A characterization of simplicial oriented geometries as groupoids with root systems

TL;DR

This paper characterizes simplicial oriented geometries as groupoids with special root systems, linking matroid theory, lattice properties, and Coxeter group structures to deepen understanding of these geometric objects.

Contribution

It provides a new characterization of simplicial oriented geometries through groupoids with root systems, confirming a conjecture and connecting various algebraic and combinatorial frameworks.

Findings

Characterization of simplicial oriented geometries as groupoids with root systems

Extension of Handa's matroid characterization to groupoid language

Establishment of lattice-theoretic properties related to Coxeter groups

Abstract

This paper shows that simplicial oriented geometries can be characterized as groupoids with root systems having certain favorable properties, as conjectured by the first author. The proof first translates Handa's characterization of oriented matroids, as acycloids which remain acycloids under iterated elementary contractions, into the language of groupoids with root systems, then establishes favorable lattice theoretic properties of a generalization of a construction which Brink and Howlett used in their study of normalizers of parabolic subgroups of Coxeter groups and uses Bj\"orner-Edelman-Ziegler's lattice theoretic characterization of simplicial oriented geometries amongst oriented geometries.