This paper investigates the question of uniqueness of the reduced oriented matroid structure arising from root systems of a Coxeter system in real vector spaces. We settle the question for finite Coxeter systems, irreducible affine Weyl groups and all rank three Coxeter systems. In these cases, the oriented matroid structure is unique unless W is of type \({\widetilde{A}}_n, n\ge 3\), in which case there are three possibilities.

中文翻译：

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从已实现的根系统定向拟阵结构

本文研究了在实际向量空间中由Coxeter系统的根系统引起的简化取向拟阵结构的唯一性问题。我们解决了有限的Coxeter系统，不可约仿射Weyl群以及所有排名第三的Coxeter系统的问题。在这些情况下，除非W为\（{\ widetilde {A}} _ n，n \ ge 3 \）类型，否则定向拟阵结构是唯一的，在这种情况下，存在三种可能性。