发表机构
Georg-August-Universität Göttingen（哥廷根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在旗流形上构造了显式的相对横gression原语，通过非交换展开和Koszul分解解决了特征形式消失的非构造性问题，并恢复了Poincaré多项式与Euler特征。
AI 中文摘要
Borel定理将旗流形上同调表示为余不变代数$H^*(K/T; \mathbb{C}) \cong S(\mathfrak{h}^\vee)/I_W$，非构造性地确立了特征形式$p_j(F_\mathfrak{t}) = \mathrm{tr}(F_\mathfrak{t}^j)$的消失；从Weil代数$W(\mathfrak{k})$的投影产生$K$上的闭原语，但无法下降。我们构造相对横gression原语$\alpha_j \in (\Lambda^{2j-1}\mathfrak{m}^\vee)^T \cong {}^K\Omega^{2j-1}(K/T)$，满足在$K/T$上$d_\mathfrak{m} \alpha_j = -p_j(F_\mathfrak{t})$。通过联络$\theta_s = \theta_\mathfrak{t} + s\theta_\mathfrak{m}$，曲率分解为$F_s = (1-s)(F_\mathfrak{t} - s\theta_\mathfrak{m}^2)$；利用非交换词和以及Euler Beta积分展开，得到对所有$j \ge 1$的普适闭式公式。这些原语表现出结构转变：$\alpha_2 = -\frac{1}{3}\mathrm{tr}(\theta_\mathfrak{m}^3)$无曲率，$\alpha_3$是$F_\mathfrak{t}$的线性函数，而在$j = 4$时，$[F_\mathfrak{t}, \theta_\mathfrak{m}^2] \neq 0$将循环迹对称性粉碎为独立项链，产生不可约不变量$\mathrm{tr}(\theta_\mathfrak{m} \wedge F_\mathfrak{t} \wedge \theta_\mathfrak{m}^2 \wedge F_\mathfrak{t})$。在$A_2$根格（$\mathfrak{sl}_3(\mathbb{C})$）上，$T$-不变上链对应闭圈，$\alpha_2 = \Omega_- - \Omega_+$是三角形的定向差，$d_\mathfrak{m}$作为边碎裂算子，见证$c_2(E)$的恰当性。虽然特征理论在Specht分解中于$k=4$遭遇重数障碍，但最小Koszul DGA分解$K_\bullet(p_2, \dots, p_k)$解决了余不变代数，恢复了Poincaré多项式$P_{K/T}(t) = \prod_{j=2}^k \frac{1-t^{2j}}{1-t^2}$和Euler特征$\chi(K/T) = k!$，且无矩阵秩歧义。
英文摘要
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.
Comments55 pages
