The $E_6$ Restricted Hyperplane Arrangement and its $E_7$ Shadow: Weyl Transport on a Minuscule Bruhat Poset

Abstract

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 $I(E_6,\mathbf{27})$.