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Automorphisms and opposition in spherical buildings of exceptional type, IV: The $E_7$ case

Yannick Neyt, James Parkinson, Hendrik Van Maldeghem, Magali Victoor

Abstract

An automorphism of a spherical building is called \textit{domestic} if it maps no chamber onto an opposite chamber. This paper forms a significant part of a large project classifying domestic automorphisms of spherical buildings of exceptional type. In previous work the classifications for $\mathsf{G}_2$, $\mathsf{F}_4$ and $\mathsf{E}_6$ have been completed, and the present work provides the classification for buildings of type $\mathsf{E}_7$. In many respects this case is the richest amongst all exceptional types.

Automorphisms and opposition in spherical buildings of exceptional type, IV: The $E_7$ case

Abstract

An automorphism of a spherical building is called \textit{domestic} if it maps no chamber onto an opposite chamber. This paper forms a significant part of a large project classifying domestic automorphisms of spherical buildings of exceptional type. In previous work the classifications for , and have been completed, and the present work provides the classification for buildings of type . In many respects this case is the richest amongst all exceptional types.
Paper Structure (41 sections, 79 theorems, 109 equations, 2 figures)

This paper contains 41 sections, 79 theorems, 109 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on polar and parapolar spaces
  3. Abstract definitions
  4. Lie incidence geometries
  5. Some parapolar spaces of exceptional type
  6. $\mathsf{E}_{6,1}(\mathbb{K})$
  7. $\mathsf{E}_{7,1}(\mathbb{K})$
  8. $\mathsf{E}_{7,7}(\mathbb{K})$
  9. Types $\mathsf{F}_{4,1}$ and $\mathsf{F}_{4,4}$
  10. Quaternion and octonion Veronese varieties
  11. Opposition
  12. Kangaroos in oriflamme geometries
  13. Kangaroos and general properties
  14. Diligent kangaroos
  15. Kangaroos in $\mathsf{E_{6,1}}(\mathbb{K})$
  16. ...and 26 more sections

Key Result

Theorem 1

Let $\Delta=\mathsf{E}_7(\mathbb{K})$ with $|\mathbb{K}|>2$. Conversely, every domestic automorphism of $\Delta$ fixing no chamber is conjugate to some collineation as in $(i)$, $(ii)$ or $(iii)$ above.

Figures (2)

  • Figure 1: The opposition diagrams of type $\mathsf{E_7}$
  • Figure 2: The Dynkin diagram of $\Delta$ showing the chamber $C$

Theorems & Definitions (159)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 2.2
  • proof
  • Remark 2.11
  • Remark 2.14
  • Definition 3.1
  • Lemma 3.2
  • proof
  • ...and 149 more