Lines and opposition in Lie incidence geometries of exceptional type

TL;DR

This work develops a unifying framework to study lines in exceptional Lie incidence geometries by exploiting the opposition relation. It classifies geometric lines across minuscule types E_{6,1} and E_{7,7}, and extends to hexagonic geometries in exceptional types, using projections, round-up triples, and residue arguments. A key outcome is the equivalence between geometric lines and minimal blocking sets in many finite cases, enabling a comprehensive blocking-set census and shedding light on automorphisms and opposition-preserving maps as collineations. The methods also connect to generalised hexagons (G_2-type) and metasymplectic spaces, providing algorithmic tools (combing) to reduce complex line configurations to ordinary lines. Collectively, the results advance the understanding of incidence geometries associated with exceptional Lie groups and have implications for projectivity groups and symmetry analyses in spherical buildings.

Abstract

We characterise sets of points of exceptional Lie incidence geometries, that is, the natural geometries arising from spherical buildings of exceptional types $\mathsf{F_4}$, $\mathsf{E_6}$, $\mathsf{E_7}$, $\mathsf{E_8}$ and $\mathsf{G_2}$, 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.

Theorems & Definitions (128)

• Lemma 2.5
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• Lemma 2.6
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• Lemma 2.7
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• Lemma 2.8
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• Lemma 2.9
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