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Parabolic subgroups of complex braid groups: the remaining case

Owen Garnier

Abstract

Recently, Marin and González-Meneses introduced a class of ``parabolic'' subgroups for generalized braid groups associated to arbitrary complex reflection groups. Using notably Garside group structures on these generalized braid groups, they proved general results on parabolic subgroups for all cases but one. This last case is that of the complex braid group B(G31), which has no Garside group structure known so far, but instead a Garside groupoid structure. Using this Garside groupoid structure, we complete the results of Marin and González-Meneses by proving their main theorems for the parabolic subgroups of the complex braid group B(G31).

Parabolic subgroups of complex braid groups: the remaining case

Abstract

Recently, Marin and González-Meneses introduced a class of ``parabolic'' subgroups for generalized braid groups associated to arbitrary complex reflection groups. Using notably Garside group structures on these generalized braid groups, they proved general results on parabolic subgroups for all cases but one. This last case is that of the complex braid group B(G31), which has no Garside group structure known so far, but instead a Garside groupoid structure. Using this Garside groupoid structure, we complete the results of Marin and González-Meneses by proving their main theorems for the parabolic subgroups of the complex braid group B(G31).
Paper Structure (17 sections, 63 theorems, 48 equations)

This paper contains 17 sections, 63 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Parabolic subgroupoids of Springer groupoids
  3. Complex braid groups, regular braids
  4. Dual braid monoid
  5. Lyashko-Looijenga maps, Springer groupoids
  6. Topological definition of Springer groupoids
  7. Parabolic subgroups and local braid groupoids
  8. Standard parabolic subgroupoids
  9. Garside definition of Springer groupoids
  10. Garside standard parabolic subgroupoids
  11. Correspondence between standard parabolic subgroupoids and relative local braid groupoids
  12. The particular case of $B(G_{31})$
  13. Diagrams for $B(G_{31})$
  14. Swaps, recurrent elements and positive conjugators
  15. Existence of parabolic closure for $B(G_{31})$
  16. ...and 2 more sections

Key Result

Theorem 1

(Theorem theo:parabolic_closure) For every element $x\in B(G_{31})$, there exists a unique minimal parabolic subgroup $\mathop{\mathrm{PC}}\nolimits(x)$ of $B(G_{31})$ containing it, and we have $\mathop{\mathrm{PC}}\nolimits(x^m)=\mathop{\mathrm{PC}}\nolimits(x)$ for every $m\neq 0$.

Theorems & Definitions (129)

  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Theorem 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Lemma 2.5
  • ...and 119 more