对称多面体场景与平行重绘

中文摘要 AI 辅助

提升与平行重绘是应用离散几何中的经典主题，分别涉及将$(d-1)$-图（即$\nmathbb{R}^{d-1}$中顶点-超平面关联几何的实现）垂直提升为$d$维多面体场景，以及在$\nmathbb{R}^d$内重绘$d$-图（或超平面排列）同时保持指定的超平面法线。在本文中，我们开发了一个统一框架，用于分析强制对称提升和强制对称平行重绘。特别地，我们表明这些理论的经典对偶性在强制对称设置中同样出现。我们建立了轨道提升矩阵和轨道共点几何矩阵，并证明它们是标准提升矩阵和共点几何（或平行重绘）矩阵的适当对称适应类比。我们还明确了与怀特利图的平行设计矩阵的轨道版本的关系。利用这些工具，我们推导了对称图成为强制对称平坦以及对称超平面排列成为强制对称鲁棒的必要条件，这些条件以与对称关联几何相关的群标记商图上的稀疏度计数表示。最后，我们讨论了关于这些条件在一般配置下充分性的猜想。

英文摘要

Liftings and parallel redrawings are classical topics in applied discrete geometry, concerned respectively with vertically lifting a $(d-1)$-picture -- a realisation of a vertex-hyperplane incidence geometry in $\mathbb{R}^{d-1}$ -- to a $d$-dimensional polyhedral scene, and with redrawing a $d$-picture (or hyperplane arrangement) within $\mathbb{R}^d$ while preserving prescribed hyperplane normals. In this paper, we develop a unified framework for the analysis of forced-symmetric liftings and forced-symmetric parallel redrawings. In particular, we show that the classical duality for these theories also appears in the forced-symmetric setting. We establish the orbit lifting matrix and the orbit concurrence geometry matrix, and show that they are the appropriate symmetry-adapted analogues of the standard lifting matrix and concurrence geometry (or parallel redrawing) matrix. We also make explicit the relationship with the orbit version of Whiteley's parallel design matrix for graphs. Using these tools, we derive necessary conditions for symmetric pictures to be forced-symmetric flat, and for symmetric hyperplane arrangements to be forced-symmetric robust, expressed as sparsity counts on the group-labelled quotient graphs associated with the symmetric incidence geometries. Finally, we discuss conjectures regarding the sufficiency of these conditions for generic configurations.

发表机构

• Katholieke Universiteit Leuven（荷语鲁汶大学）
• Lancaster University（兰卡斯特大学）
• Umeå University（于默奥大学）

机构由 AI 辅助整理，请以论文原文为准。

补充信息

备注

32 pages, 6 figures