{
  "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 $I(E_6,\\mathbf{27})$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10701",
  "author_total": 2,
  "authors": [
    {
      "name": "Saber Ahmed"
    },
    {
      "name": "Mboyo Esole"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3a2rep400vrtx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10701",
  "fulltext_url": null,
  "id": "3a2rep400vrtx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10701",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "hep-th"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7212314093",
  "pdf_url": "https://arxiv.org/pdf/2609.10701",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3a2rep400vrtx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:26Z",
      "native_id": "2609.10701",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10701"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    }
  ],
  "title": "The $E_6$ Restricted Hyperplane Arrangement and its $E_7$ Shadow: Weyl Transport on a Minuscule Bruhat Poset",
  "title_text": "The E_6 Restricted Hyperplane Arrangement and its E_7 Shadow: Weyl Transport on a Minuscule Bruhat Poset",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/3a2rep400vrtx",
  "venue": "arXiv",
  "versions": []
}