Lines and Opposition in Lie Incidence Geometries of Exceptional Type

2026/02/28 by Sira Busch, Hendrik Van Maldeghem
Mathematics · #math.CO #math.GR #msc:05B25 #msc:51E24 #msc:51E21

paper · pdf

50 pages

arxiv created 2026/07/30 · arxiv updated 2026/07/31

Abstract

We characterise sets of points of exceptional Lie incidence geometries, that is, the natural geometries arising from spherical buildings of exceptional types F₄, E₆, E₇, E₈ and G₂, that form a line using the opposition relation. With that, we obtain a classification of so-called ``geometric lines'' in many of these geometries. Furthermore, our results lead to a characterisation of geometric lines in finite exceptional Lie incidence geometries as minimal blocking sets, that is, point sets of the size of a line admitting no object opposite to all of their members, in most cases, and we classify all exceptions. As a further consequence, we obtain a characterisation of automorphisms of exceptional spherical buildings as certain opposition preserving maps.

Citations

• Automorphisms and opposition in spherical buildings of exceptional type, III. Metasymplectic spaces
• On certain blocking sets and the minimum weight of the code of generalised polygons
• Groups of Projectivities in Spherical Buildings of Non-Simply Laced Type
• Groups of Projectivities and Levi Subgroups in Spherical Buildings of Simply Laced Type
• Root shadow spaces

Related

• A Characterization of Lines in Finite Lie Incidence Geometries of Classical Type
• Automorphisms and opposition in spherical buildings of exceptional type, IV: The E₇ case
• Automorphisms and opposition in spherical buildings of exceptional type,\n I
• Automorphisms and opposition in spherical buildings of exceptional type, II: Moufang hexagons
• Automorphisms and opposition in spherical buildings of exceptional type, V: The E₈ case