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The E6 Restricted Hyperplane Arrangement and its E7 Shadow: Weyl Transport on a Minuscule Bruhat Poset

  • Saber Ahmed
  • Mboyo Esole
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Abstract

We study the restricted fan cut inside the dual fundamental Weyl chamber by the weights of a $27$-dimensional minuscule representation of E6; 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 56 of E7, defined entirely inside E7, has Hasse diagram equal to the chamber adjacency graph of the E6 arrangement. Three canonical 7+7 splittings of it, of types A7, D7, and E7, are the visible traces of Levi-center u(1)-charge decompositions of the 56 and reproduce the three level-$8$ decompositions on the E6 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 E7 shadow records not merely the chamber graph but, once matched with the independent E6 classification, the full local wall architecture of I(E6,27).

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Cite

BibTeXRIS
BibTeX
@misc{ahmed2026restricted,
  title = {{The $E_6$ Restricted Hyperplane Arrangement and its $E_7$ Shadow: Weyl Transport on a Minuscule Bruhat Poset}},
  author = {Saber Ahmed and Mboyo Esole},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.10701},
  eprint = {2609.10701},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10701},
}
RIS
TY  - UNPB
TI  - The E_6 Restricted Hyperplane Arrangement and its E_7 Shadow: Weyl Transport on a Minuscule Bruhat Poset
AU  - Saber Ahmed
AU  - Mboyo Esole
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.10701
UR  - https://arxiv.org/abs/2609.10701
AB  - 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 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 u(1)-charge decompositions of the 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_6,27).
ER  -