Vol. 17, No. 2, 2019

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Opposition diagrams for automorphisms of small spherical buildings

James Parkinson and Hendrik Van Maldeghem

Vol. 17 (2019), No. 2, 141–188
Abstract

An automorphism θ of a spherical building Δ is called capped if it satisfies the following property: if there exist both type J1 and J2 simplices of Δ mapped onto opposite simplices by θ then there exists a type J1 J2 simplex of Δ mapped onto an opposite simplex by θ. In previous work we showed that if Δ is a thick irreducible spherical building of rank at least 3 with no Fano plane residues then every automorphism of Δ is capped. In the present work we consider the spherical buildings with Fano plane residues (the 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 F4 and E6.

Keywords
spherical building, opposition diagram, capped automorphism, domestic automorphism, displacement
Mathematical Subject Classification 2010
Primary: 20E42, 51E24
Milestones
Received: 25 March 2018
Revised: 14 January 2019
Accepted: 11 February 2019
Published: 31 May 2019
Authors
James Parkinson
The University of Sydney
Sydney NSW
Australia
Hendrik Van Maldeghem
Vakgroep Wiskunde: Algebra en Meetkunde
University of Ghent
Ghent
Belgium