Nice 划分、超可解性与图形排列形变中的自由性

中文摘要 AI 辅助

我们研究图形排列的仿射形变 $\mathcal{A}(G_{\mathcal{S}})$ 及其锥。这里 $G=([n],E(G))$ 是简单图，$\mathcal{S}=(S_{ij})$ 是有限增益集族，$\mathcal{A}(G_{\mathcal{S}})$ 由超平面 $x_i-x_j=a$（其中 $a\in S_{ij}$）组成。我们称 $G_{\mathcal{S}}$ 为分块可容许的，如果每个块都有一个顶点排序 $v_1,\ldots,v_m$，使得对每个 $k$ 及所有不同的 $i,j>k$，有 $S_{v_kv_j}-S_{v_kv_i}\subseteq S_{v_iv_j}$。每个这样的排序都是完美消去序。我们证明，对任意 $G$，以下条件等价：(i) $G_{\mathcal{S}}$ 是分块可容许的；(ii) 锥 $c\mathcal{A}(G_{\mathcal{S}})$ 是超可解的；(iii) $\mathcal{A}(G_{\mathcal{S}})$ 允许一个 nice 划分。我们给出这些等价性的直接排列理论证明。对于一个块，可容许性与超可解性之间的等价性已包含在 Zaslavsky 对超可解图形提升格的刻画中。我们进一步证明，块上的每个 nice 划分都由一个可容许排序诱导，且其诱导的边类是以不同顶点为中心的星。在这些等价条件下，我们构造通过无穷远超平面的极大模链。我们还证明，如果 $c\mathcal{A}(G_{\mathcal{S}})$ 是自由的，则 $G$ 是弦图。

英文摘要

We study affine deformations $\mathcal{A}(G_{\mathcal{S}})$ of graphic arrangements and their cones. Here $G=([n],E(G))$ is a simple graph, $\mathcal{S}=(S_{ij})$ is a family of finite gain sets, and $\mathcal{A}(G_{\mathcal{S}})$ consists of the hyperplanes $x_i-x_j=a$ with $a\in S_{ij}$. We call $G_{\mathcal{S}}$ blockwise admissible if every block has a vertex ordering $v_1,\ldots,v_m$ satisfying $S_{v_kv_j}-S_{v_kv_i}\subseteq S_{v_iv_j}$ for every $k$ and all distinct $i,j>k$. Every such ordering is a perfect elimination ordering. We prove, for arbitrary $G$, that the following are equivalent: (i) $G_{\mathcal{S}}$ is blockwise admissible; (ii) the cone $c\mathcal{A}(G_{\mathcal{S}})$ is supersolvable; and (iii) $\mathcal{A}(G_{\mathcal{S}})$ admits a nice partition. We give a direct arrangement-theoretic proof of these equivalences. For a block, the equivalence between admissibility and supersolvability is already contained in Zaslavsky's characterization of supersolvable graphic-lift lattices. We further show that every nice partition on a block is induced by an admissible ordering and that its induced edge classes are stars with distinct centers. Under these equivalent conditions, we construct a maximal modular chain through the hyperplane at infinity. We also prove that if $c\mathcal{A}(G_{\mathcal{S}})$ is free, then $G$ is chordal.

发表机构

• College of Science National University of Defense Technology（国防科技大学理学院）
• School of Mathematics Hunan University（湖南大学数学学院）

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