Gorsky–Le–Shen–Simental and Galashin–Lam–Sherman-Bennett–Speyer. The first combinatorial model is called Demazure weaves, which has its origin in symplectic geometry. The second combinatorial model is called a 3D plabic graph. I will review some necessary concepts and constructions about both models. Next, I will explain the T-shift algorithm on a 3D plabic graph, which is a way to extract specific Demazure weaves from a given 3D plabic graph. Such Demazure weaves are called T-shifted weaves. Our method significantly generalizes the known T-shift procedure on plabic graphs for positroid varieties. Lastly, I will mention the symplectic geometry reasoning behind this T-shift procedure. This is a joint work with Roger Casals and Daping Weng.

On the second day, I will talk about a splicing map for double Bott–Samelson varieties, generalizing the previous work of Gorsky–Scroggin on the case of top-dimensional positroid variety. Also, we will identify skew Schubert varieties as a special case of double Bott–Samelson varieties and braid varieties and study the combinatorial aspect of a splicing map for skew Schubert varieties. This is a joint work with Eugene Gorsky, Tonie Scroggin, and José​ Simental.