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Uniclass automorphisms of spherical buildings

Yannick Neyt, James Parkinson, Hendrik Van Maldeghem

Abstract

An automorphism of a building is called uniclass if the Weyl distance between any chamber and its image lies in a single (twisted) conjugacy class of the Coxeter group. In this paper we characterise uniclass automorphisms of spherical buildings in terms of their fixed structure. For this purpose we introduce the notion of a Weyl substructure in a spherical building. We also link uniclass automorphisms to the Freudenthal--Tits magic square.

Uniclass automorphisms of spherical buildings

Abstract

An automorphism of a building is called uniclass if the Weyl distance between any chamber and its image lies in a single (twisted) conjugacy class of the Coxeter group. In this paper we characterise uniclass automorphisms of spherical buildings in terms of their fixed structure. For this purpose we introduce the notion of a Weyl substructure in a spherical building. We also link uniclass automorphisms to the Freudenthal--Tits magic square.
Paper Structure (33 sections, 63 theorems, 62 equations, 3 tables)

This paper contains 33 sections, 63 theorems, 62 equations, 3 tables.

Table of Contents

  1. Background and preliminary results
  2. Coxeter groups and twisted conjugacy classes
  3. An involution on the set of twisted classes
  4. Twisted involutions
  5. Admissible diagrams
  6. Bi-capped classes
  7. Buildings
  8. Automorphisms of spherical buildings and opposition diagrams
  9. Displacement spectra
  10. Fundamental properties of displacement spectra
  11. Uniclass automorphisms
  12. Fixed and opposition diagrams of uniclass automorphisms
  13. The finite case
  14. The non-thick case
  15. Uniclass automorphisms are capped
  16. ...and 18 more sections

Key Result

Theorem 1

Let $\theta$ be a nontrivial automorphism of a thick irreducible spherical building $\Delta$ of rank at least $2$. Then $\theta$ is uniclass if and only if $\theta$ is either anisotropic, or: Moreover, for each uniclass automorphism the twisted conjugacy class $\mathop{\mathrm{\mathsf{Disp}}}\nolimits(\theta)$ is explicitly determined (see Table AllWS).

Theorems & Definitions (137)

  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Lemma 1.1
  • proof
  • Proposition 1.2
  • proof
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • ...and 127 more