Cluster structures on braid varieties

TL;DR

This paper establishes cluster algebra structures on braid varieties associated with any simple Lie group, using weave calculus and tropicalization, and proves several properties including their equality, acyclicity, and automorphisms.

Contribution

It introduces explicit cluster structures on braid varieties, confirming conjectures and providing computational tools for their analysis.

Findings

Cluster $ ext{A}$-structures and Poisson structures are constructed on braid varieties.

Upper cluster algebras are shown to equal their cluster algebras.

The paper explicitly determines DT-transformations as twist automorphisms.

Abstract

We show the existence of cluster $\mathcal{A}$-structures and cluster Poisson structures on any braid variety, for any simple Lie group. The construction is achieved via weave calculus and a tropicalization of Lusztig's coordinates. Several explicit seeds are provided and the quiver and cluster variables are readily computable. We prove that these upper cluster algebras equal their cluster algebras, show local acyclicity, and explicitly determine their DT-transformations as the twist automorphisms of braid varieties. The main result also resolves the conjecture of B. Leclerc on the existence of cluster algebra structures on the coordinate rings of open Richardson varieties.