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Minimal generating set of cactus groups

Eddy Godelle

TL;DR

The paper studies cactus groups $C(W,S)$ associated with Coxeter systems and proves that for non-commutative $W$ the lower central series does not stabilize. It develops cross-section and transversal-section frameworks to obtain minimal presentations with generators from a cross section, and provides complete presentations for finite irreducible Coxeter groups not of type $E$, with explicit constructions for types $A_n,B_n,D_n,I_n,F_4,H_3,H_4$ and no transversal section in type $E$. It also analyzes abelianisation, showing $C(W,S)^{ab} \cong \mathbb{Z}_2^m$ where $m$ counts equivalence classes of $\mathcal{F}(W,S)$ under $\equiv$, and demonstrates that non-Abelian $W$ yields a $\mathbb{Z}_2*\mathbb{Z}_2$ quotient, contributing to the non-stabilization result. The results yield minimal generating sets and explicit presentations for most finite irreducible Coxeter types, facilitating connections to representation theory and potential links to Calogero-Moser spaces and Kazhdan-Lusztig cells.

Abstract

We prove that the lower central series of the cactus group associated with a non commutative Coxeter group never stabilizes. We also compute a minimal presentation in terms of generators for the cactus group associated with a finite Coxeter groups, except in type E.

Minimal generating set of cactus groups

TL;DR

The paper studies cactus groups associated with Coxeter systems and proves that for non-commutative the lower central series does not stabilize. It develops cross-section and transversal-section frameworks to obtain minimal presentations with generators from a cross section, and provides complete presentations for finite irreducible Coxeter groups not of type , with explicit constructions for types and no transversal section in type . It also analyzes abelianisation, showing where counts equivalence classes of under , and demonstrates that non-Abelian yields a quotient, contributing to the non-stabilization result. The results yield minimal generating sets and explicit presentations for most finite irreducible Coxeter types, facilitating connections to representation theory and potential links to Calogero-Moser spaces and Kazhdan-Lusztig cells.

Abstract

We prove that the lower central series of the cactus group associated with a non commutative Coxeter group never stabilizes. We also compute a minimal presentation in terms of generators for the cactus group associated with a finite Coxeter groups, except in type E.
Paper Structure (10 sections, 9 theorems, 13 equations)

This paper contains 10 sections, 9 theorems, 13 equations.

Table of Contents

  1. Abelinanisation
  2. Cross section
  3. Cross sections for finite irreducible Coxeter groups
  4. Type $A_n$
  5. Type $B_n$
  6. Type $D_n$
  7. Type $E_6$, $E_7$, $E_8$
  8. Type $I_n$
  9. Type $F_4$
  10. Type $H_3$ et $H_4$

Key Result

Proposition 2

Let $(W,S)$ be a Coxeter system. Consider the equivalence relation $\equiv$ on $\mathcal{F}(W,S)$ defined as the transitive closure of the binary relation $\equiv_0$ defined by $Y\equiv_0 Z$ is there exists $X$ so that $Z = \omega_X\,Y\, \omega_X$ in $W$. Denote by $m$ the number of equivalent class

Theorems & Definitions (19)

  • Definition 1
  • Proposition 2
  • Theorem 3
  • Definition 4
  • Proposition 5
  • Theorem 6
  • proof : Proof of Proposition \ref{['ThmA3']}
  • proof : Proof of Theorem \ref{['ThmBC']}
  • Lemma 2.1
  • proof
  • ...and 9 more