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arXiv · 2607.02985

Cellular $\mathbb{A}^1$-Homology from Bruhat Boundary Matrices of Split Semisimple Flag Varieties

Haoyang Liu·Tianle Liu
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Abstract

Let $k$ be a perfect field of characteristic different from 2, we compute the cellular $\mathbb{A}^1$-homology of the flag varieties $G/P_\Theta$ attached to split semisimple simply connected groups over $k$ and describe the differentials in the cellular $\mathbb{A}^1$-chain complex concretely. The construction applies uniformly to the type $A$ coefficient formula, to the type $B_n,C_n,D_n$ for $n\leq 7$, and to the exceptional types for $F_4,E_6,E_7$. Under real realization over $k=\mathbb{R}$, this computation recovers the corresponding results of real flag manifolds. We also provide a detailed computation for $SL_3/B$ and an application to the full split flag variety of type $F_4$.

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Haoyang Liu, Tianle Liu. 2026-07-03. Cellular $\mathbb{A}^1$-Homology from Bruhat Boundary Matrices of Split Semisimple Flag Varieties. https://arxiv.org/abs/2607.02985

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