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

Abstract: Let $k$ be a perfect field of characteristic different from 2, we compute the cellular $\mathbb{A}<sup>1$-homology of the flag varieties $G/P_Θ$ attached to split semisimple simply connected groups over $k$ and describe the differentials in the cellular $\mathbb{A}<sup>1$-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 $F_4$.

• The paper provides an explicit chain-level computation linking classical signed Bruhat boundary matrices to motivic A¹-differentials in split flag varieties.
• It employs Smith normal form to decompose Milnor–Witt K-theory-based homology sheaves, distinguishing free summands from torsion elements reflective of classical 2-torsion.
• The approach applies uniformly to types A, B, C, D (n≤7) and exceptional groups, offering a blueprint for algorithmic motivic invariant computations in algebraic geometry.

Cellular $\mathbb{A}^1$-Homology from Bruhat Boundary Matrices: Flag Varieties in the Range of Computable Signed Boundaries

Overview and Objectives

The paper "Cellular $\mathbb{A}^1$-Homology from Bruhat Boundary Matrices of Split Semisimple Flag Varieties" (2607.02985) establishes concrete chain-level computations for motivic (cellular $\mathbb{A}^1$-) homology sheaves of split flag varieties $G/P_\Theta$ over a perfect field $k$ of characteristic $\neq 2$. The main advance is an explicit dictionary between classical signed boundary matrices in Bruhat cell decompositions (as constructed in topology for integral homology of real flag manifolds) and the differentials of the cellular $\mathbb{A}^1$-complex of Morel and Sawant, with coefficients in Milnor--Witt $K$-theory.

The work treats types where explicit signed boundary matrices are available: type $A$, classical types $B_n$, $\mathbb{A}^1$0, $\mathbb{A}^1$1 for $\mathbb{A}^1$2, and exceptional types $\mathbb{A}^1$3, $\mathbb{A}^1$4, $\mathbb{A}^1$5. It provides a uniform motivic lift of the integral combinatorial data, applicable in arbitrary characteristic $\mathbb{A}^1$6 and for all perfect base fields.

Structure of the Approach

Cellular $\mathbb{A}^1$7-Homology and Bruhat Decompositions

The Morel--Sawant cellular theory endows cellular filtrations by affine spaces (such as Bruhat cells) with an explicit $\mathbb{A}^1$8-homology complex, whose chain groups are free modules over strictly $\mathbb{A}^1$9-invariant sheaves, specifically Milnor--Witt $\mathbb{A}^1$0-theory sheaves. The crucial nontrivial input is the computation of the boundary maps, which encode orientation-sensitive data even in the split case due to nontrivial signs and combinatorial commutator relations in the Coxeter/Weyl group.

While real (topological) flag manifolds have signed integral boundary maps determining their homology, the motivic analogue—refining the universal coefficients, introducing $\mathbb{A}^1$1-torsion, and accounting for quadratic orientation—requires a careful motivic lift of these signed chains.

Explicit Motivic Chain Model

The authors provide the following algorithm:

• Cells and Chain Indexing: The Bruhat decomposition $\mathbb{A}^1$2 induces a filtration whose (co)dimension grading matches the length function relative to the maximal element and the parabolic subgroup. Cells are indexed in reverse Bruhat order for compatibility with the Morel--Sawant convention.
• Signed Boundary Complex: The ordinary (real) signed boundary map, denoted $\mathbb{A}^1$3, is taken from known combinatorial formulas and algorithmic tables for types $\mathbb{A}^1$4, $\mathbb{A}^1$5, $\mathbb{A}^1$6, $\mathbb{A}^1$7 ($\mathbb{A}^1$8), $\mathbb{A}^1$9, $G/P_\Theta$0, $G/P_\Theta$1.
• Motivic Differential: The motivic boundary in the chain complex is given by multiplying the (reindexed) classical boundary matrix entries by $G/P_\Theta$2, producing a complex over the category of strictly $G/P_\Theta$3-invariant sheaves.

Formally, the boundary differential is

$G/P_\Theta$4

for $G/P_\Theta$5, where $G/P_\Theta$6 is a reindexed form of $G/P_\Theta$7 matching the codimension indexing, and vanishes in degree $G/P_\Theta$8.

• Coefficient Computations: Each boundary component corresponding to a Bruhat cover is described explicitly, incorporating the orientation sign (as a power of $G/P_\Theta$9), the local degree (via Milnor--Witt theory), and a motivic refinement of the coroot height factor—mirroring the classical $k$0-torsion in real integral homology.

Chain-Level Lifting and Smith Normal Form

An explicit Smith normal form analysis is applied to the integral chain complex $k$1, ensuring an algorithmic and functorial identification of torsion and free summands at the sheaf level. Motivically, elementary torsion summands ($k$2 in ordinary homology) refine to cokernel and kernel sheaves for multiplication by $k$3. The connection between real homology $k$4-torsion and motivic $k$5-torsion is made precise, and concrete computation recipes are supplied.

Main Results

Motivic Boundary Formula

Let $k$6 be a Bruhat cover. Then, the motivic boundary coefficient (Theorem/Corollary) is

$k$7

where $k$8 is the deleted position, $k$9 is the crossing root, and the various terms enumerate the orientation, local degree, and motivic coroot-height refinement. The Smith normal form algorithm translates the matrix data to direct sum decompositions of the resulting motivic homology sheaves, fully functorial in $\neq 2$0.

Smith Decomposition and Homology Sheaf Structure

The explicit computation yields, for any $\neq 2$1,

$\neq 2$2

where the elementary divisors $\neq 2$3 arise from the Smith decomposition of $\neq 2$4, and the $\neq 2$5 terms correspond to the structure of the Milnor--Witt ring. In classical and exceptional types where only $\neq 2$6-torsion is present (all nonzero $\neq 2$7), the formula simplifies and directly reflects the Betti and $\neq 2$8-torsion topological data over $\neq 2$9.

Type $\mathbb{A}^1$0 and Beyond

For type $\mathbb{A}^1$1 (and all types where only $\mathbb{A}^1$2-torsion occurs in topological real homology), the result confirms and explains prior computations of Chow--Witt rings and motivic cohomology, in particular the appearance of only $\mathbb{A}^1$3-torsion in integral (co)homology. The motivic chain-level result is thus compatible and extensions of Hudson–Matszangosz–Wendt [see J. Topol. 17 (2024), e70004], but now describes all cases with computable signed boundary tables.

Case Studies: $\mathbb{A}^1$4 and $\mathbb{A}^1$5

An explicit calculation for $\mathbb{A}^1$6 is carried out, with standard cell indexing and orientations, confirming that the only nontrivial motivic boundaries are $\mathbb{A}^1$7 in codimension $\mathbb{A}^1$8, refining the known $\mathbb{A}^1$9 entries for the real flag manifold. For the full flag variety of type $K$0, the classical combinatorics are used to obtain all Betti numbers and $K$1-torsion ranks, and the motivic homology sheaves are listed, demonstrating the tractability of the approach, even for exceptional types.

Implications and Outlook

This work provides a blueprint for the functorial computation of motivic (cellular $K$2-) homology of flag varieties in all types admitting effective signed boundary formulas. It demonstrates that, for split semisimple groups with such available data, real combinatorics and toric orientation conventions can be fully lifted to the motivic context, capturing the finer structure of quadratic forms and motivic $K$3-torsion.

Practically, this enables explicit calculation of not only (co)homology sheaves but also Chow--Witt and other generalized motivic invariants for broad families of homogeneous varieties. The method is compatible with established computer algebra implementations, and can absorb new signed boundary computations for other types as they become available.

Theoretically, the chain-level identification sharpens the understanding of how orientation, root data, and Milnor--Witt theory interact in geometric representation theory and the study of homogeneous spaces. The motivic refinement preserves arithmetic and quadratic information invisible to classical homology, opening further lines of inquiry in the structure of algebraic cycles, quadratic refinements, and their relationships across fields.

Conclusion

By providing an explicit, algorithmic correspondence between the integral signed Bruhat boundary matrices for split flag varieties and the motivic differentials in cellular $K$4-homology, the paper (2607.02985) achieves a chain-level description of homological invariants with full functoriality in the base field (excluding characteristic $K$5). This result expands the calculational toolkit for algebraic geometers and topologists studying flag varieties and deepens the connection between classical and quadratic/motivic invariants in algebraic geometry. As boundary data and computational techniques expand, the method will permeate broader classes of groups and varieties.