Cluster structures on braid varieties

Let G be a semisimple group over C. Let $\beta$ be a positive braid whose Demazure product is the longest Weyl group element. The braid variety $M(\beta)$ generalizes many well known varieties, including positroid cells, open Richardson varieties, and double Bruhat cells. We provide a concrete construction of the cluster structures on $M(\beta)$, using the weaves of Casals and Zaslow and a new combinatorial construction called Lusztig cycles. We show that the coordinate ring of $M(\beta)$ is a cluster algebra, which confirms a conjecture of Leclerc as special cases. As an application, we show that $M(\beta)$ admits a natural Poisson structure and can be further quantized. This talk is based on joint work with Roger Casals, Eugene Gorsky, Mikhail Gorsky, Ian Le, and Jose Simental.