Trialities and 1-Systems of Q +(7,q)

Abstract

The image of a 1-system of Q+ (7, q) under a triality of the D4-geometry, attached to Q+ (7, q), will be investigated. Attention will mainly be paid to the case of a locally hermitian, semiclassical 1-system of a Q(6, q), embedded in Q+(7, q). It is found that its image under a triality is always locally hermitian and semiclassical as well. Moreover, it is a proper 1-system of Q+(7, q) whenever the original 1-system of Q(6, q) is not a spread of some generalized hexagon H(q) on Q(6, q). Finally, some results concerning isomorphisms will be obtained.