## Opposition diagrams for automorphisms of small spherical buildings

### J. Parkinson, H. Van Maldeghem

#### Abstract

An automorphism \(\theta\) of a spherical building \(\Delta\) is
called *capped* if it satisfies the following property: if
there exist both type \(J_1\) and \(J_2\) simplices of
\(\Delta\) mapped onto opposite simplices by \(\theta\) then
there exists a type \(J_1\cup J_2\) simplex of \(\Delta\) mapped
onto an opposite simplex by \(\theta\). In previous work we
showed that if \(\Delta\) is a thick irreducible spherical
building of rank at least \(3\) with no Fano plane residues then
every automorphism of \(\Delta\) 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 \(\mathsf{F}_4\) and \(\mathsf{E}_6\).

Keywords:
spherical buildings, incidence geometry,.

AMS Subject Classification:
Primary 20E42; secondary 51E24.