发表机构
Charles University; VSB–Technical University of Ostrava; University of Ostrava（查理大学; 俄斯特拉发理工大学; 俄斯特拉发大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对有限Coxeter系统定义抛物bar复形，证明其Hochster型分解，建立相关函子性、对偶性等结果，将其应用于Weyl群的同调研究并恢复特殊模型。
AI 中文摘要
设$(W,S)$为有限Coxeter系统，$\boldsymbol{\textit{K}}\boldsymbol{\textit{⊆}}2^S$为单纯复形。我们定义抛物bar复形$\boldsymbol{\textit{B}}_{\boldsymbol{\textit{K}}}(W)$，并证明其可按$w\boldsymbol{\textit{∈}}W$分解，其中$w$项是由右下降集$\boldsymbol{\textit{Des}}_R(w)$确定的相对序复形链复形。显式积分Morse归约将该项（至多差Schubert位移$2\boldsymbol{\textit{ℓ}}(w)+1$）与诱导子复形$\boldsymbol{\textit{K}}_{\boldsymbol{\textit{Des}}_R(w)}$的增广链等同。对Weyl群而言，该复形是$\boldsymbol{\textit{X}}_{\boldsymbol{\textit{K}}}(G)\boldsymbol{\textit{=}}\boldsymbol{\textit{hocolim}}_{I\boldsymbol{\textit{∈}}\boldsymbol{\textit{K}}}G/G_I$的胞腔链复形，因此其同调是下降加权的Hochster分解。我们证明了索引复形包含的函子性与同伦检测定理、广义同调球的Alexander对偶对称性，且对单连通$G$，有刻画该族同调球成员中边界单形的刚性定理。对$\boldsymbol{\textit{G}}\boldsymbol{\textit{=}}(\boldsymbol{\textit{SU}}(2))^r$，该构造同伦等价于$(\boldsymbol{\textit{D}}^3,\boldsymbol{\textit{S}}^2)^{\boldsymbol{\textit{K}}}$，而拟阵独立复形给出Tutte多项式特化。边界单形情形恢复了单位伴随球的两生成元积分Morse模型。
英文摘要
Let $G$ be a compact, connected, simply connected semisimple Lie group with Weyl group $W$ and simple reflections $S$. For a simplicial complex $\mathcal K$ on $S$, form the homotopy colimit $X_{\mathcal K}(G)=\operatorname*{hocolim}_{I\in\mathcal K}G/G_I$ of standard partial flag manifolds. We compute its integral homology. If $\operatorname{Des}_R(w)$ is the right descent set of $w\in W$ and $\ell(w)$ its Coxeter length, then $$ H_n(X_{\mathcal K}(G);\mathbb Z)\cong \bigoplus_{w\in W}\widetilde H_{n-2\ell(w)-1}(\mathcal K_{\operatorname{Des}_R(w)};\mathbb Z). $$ Thus the induced subcomplexes of $\mathcal K$ supply the topological data, while the Weyl group determines which subcomplex occurs and the Schubert-degree shift. The proof gives a chain-level splitting and an integral Morse reduction. We derive homotopy detection, duality and rigidity results, and recover polyhedral products, matroid--Tutte formulas, and the adjoint sphere as special cases.
