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

Let $k$ be a perfect field of characteristic different from 2, we compute the cellular ${\mathbb{A}}^1$-homology of the flag varieties $G\mathrm{/}{P}_{\mathrm{\Theta }}$ 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 $n\le 7$, 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 $S{L}_3\mathrm{/}B$ and an application to the full split flag variety of type $F_4$.