Opposition diagrams for automorphisms of small spherical buildings

TL;DR

This paper studies automorphisms of small spherical buildings, introduces opposition diagrams to analyze their structure, and classifies certain automorphisms, especially in buildings with Fano plane residues.

Contribution

It develops opposition diagrams for small spherical buildings with Fano plane residues and classifies domestic automorphisms in specific types.

Findings

Existence of uncapped automorphisms in small buildings.

Introduction of enhanced opposition diagrams.

Complete classification of domestic automorphisms in types F4 and E6.

Abstract

An automorphism $\theta$ of a spherical building $\Delta$ is called \textit{capped} if it satisfies the following property: if there exist both type $J_1$ and $J_2$ simplices of $\Delta$ mapped onto opposite simplices by $\theta$ then there exists a type $J_1\cup J_2$ simplex of $\Delta$ mapped onto an opposite simplex by $\theta$. In previous work we showed that if $\Delta$ is a thick irreducible spherical building of rank at least $3$ with no Fano plane residues then every automorphism of $\Delta$ is capped. In the present work we consider the spherical buildings with Fano plane residues (the \textit{small buildings}). We show that uncapped automorphisms exist in these buildings and develop an enhanced notion of "opposition diagrams" to capture the structure of these automorphisms. Moreover we provide applications to the theory of "domesticity" in spherical buildings, including the complete classification of domestic automorphisms of small buildings of types $\mathsf{F}_4$ and $\mathsf{E}_6$.