On automorphisms of flag spaces

TL;DR

This paper characterizes the automorphisms of the flag space in a 3D projective space, showing they are exactly those induced by collineations and dualities, and discusses automorphisms of the associated flag variety over a commutative field.

Contribution

It provides a complete characterization of automorphisms of the flag space in three-dimensional projective geometry, linking them to fundamental geometric transformations.

Findings

Automorphisms are precisely those from collineations and dualities.

No other automorphisms exist beyond these transformations.

Automorphisms of the flag variety are characterized over a commutative field.

Abstract

We show that the automorphisms of the flag space associated with a 3-dimensional projective space can be characterized as bijections preserving a certain binary relation on the set of flags in both directions. From this we derive that there are no other automorphisms of the flag space than those coming from collineations and dualities of the underlying projective space. Further, for a commutative ground field, we discuss the corresponding flag variety and characterize its group of automorphic collineations.