Explicit Transgression Primitives on Flag Manifolds
Abstract
Borel's theorem presents the flag manifold cohomology as the coinvariant algebra H*(K/T; ℂ) ≅ S(h∨)/IW, establishing the vanishing of characteristic forms pj(Ft) = tr(Ftj) non-constructively; projection from the Weil algebra W(k) yields closed primitives on K failing to descend. We construct relative transgression primitives αj ∈ (Λ2j−1m∨)T ≅ KΩ2j−1(K/T) satisfying dm αj = −pj(Ft) on K/T. Via connections θs = θt + sθm, the curvature factors as Fs = (1−s)(Ft − sθm2); expanding via non-commutative word sums and Euler's Beta integral yields a universal closed formula for all j ≥ 1. These primitives exhibit a structural transition: α2 = −1/3tr(θm3) is curvature-free, α3 is linear in Ft, while at j = 4, [Ft, θm2] ≠ 0 shatters cyclic trace symmetry into independent necklaces, producing the irreducible invariant tr(θm ∧ Ft ∧ θm2 ∧ Ft). On the A2 root lattice (sl3(ℂ)), T-invariant cochains correspond to closed loops, α2 = Ω- − Ω+ is the oriented difference of triangles, and dm acts as an edge-fracturing operator witnessing the exactness of c2(E). While character theory encounters a multiplicity barrier at k=4 in Specht decompositions, a minimal Koszul DGA resolution K•(p2, …, pk) resolves the coinvariant algebra, recovering the Poincaré polynomial PK/T(t) = ∏j=2k (1−t2j)/(1−t2) and Euler characteristic χ(K/T) = k! without matrix-rank ambiguities.
Related
- The intersection of two real flag manifolds in a complex flag manifold, and the canonical form of a compact symmetric triad
- Differential calculus on quantized irreducible flag manifolds
- Rational homotopy theory of flag manifolds
- CR Orbits on the complex flag manifolds of G2
- On topologically transitive subsets of flag manifolds
BibTeX
@misc{weiler2026explicit,
title = {{Explicit Transgression Primitives on Flag Manifolds}},
author = {Greg Weiler},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00300},
eprint = {2610.00300},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00300},
}
RIS
TY - UNPB
TI - Explicit Transgression Primitives on Flag Manifolds
AU - Greg Weiler
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00300
UR - https://arxiv.org/abs/2610.00300
AB - Borel's theorem presents the flag manifold cohomology as the coinvariant algebra H^*(K/T; ℂ) ≅ S(h^∨)/I_W, establishing the vanishing of characteristic forms p_j(F_t) = tr(F_t^j) non-constructively; projection from the Weil algebra W(k) yields closed primitives on K failing to descend. We construct relative transgression primitives α_j ∈ (Λ^2j−1m^∨)^T ≅ ^KΩ^2j−1(K/T) satisfying d_m α_j = −p_j(F_t) on K/T. Via connections θ_s = θ_t + sθ_m, the curvature factors as F_s = (1−s)(F_t − sθ_m^2); expanding via non-commutative word sums and Euler's Beta integral yields a universal closed formula for all j ≥ 1. These primitives exhibit a structural transition: α_2 = −1/3tr(θ_m^3) is curvature-free, α_3 is linear in F_t, while at j = 4, [F_t, θ_m^2] ≠ 0 shatters cyclic trace symmetry into independent necklaces, producing the irreducible invariant tr(θ_m ∧ F_t ∧ θ_m^2 ∧ F_t). On the A_2 root lattice (sl_3(ℂ)), T-invariant cochains correspond to closed loops, α_2 = Ω_- − Ω_+ is the oriented difference of triangles, and d_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_•(p_2, …, p_k) resolves the coinvariant algebra, recovering the Poincaré polynomial P_K/T(t) = ∏_j=2^k (1−t^2j)/(1−t^2) and Euler characteristic χ(K/T) = k! without matrix-rank ambiguities.
ER -