Coxeter Descents and Parabolic Homotopy Colimits: A Hochster-type decomposition and integral Morse reduction

Abstract

Let $(W,S)$ be a finite Coxeter system and let $\mathcal K\subseteq 2^S$ be a simplicial complex. We define a parabolic bar complex ${\mathcal B}_{\mathcal K}(W)$ and prove a decomposition indexed by $w\in W$ in which the $w$-summand is a relative order-complex chain complex determined by the right descent set ${\operatorname{Des}_{R}}(w)$. An explicit integral Morse reduction identifies this summand, up to the Schubert shift $2\ell(w)+1$, with the augmented chains of the induced subcomplex $\mathcal K_{{\operatorname{Des}_{R}}(w)}$. For Weyl groups the complex is the cellular chain complex of \[ {X_{\mathcal K}}(G)={\operatorname*{hocolim}}_{I\in\mathcal K} G/G_I, \] so its homology is a descent-weighted Hochster decomposition. We prove functoriality and a homotopy-detection theorem for inclusions of indexing complexes, an Alexander-duality symmetry for generalized homology spheres, and, for simple $G$, a rigidity theorem characterizing the boundary simplex among the homology-sphere members of the family. For $G=(SU(2))^r$ the construction agrees up to homotopy with $(D^3,S^2)^{\mathcal K}$, and matroid independence complexes give a Tutte-polynomial specialization. The boundary-simplex case recovers the two-generator integral Morse model of the unit adjoint sphere.