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 $G/P_\Theta$ 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 $F_4$.