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PreprintCC BY-NC-ND 4.0arXiv

Parabolic Homotopy Colimits and Coxeter Descents

  • Yifan Zhang
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Abstract

Let G be a compact, connected, simply connected semisimple Lie group with Weyl group W and simple reflections S. For a simplicial complex š’¦ on S, form the homotopy colimit Xš’¦(G)=*hocolimIāˆˆš’¦G/GI of standard partial flag manifolds. We compute its integral homology. If DesR(w) is the right descent set of wāˆˆ W and ā„“(w) its Coxeter length, then Hn(Xš’¦(G);ā„¤)ā‰… ā؁wāˆˆ WHĢƒnāˆ’2ā„“(w)āˆ’1(š’¦DesR(w);ā„¤). Thus the induced subcomplexes of š’¦ 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.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026parabolic,
  title = {{Parabolic Homotopy Colimits and Coxeter Descents}},
  author = {Yifan Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.01882},
  eprint = {2609.01882},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.01882},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Parabolic Homotopy Colimits and Coxeter Descents
AU  - Yifan Zhang
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.01882
UR  - https://arxiv.org/abs/2609.01882
AB  - Let G be a compact, connected, simply connected semisimple Lie group with Weyl group W and simple reflections S. For a simplicial complex š’¦ on S, form the homotopy colimit X_š’¦(G)=*hocolim_Iāˆˆš’¦G/G_I of standard partial flag manifolds. We compute its integral homology. If Des_R(w) is the right descent set of wāˆˆ W and ā„“(w) its Coxeter length, then H_n(X_š’¦(G);ā„¤)ā‰… ā؁_wāˆˆ WHĢƒ_nāˆ’2ā„“(w)āˆ’1(š’¦_Des_R(w);ā„¤). Thus the induced subcomplexes of š’¦ 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.
ER  -