Parabolic Homotopy Colimits and Coxeter Descents

Abstract

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 $X_{\mathcal{K}}(G)={\mathrm{hocolim}}_{I\in \mathcal{K}}G/G_I$ of standard partial flag manifolds. We compute its integral homology. If ${\mathrm{Des}}_R(w)$ is the right descent set of $w\in W$ and $\ell (w)$ its Coxeter length, then $$H_n(X_{\mathcal{K}}(G);\mathbb{Z})\cong \underset{w\in W}{⨁}{\tilde{H}}_{n-2\ell (w)-1}({\mathcal{K}}_{{\mathrm{Des}}_R(w)};\mathbb{Z}).$$ 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.