The sections of the Weyl group

TL;DR

This paper classifies all sections of finite Weyl groups satisfying braid relations for almost-simple groups over algebraically closed fields, revealing a rich partial order structure and exploring applications.

Contribution

It provides a complete classification of such sections and uncovers their partial order structure, which was not previously known.

Findings

Identified all sections satisfying braid relations in finite Weyl groups.

Established a partial order structure on these sections.

Presented applications of the classification and structure.

Abstract

We compute all sections of the finite Weyl group, that satisfy the braid relations, in the case that G is an almost-simple connected reductive group defined over an algebraically closed field. We then demonstrate that this set of sections has an interesting partially ordered structure, and also give some applications.