Cluster structures on braid varieties

Given a simple algebraic group \(G\) and an element \(\beta\) of its positive braid monoid we consider an affine, smooth algebraic variety \(X(\beta)\) that generalizes some well-known varieties in Lie theory, including open Richardson varieties and double Bott-Samelson cells. In this talk, we will construct a cluster algebra structure on the coordinate ring of \(X(\beta)\) using combinatorial objects called algebraic weaves and tropicalization of Lusztig’s coordinates. We will also give properties of this cluster structure, including local acyclicity and the existence of reddening sequences. This is based on joint work with R. Casals, E. Gorsky, M. Gorsky, L. Shen and I. Le.