On parabolic subgroups of Artin-Tits groups

TL;DR

This paper investigates the structure of parabolic subgroups in Artin-Tits groups, proving a key conjecture in specific cases and providing simplified proofs of classical results, advancing understanding of subgroup intersections.

Contribution

It proves the conjecture that intersections of parabolic subgroups are parabolic in certain cases and shows its equivalence to a weaker conjecture, offering new algebraic proofs.

Findings

Proved the conjecture in a specific case.

Established the equivalence of two conjectures about parabolic subgroups.

Provided simplified algebraic proofs of classical results.

Abstract

Abstract. We address the conjecture which states that an intersection of parabolic subgroups of an Artin-Tits group is a parabolic subgroup. We prove that the conjecture is equivalent to a, a priori, weaker conjecture. We also prove the conjecture in a specific case. Along the way, we provide short and almost self-contain algebraical proofs of several classical results on Artin-Tits groups, such as those of Van der Lek on intersection of standard parabolic subgroups.