Groups of Projectivities and Levi Subgroups in Spherical Buildings of Simply Laced Type

TL;DR

This paper studies projectivity groups in spherical buildings of simply laced type, revealing their structure, relation to Levi subgroups, and conditions for their coincidence, with special cases for exceptional types.

Contribution

It introduces and analyzes special and general projectivity groups, linking them to Levi subgroups and providing criteria for their equality in spherical buildings.

Findings

Permutation groups of projectivity groups are determined for irreducible residues.

Special projectivity groups encode Levi subgroup actions on residues.

Conditions for coincidence of special and general projectivity groups are established.

Abstract

We introduce the special and general projectivity groups attached to a simplex $F$ of a thick irreducible spherical building of simply laced type. If the residue of $F$ is irreducible, we determine the permutation group of both projectivity groups of $F$, acting on the residue of $F$ and show that the special projectivity group determines the precise action of the Levi subgroup of a parabolic subgroup on the corresponding residue. This reveals three special cases for the exceptional types $\mathsf{E_6,E_7,E_8}$. Furthermore, we establish a general diagrammatic rule to decide when exactly the special and general projectivity groups of $F$ coincide.