$E_6$ 限制超平面排列及其 $E_7$ 影子：在微小子布鲁哈特偏序集上的外尔输运
The $E_6$ Restricted Hyperplane Arrangement and its $E_7$ Shadow: Weyl Transport on a Minuscule Bruhat Poset
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中文总结 AI 辅助
本研究通过 $E_7$ 的微小子表示输运，刻画了 $E_6$ 的 $27$ 维权诱导的限制超平面排列，确定了其腔室、极射线及完整壁架构，并揭示了表示论与三次曲面几何的深层联系。
中文摘要 AI 辅助
我们研究由 $E_6$ 的 $27$ 维微小子表示之权在双基本外尔腔内部所切割出的限制扇形；这两个这样的表示互为对偶并给出相同的排列。$27$ 个权中仅有 $11$ 个的核与该腔内部相交，我们证明它们决定了整个扇形。该扇形恰好有 $14$ 个腔室和 $18$ 条极射线，每个腔室都是六维单纯锥，我们确定了所有面、射线及关联关系。腔室数目此前由 Diaconescu 和 Entin 得到；单纯结构、极射线及关联数据是新的。三次曲面上 $27$ 条直线的几何随后解释并组织了所得的腔室架构。我们还枚举了所有面，计算了两个特征多项式——该排列不是超可解的——连同格指标和射影腔室体积，并描述了有向拟阵。我们的主要结果是表示论方面的。$E_7$ 的微小子 $\mathbf{56}$ 的一个特殊的 $14$ 元素可见子偏序集，完全在 $E_7$ 内部定义，其哈斯图等于 $E_6$ 排列的腔室邻接图。它的三个规范 $7+7$ 分割，类型分别为 $A_7$、$D_7$ 和 $E_7$，是 $\mathbf{56}$ 的 Levi 中心 $\mathfrak{u}(1)$ 荷分解的可见痕迹，并在 $E_6$ 一侧重现了三个 level-$8$ 分解。更强地，其覆盖上的单根标签，由最小长度陪集代表输运，逐腔室地恢复了全部六个面，并在全局上恢复了 $11$ 个活跃权超平面和双外尔腔的边界壁。因此，$E_7$ 影子不仅记录了腔室图，而且在与独立的 $E_6$ 分类匹配后，记录了 $I(E_6,\mathbf{27})$ 的完整局部壁架构。
英文摘要
We study the restricted fan cut inside the dual fundamental Weyl chamber by the weights of a $27$-dimensional minuscule representation of $E_6$; the two such representations are dual and give the same arrangement. Only $11$ of the $27$ weights have kernels meeting its interior, and we prove that they determine the entire fan. It has exactly $14$ chambers and $18$ extreme rays, every chamber is a six-dimensional simplicial cone, and we determine all facets, rays, and incidence relations. The chamber count was previously obtained by Diaconescu and Entin; the simplicial structure, extreme rays, and incidence data are new. The geometry of the $27$ lines on a cubic surface then explains and organizes the resulting chamber architecture. We also enumerate all faces, compute both characteristic polynomials---the arrangement is not supersolvable---together with lattice indices and projective chamber volumes, and describe the oriented matroid. Our main result is representation-theoretic. A distinguished $14$-element visible subposet of the minuscule $\mathbf{56}$ of $E_7$, defined entirely inside $E_7$, has Hasse diagram equal to the chamber adjacency graph of the $E_6$ arrangement. Three canonical $7+7$ splittings of it, of types $A_7$, $D_7$, and $E_7$, are the visible traces of Levi-center $\mathfrak{u}(1)$-charge decompositions of the $\mathbf{56}$ and reproduce the three level-$8$ decompositions on the $E_6$ side. More strongly, the simple-root labels on its covers, transported by minimal-length coset representatives, recover chamber by chamber all six facets and, globally, the $11$ active weight hyperplanes and the boundary walls of the dual Weyl chamber. Thus the $E_7$ shadow records not merely the chamber graph but, once matched with the independent $E_6$ classification, the full local wall architecture of $I(E_6,\mathbf{27})$.
发表机构
- Hamilton College（汉密尔顿学院）
- Northeastern University（东北大学）
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