欧几里得∨-系统与实PK排列
Euclidean $\vee$-systems and real PK arrangements
AI总结：
建立超平面排列几何中两种结构的对应，证每个不可约欧几里得∨ -系统确定实PK排列，反之亦然，还给出相关结论如模空间同胚、超平面排列是单纯形等。
AI中文摘要：
我们在超平面排列几何中出现的两种结构之间建立了一种对应关系：欧几里得∨ -系统和实多面体凯勒（PK）排列。我们证明每个不可约欧几里得∨ -系统确定一个实PK排列，反之，每个实PK排列都以这种方式产生。因此，我们表明在固定射影类中的欧几里得∨ -系统的模空间与多面体的相对内部同胚。我们还直接证明了与欧几里得∨ -系统相关的超平面排列是单纯形的。在目前已知的单纯线排列中，我们精确地确定了那些由∨ -系统产生的排列。因此，我们证明了对于最多有27个向量的不可约三阶欧几里得∨ -系统，施赖伯 - 韦谢洛夫目录是完整的。
英文摘要：
We establish a correspondence between two structures arising in the geometry of hyperplane arrangements: Euclidean $\vee$-systems and real polyhedral Kähler (PK) arrangements. We prove that every irreducible Euclidean $\vee$-system determines a real PK arrangement, and conversely that every real PK arrangement arises this way. As a consequence, we show that, up to equivalence, there are exactly three irreducible rank-three Euclidean $\vee$-systems whose vectors have equal length; their arrangements are the mirrors of the reflection groups of the regular tetrahedron, cube, and icosahedron. The correspondence also yields a description of the moduli space of Euclidean $\vee$-systems in a fixed projective class: it is homeomorphic to the relative interior of a polytope. We also give a direct proof that the hyperplane arrangement associated with a Euclidean $\vee$-system is simplicial. Among the currently known simplicial line arrangements, we identify precisely those that arise from $\vee$-systems. As a consequence, we prove that the Schreiber--Veselov catalog is complete for irreducible rank-three Euclidean $\vee$-systems with at most $27$ vectors.
