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On automorphisms of flag spaces

Hans Havlicek, Klaus List, Corrado Zanella

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.

On automorphisms of flag spaces

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.

Paper Structure

This paper contains 4 sections, 7 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. The flag space
  3. Plücker Transformations
  4. The flag variety

Key Result

Proposition 1

The pencils of flags are exactly the maximal sets of mutually related flags.

Theorems & Definitions (16)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Theorem 1
  • proof
  • Proposition 4
  • proof
  • ...and 6 more