Comparing cluster algebras on braid varieties

TL;DR

This paper demonstrates that two independently constructed cluster algebra structures on braid varieties are actually the same, unifying different approaches and deepening understanding of their algebraic and geometric properties.

Contribution

It proves the equivalence of two cluster algebra constructions on braid varieties, connecting weave-based and Deodhar geometry approaches.

Findings

The two cluster algebra structures coincide.

The study unifies combinatorial and geometric perspectives.

Results apply to various classes of varieties including Bruhat and positroid varieties.

Abstract

Braid varieties parametrize linear configurations of flags with transversality conditions dictated by positive braids. They include and generalize reduced double Bruhat cells, positroid varieties, open Bott-Samelson varieties, and Richardson varieties, among others. Recently, two cluster algebra structures were independently constructed in the coordinate rings of braid varieties: one using weaves and the other using Deodhar geometry. The main result of the article is that these two cluster algebras coincide. More generally, our comparative study matches the different concepts and results from each approach to the other, both on the combinatorial and algebraic geometric aspects.