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Cellular 𝔸1-Homology from Bruhat Boundary Matrices of Split Semisimple Flag Varieties

  • Haoyang Liu
  • Tianle Liu
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Abstract

We determine the cellular 𝔸1-chain complex of full and partial flag varieties of split semisimple simply connected groups over a perfect field of characteristic different from 2. With compatible integral normal orientations and reindexing by opposite Bruhat cells, the differential in each degree at least two is represented by the motivic Hopf map η times one half of the signed cellular boundary matrix of the corresponding real flag manifold. Consequently, the Smith data of the resulting integer complex give all positive-degree cellular homology sheaves in terms of Milnor--Witt K-theory and the kernels and cokernels of integer multiples of η. We obtain a uniform computation of the second cellular 𝔸1-homology of SLn/B for n≥3, together with complete calculations for SL3/B and the full flags of types F4 and G2.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026cellular,
  title = {{Cellular $\mathbbA^1$-Homology from Bruhat Boundary Matrices of Split Semisimple Flag Varieties}},
  author = {Haoyang Liu and Tianle Liu},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2607.02985},
  eprint = {2607.02985},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.02985},
}
RIS
TY  - UNPB
TI  - Cellular 𝔸^1-Homology from Bruhat Boundary Matrices of Split Semisimple Flag Varieties
AU  - Haoyang Liu
AU  - Tianle Liu
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2607.02985
UR  - https://arxiv.org/abs/2607.02985
AB  - We determine the cellular 𝔸^1-chain complex of full and partial flag varieties of split semisimple simply connected groups over a perfect field of characteristic different from 2. With compatible integral normal orientations and reindexing by opposite Bruhat cells, the differential in each degree at least two is represented by the motivic Hopf map η times one half of the signed cellular boundary matrix of the corresponding real flag manifold. Consequently, the Smith data of the resulting integer complex give all positive-degree cellular homology sheaves in terms of Milnor--Witt K-theory and the kernels and cokernels of integer multiples of η. We obtain a uniform computation of the second cellular 𝔸^1-homology of SL_n/B for n≥3, together with complete calculations for SL_3/B and the full flags of types F_4 and G_2.
ER  -