Two interesting oriented matroids
Two interesting oriented matroids
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两个有趣的定向拟阵
作者：
Jürgen Richter
有向拟阵是真实的向量空间中构形的一种组合模型。可实现性问题在理论中扮演着核心角色：给定一个定向拟阵，找到一个相关的向量配置。在本文中，我们提出了两个密切相关的秩为3的定向拟阵Ω14和Ω − 14，它们有14个元素，在可实现性方面具有有趣的性质。Ω14和Ω − 14只在一个基方向上不同。可实现的定向拟阵Ω14至少有两个有趣的性质：第一，它有一个没有度量实现的组合对称;第二，它有一个不连通的实现空间。换句话说，Ω14有不同的实现，它们不能连续变形为彼此，同时保持在同一个合痕类中。定向拟阵Ω−14是不可实现的，但它没有双二次最终多项式。换句话说，唯一已知的有效算法方法无法证明Ω−14的不可实现性。
Oriented matroids are a combinatorial model for configurations in real vector spaces. A central role in the theory is played by the realizability problem: Given an oriented matroid, find an associated vector configuration. In this paper we present two closely related oriented matroids Ω14 and Ω − 14 of rank 3 with 14 elements that have interesting properties with respect to realizability. Ω14 and Ω − 14 differ in exactly one basis orientation. The realizable oriented matroid Ω14 has at least two interesting properties: First it has a combinatorial symmetry that has no metric realization, and second it has a disconnected realization space. In other words, there are different realizations of Ω14 that cannot be continuously deformed into each other while staying in the same isotopy class. The oriented matroid Ω−14 is non-realizable but it has no bi-quadratic final polynomial. In other words, the only known effective algorithmic method fails to prove the non-realizability of Ω−14.