On the minimal positive standardizer of a parabolic subgroup of an Artin-Tits group

TL;DR

This paper introduces an algorithm for computing the minimal standardizer of a parabolic subgroup in Artin-Tits groups, generalizing a geometric problem related to braid groups to an algebraic context.

Contribution

It provides a geometric algorithm for the minimal standardizer and extends the concept to algebraic structures of Artin-Tits groups, linking it to the $pn$-normal form.

Findings

Algorithm effectively computes the minimal standardizer.

Generalization from geometric to algebraic setting.

Connection established between standardizer and $pn$-normal form.

Abstract

The minimal standardizer of a curve system on a punctured disk is the minimal braid that transforms it into a system formed only by round curves. We give an algorithm to compute it in a geometrical way. Then, we generalize this problem algebraically to parabolic subgroups of Artin-Tits groups of spherical type and we show that, to compute the minimal standardizer of a parabolic subgroup, it suffices to compute the $pn$-normal form of a particular central element.