{
  "abstract": "Borel's theorem presents the flag manifold cohomology as the coinvariant algebra $H^*(K/T; \\mathbb{C}) \\cong S(\\mathfrak{h}^\\vee)/I_W$, establishing the vanishing of characteristic forms $p_j(F_\\mathfrak{t}) = \\mathrm{tr}(F_\\mathfrak{t}^j)$ non-constructively; projection from the Weil algebra $W(\\mathfrak{k})$ yields closed primitives on $K$ failing to descend. We construct relative transgression primitives $α_j \\in (Λ^{2j-1}\\mathfrak{m}^\\vee)^T \\cong {}^KΩ^{2j-1}(K/T)$ satisfying $d_\\mathfrak{m} α_j = -p_j(F_\\mathfrak{t})$ on $K/T$. Via connections $θ_s = θ_\\mathfrak{t} + sθ_\\mathfrak{m}$, the curvature factors as $F_s = (1-s)(F_\\mathfrak{t} - sθ_\\mathfrak{m}^2)$; expanding via non-commutative word sums and Euler's Beta integral yields a universal closed formula for all $j \\ge 1$. These primitives exhibit a structural transition: $α_2 = -\\frac{1}{3}\\mathrm{tr}(θ_\\mathfrak{m}^3)$ is curvature-free, $α_3$ is linear in $F_\\mathfrak{t}$, while at $j = 4$, $[F_\\mathfrak{t}, θ_\\mathfrak{m}^2] \\neq 0$ shatters cyclic trace symmetry into independent necklaces, producing the irreducible invariant $\\mathrm{tr}(θ_\\mathfrak{m} \\wedge F_\\mathfrak{t} \\wedge θ_\\mathfrak{m}^2 \\wedge F_\\mathfrak{t})$. On the $A_2$ root lattice ($\\mathfrak{sl}_3(\\mathbb{C})$), $T$-invariant cochains correspond to closed loops, $α_2 = Ω_- - Ω_+$ is the oriented difference of triangles, and $d_\\mathfrak{m}$ acts as an edge-fracturing operator witnessing the exactness of $c_2(E)$. While character theory encounters a multiplicity barrier at $k=4$ in Specht decompositions, a minimal Koszul DGA resolution $K_\\bullet(p_2, \\dots, p_k)$ resolves the coinvariant algebra, recovering the Poincaré polynomial $P_{K/T}(t) = \\prod_{j=2}^k \\frac{1-t^{2j}}{1-t^2}$ and Euler characteristic $χ(K/T) = k!$ without matrix-rank ambiguities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00300",
  "author_total": 1,
  "authors": [
    {
      "name": "Greg Weiler"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4wbdqmb7qf7d6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00300",
  "fulltext_url": null,
  "id": "4wbdqmb7qf7d6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00300",
  "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",
      "math.DG"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2610|Homotopy and Cohomology in Algebraic Topology"
    ]
  ],
  "openalex": "W7216035769",
  "pdf_url": "https://arxiv.org/pdf/2610.00300",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4wbdqmb7qf7d6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.00300",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00300"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Explicit Transgression Primitives on Flag Manifolds",
  "title_text": "Explicit Transgression Primitives on Flag Manifolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/4wbdqmb7qf7d6",
  "venue": "arXiv",
  "versions": []
}