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Braid variety cluster structures, II: general type

Pavel Galashin, Thomas Lam, Melissa Sherman-Bennett

TL;DR

This work proves that braid varieties for any complex simple algebraic group $G$ carry a cluster algebra structure, with coordinate rings ${\mathbb C}[\accentset{\circ}{R}_\beta]$ isomorphic to ${\mathcal A}(\Sigma_\beta)$ for seeds built from Deodhar geometry. Central to the construction are the Deodhar torus $T_\beta$, cluster variables $x_c$ tied to vanishing orders on Deodhar hypersurfaces, and a 2-form $\omega_\beta$ encoding the exchange matrix. The authors develop a deletion-contraction induction and study double braid moves, showing seeds mutate (or are quasi-equivalent) along moves, which yields the cluster structure for all braid varieties in simply-laced types and, via folding, for multiply-laced types. Folding together with the double-braid move analysis allows a unified treatment across all simple groups, and a combinatorial algorithm (implemented) computes the seeds purely from root-system data. The results imply open Richardson varieties and related strata inherit cluster structures and exhibit a curious Lefschetz property in cohomology, highlighting deep connections between geometric, combinatorial, and representation-theoretic aspects of braid varieties.

Abstract

We show that braid varieties for any complex simple algebraic group $G$ are cluster varieties. This includes open Richardson varieties inside the flag variety $G/B$.

Braid variety cluster structures, II: general type

TL;DR

This work proves that braid varieties for any complex simple algebraic group carry a cluster algebra structure, with coordinate rings isomorphic to for seeds built from Deodhar geometry. Central to the construction are the Deodhar torus , cluster variables tied to vanishing orders on Deodhar hypersurfaces, and a 2-form encoding the exchange matrix. The authors develop a deletion-contraction induction and study double braid moves, showing seeds mutate (or are quasi-equivalent) along moves, which yields the cluster structure for all braid varieties in simply-laced types and, via folding, for multiply-laced types. Folding together with the double-braid move analysis allows a unified treatment across all simple groups, and a combinatorial algorithm (implemented) computes the seeds purely from root-system data. The results imply open Richardson varieties and related strata inherit cluster structures and exhibit a curious Lefschetz property in cohomology, highlighting deep connections between geometric, combinatorial, and representation-theoretic aspects of braid varieties.

Abstract

We show that braid varieties for any complex simple algebraic group are cluster varieties. This includes open Richardson varieties inside the flag variety .
Paper Structure (42 sections, 49 theorems, 105 equations, 2 tables)

This paper contains 42 sections, 49 theorems, 105 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Double braid varieties
  3. Cluster variables and Deodhar geometry
  4. Exchange matrix
  5. Deletion-contraction induction
  6. Deodhar geometry
  7. Background
  8. Weighted flags
  9. Distinguished subexpressions
  10. Double braid varieties and the Deodhar torus
  11. $H$-valued functions
  12. Grid and chamber minors
  13. Almost positive sequences and Deodhar hypersurfaces
  14. Cluster variables
  15. A two-form on the braid variety, and a seed
  16. ...and 27 more sections

Key Result

Theorem 1.1

The coordinate ring of $\accentset{\circ}{R}_\beta$ is isomorphic to the cluster algebra ${\mathcal{A}}_{\beta} = {\mathcal{A}}(\Sigma_\beta)$.

Theorems & Definitions (110)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Lemma 2.7
  • ...and 100 more