Abstract
An automorphism of a spherical building is called capped if it satisfies the following property: if there exist both type and simplices of mapped onto opposite simplices by then there exists a type simplex of mapped onto an opposite simplex by . In previous work we showed that if is a thick irreducible spherical building of rank at least with no Fano plane residues then every automorphism of 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 and .
Citation
James Parkinson. Hendrik Van Maldeghem. "Opposition diagrams for automorphisms of small spherical buildings." Innov. Incidence Geom. Algebr. Topol. Comb. 17 (2) 141 - 188, 2019. https://doi.org/10.2140/iig.2019.17.141
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