Pure Braid Group Presentations via Longest Elements

TL;DR

This paper introduces a new simplified Coxeter-theoretic presentation of the pure braid group using generators from longest elements, with relations characterized by commutators and specific palindromic relations, and explores its limitations.

Contribution

It provides a novel, simplified pure braid group presentation based on Coxeter theory, and addresses the generation limitations for Coxeter arrangements.

Findings

New presentation of pure braid group using longest elements

Relations are only commutators or palindromic length 5 box relations

The set does not generate for all Coxeter arrangements

Abstract

This paper gives a new, simplified presentation of the classical pure braid group. The generators are given by the squares of the longest elements over connected subgraphs, and we prove that the only relations are either commutators or certain palindromic length 5 box relations. This presentation is motivated by twist functors in algebraic geometry, but the proof is entirely Coxeter-theoretic. We also prove that the analogous set does not generate for all Coxeter arrangements, which in particular answers a question of Donovan and Wemyss.