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Geometries with trialities arising from linear spaces

Rémi Delaby, Dimitri Leemans, Philippe Tranchida

TL;DR

The paper constructs a triangle complex Δ(Γ) from any rank-2 geometry Γ, producing a rank-3 incidence system that inherently possesses a triality cycling three copies of Γ’s flags. It develops a precise flag-transitivity criterion linking Δ(Γ)’s symmetry to Γ’s action on triples of non-collinear points, and classifies the linear-spaces Γ for which Δ(Γ) is flag-transitive. The key results yield an infinite family of thick, flag-transitive, residually connected geometries with trialities but no dualities, notably arising from affine planes, and provide a complete roster of when Δ(Γ) is firm and residually connected. The work bridges incidence-geometry constructions with group-action analyses, delivering concrete geometric interpretations alongside the group-theoretic structure of correlations and trialities.

Abstract

A triality is a sort of super-symmetry that exchanges the types of the elements of an incidence geometry in cycles of length three. Although geometries with trialities exhibit fascinating behaviors, their construction is challenging, making them rare in the literature. To understand trialities more deeply, it is crucial to have a wide variety of examples at hand. In this article, we introduce a general method for constructing various rank-three incidence systems with trialities. Specifically, for any rank two incidence system $Γ$, we define its triangle complex $Δ(Γ)$, a rank three incidence system whose elements consist of three copies of the flags (pairs of incident elements) of $Γ$. This triangle complex always admits a triality that cyclically permutes the three copies. We then explore in detail the properties of the triangle complex when $Γ$ is a linear space, including flag-transitivity, the existence of dualities, and connectivity properties. As a consequence of our work, this construction yields the first infinite family of thick, flag-transitive and residually connected geometries with trialities but no dualities.

Geometries with trialities arising from linear spaces

TL;DR

The paper constructs a triangle complex Δ(Γ) from any rank-2 geometry Γ, producing a rank-3 incidence system that inherently possesses a triality cycling three copies of Γ’s flags. It develops a precise flag-transitivity criterion linking Δ(Γ)’s symmetry to Γ’s action on triples of non-collinear points, and classifies the linear-spaces Γ for which Δ(Γ) is flag-transitive. The key results yield an infinite family of thick, flag-transitive, residually connected geometries with trialities but no dualities, notably arising from affine planes, and provide a complete roster of when Δ(Γ) is firm and residually connected. The work bridges incidence-geometry constructions with group-action analyses, delivering concrete geometric interpretations alongside the group-theoretic structure of correlations and trialities.

Abstract

A triality is a sort of super-symmetry that exchanges the types of the elements of an incidence geometry in cycles of length three. Although geometries with trialities exhibit fascinating behaviors, their construction is challenging, making them rare in the literature. To understand trialities more deeply, it is crucial to have a wide variety of examples at hand. In this article, we introduce a general method for constructing various rank-three incidence systems with trialities. Specifically, for any rank two incidence system , we define its triangle complex , a rank three incidence system whose elements consist of three copies of the flags (pairs of incident elements) of . This triangle complex always admits a triality that cyclically permutes the three copies. We then explore in detail the properties of the triangle complex when is a linear space, including flag-transitivity, the existence of dualities, and connectivity properties. As a consequence of our work, this construction yields the first infinite family of thick, flag-transitive and residually connected geometries with trialities but no dualities.

Paper Structure

This paper contains 9 sections, 23 theorems, 12 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The triangle complex of a rank two geometry
  4. A criterion for the flag-transitivity of $\Delta(\Gamma)$
  5. The cases where $n=2$ or $m=2$
  6. Flag-transitive linear spaces and triangles
  7. Connectedness and residual connectedness
  8. Firm, residually connected and flag-transitive geometries with a triality
  9. Conclusion

Key Result

Theorem 1.1

Assume $\Gamma$ is a flag-transitive linear space of $v$ points with a group $G$ acting transitively on the set $T(\Gamma)$ of ordered triples of non-collinear points of $\Gamma$. Then one of the following occurs: The converse is also true. For any pair $(\Gamma,G)$ satisfying the conditions of one of the cases given above, the action of $G$ is transitive on $T(\Gamma)$.

Figures (11)

  • Figure 1: An alternating path from $F_1$ to $F_2$ of length $6$.
  • Figure 2: An alternating path from $\phi(F_1)$ to $\phi(F_2)$ of length $3$.
  • Figure 3: A path from $(p_1,L_1,1)$ to $(p_1',L_1,1)$.
  • Figure 4: A path from $(p_1,L_1,1)$ to $(p_1,L_1',1)$.
  • Figure 5: The case where $L_2 = L_2'$.
  • ...and 6 more figures

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 35 more