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Springer categories for regular centralizers in well-generated complex braid groups

Owen Garnier

TL;DR

The paper develops a comprehensive Garside-theoretic framework for Springer categories, extending dual braid monoid ideas to centralizers of Springer regular elements in well-generated complex reflection groups. It defines and analyzes periodic/divided Garside structures, describes how braided reflections correspond to atomic loops inside Springer groupoids, and proves a Hurwitz-type presentation for Springer categories. The work yields a pure Garside-theoretic proof of centralizer/center results for finite-index subgroups in complex braid groups, and supplies explicit presentations for the complex braid group B(G_{31}) via Reidemeister–Schreier-type techniques in the groupoid setting. It also relates Springer categories to Lyashko–Looijenga data and noncrossing partitions, and provides concrete orbit-based presentations that corroborate and extend known conjectures for G_{31}. Overall, the paper broadens the toolkit for understanding braid groups of centralizers beyond well-generated cases, with concrete computational realizations and new structural results about centers and conjugacy within these groups.

Abstract

In his proof of the K(pi,1) conjecture for complex reflection arrangements, Bessis defined Garside categories suitable for studying braid groups of centralizers of Springer regular elements in well-generated complex reflection groups. We provide a detailed study of these categories, which we call Springer categories. We describe in particular the conjugacy of braided reflections of regular centralizer in terms of the Garside structure of the associated Springer category. In so doing we obtain a pure Garside theoretic proof of a theorem of Digne, Marin and Michel on the center of finite index subgroups in complex braid groups in the case of a regular centralizer in a well-generated group. We also provide a "Hurwitz-like" presentation of Springer categories. To this aim we provide additional insights on noncrossing partitions in the infinite series. Lastly, we use this "Hurwitz-like" presentation, along with a generalized Reidemeister-Schreier method we introduce for groupoids, to deduce nice presentations of the complex braid group B(G31).

Springer categories for regular centralizers in well-generated complex braid groups

TL;DR

The paper develops a comprehensive Garside-theoretic framework for Springer categories, extending dual braid monoid ideas to centralizers of Springer regular elements in well-generated complex reflection groups. It defines and analyzes periodic/divided Garside structures, describes how braided reflections correspond to atomic loops inside Springer groupoids, and proves a Hurwitz-type presentation for Springer categories. The work yields a pure Garside-theoretic proof of centralizer/center results for finite-index subgroups in complex braid groups, and supplies explicit presentations for the complex braid group B(G_{31}) via Reidemeister–Schreier-type techniques in the groupoid setting. It also relates Springer categories to Lyashko–Looijenga data and noncrossing partitions, and provides concrete orbit-based presentations that corroborate and extend known conjectures for G_{31}. Overall, the paper broadens the toolkit for understanding braid groups of centralizers beyond well-generated cases, with concrete computational realizations and new structural results about centers and conjugacy within these groups.

Abstract

In his proof of the K(pi,1) conjecture for complex reflection arrangements, Bessis defined Garside categories suitable for studying braid groups of centralizers of Springer regular elements in well-generated complex reflection groups. We provide a detailed study of these categories, which we call Springer categories. We describe in particular the conjugacy of braided reflections of regular centralizer in terms of the Garside structure of the associated Springer category. In so doing we obtain a pure Garside theoretic proof of a theorem of Digne, Marin and Michel on the center of finite index subgroups in complex braid groups in the case of a regular centralizer in a well-generated group. We also provide a "Hurwitz-like" presentation of Springer categories. To this aim we provide additional insights on noncrossing partitions in the infinite series. Lastly, we use this "Hurwitz-like" presentation, along with a generalized Reidemeister-Schreier method we introduce for groupoids, to deduce nice presentations of the complex braid group B(G31).
Paper Structure (38 sections, 85 theorems, 164 equations, 5 algorithms)

This paper contains 38 sections, 85 theorems, 164 equations, 5 algorithms.

Table of Contents

  1. Preliminaries on Garside categories
  2. Definitions
  3. Normal forms, groupoid and conjugacy
  4. Periodic elements and divided categories
  5. The case $p=1$
  6. Complex reflection groups, braid groups and braid groupoids
  7. Reminders on complex reflection groups and their braid groups
  8. Definitions, basic invariants
  9. Braid groups, braided reflections
  10. Regular elements, regular braids
  11. Dual braid monoid for well-generated groups
  12. Lyashko-Looijenga map, cyclic labels
  13. Regular numbers and topological groupoids
  14. Springer categories which are monoids
  15. The case $W=G(1,1,n+1)$ for $n\geqslant 5$
  16. ...and 23 more sections

Key Result

Theorem 1

Let $u$ be an object of $\mathcal{C}$. Any atomic loop inside $\mathcal{C}(u,u)$ is a braided reflection of the group $B(W_g)\simeq\mathcal{G}(u,u)$. Conversely, any braided reflection $\sigma\in B(W_g)\simeq \mathcal{G}(u,u)$ is conjugate in the Springer groupoid to some atomic loop.

Theorems & Definitions (188)

  • Theorem 1: Theorem \ref{['prop:braidreflectionconjugatetoatomicloop']}
  • Theorem 2: Theorem \ref{['theo:conj_of_ato_loops']}
  • Corollary 3
  • Corollary 4: Corollary \ref{['cor:center_finite_index_subgroups']}
  • Theorem 5: Theorem \ref{['prop:hurwitzpresentationforc31']}
  • Proposition 6: Reidemeister-Schreier method for groupoids, Proposition \ref{['prop:reidemeister_schreier']}
  • Theorem 7: Theorem \ref{['theo:isomorphism_presentation_b31']} and Section \ref{['sec:4.3.1']}
  • Lemma 1.1
  • proof
  • Definition 1.2
  • ...and 178 more