Characterizations of Trialities of Type $\mathrm {I}_{\mathsf {id}}$ in Buildings of Type $\mathsf {D}_4$

Abstract

A triality of type $\mathrm {I}_{\mathsf {id}}$ in a building $\varDelta$ of type $\mathsf {D}_4$ is a type rotating automorphism of order 3 whose structure of fixed flags is the building of type $\mathsf {G}_2$ related to Dickson’s simple groups (in geometric term, this building is the split Cayley generalized hexagon over the field in question). Such a triality exists over any field and is unique up to conjugacy. In this paper, we present two characterizations of such trialities among all type rotating automorphisms (hence not necessarily of order 3). We prove that, if for a type rotating automorphism $\theta$ of $\varDelta$, no non-fixed line and its image are contained in adjacent chambers, and $\theta$ fixes at least one line, then $\theta$ is a triality of type $\mathrm {I}_{\mathsf {id}}$ (here, lines are vertices of type 2, with Bourbaki labeling). Also, if a type rotating automorphism $\theta$ of $\varDelta$ never maps a line to an opposite line, then it is also a triality of type $\mathrm {I}_{\mathsf {id}}$. We moreover show that this condition is equivalent with $\theta$ not map** any chamber to an opposite one. The latter completes the programme for type rotating automorphisms of buildings of type $\mathsf {D}_4$ of determining all domestic automorphisms of spherical buildings.