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Incidence geometries with trialities coming from maps with Wilson trialities

Dimitri Leemans and Klara Stokes

Vol. 20 (2023), No. 2-3, 325–340
Abstract

Triality is a classical notion in geometry that arose in the context of the Lie groups of type D4. Another notion of triality, Wilson triality, appears in the context of reflexible maps. We build a bridge between these two notions, showing how to construct an incidence geometry with a triality from a map that admits a Wilson triality. We also extend a result by Jones and Poulton, showing that for every prime power q, the group L2(q3) has maps that admit Wilson trialities but no dualities.

In memoriam Jacques Tits, whose mathematicshave been a great source of inspiration in our work

Keywords
triality, maps, incidence geometry, projective special linear groups
Mathematical Subject Classification
Primary: 20C33, 51A10, 51E24
Milestones
Received: 31 August 2022
Revised: 17 January 2023
Accepted: 15 February 2023
Published: 13 September 2023
Authors
Dimitri Leemans
Département de Mathématique
Université libre de Bruxelles
Algèbre et Combinatoire
Brussels
Belgium
Klara Stokes
Department of Mathematics and Mathematical Statistics
Umeå Universitet
Umeå
Sweden