Effective Bialynicki-Birula-Brosnan motivic decompositions

Abstract: Let $G$ be an isotropic reductive group and $X$ be a projective $G$-homogeneous variety. Using results from Bialynicki-Birula, Hesselink and Iversen, Brosnan showed that if $G$ is of inner type, the motive of $X$ can be expressed as a direct sum of Tate twists of motives of projective homogeneous varieties for the anisotropic kernel of $G$. We provide a SageMath implementation of this decomposition, based on a depth-first search of the Cayley graph of the Weyl group of $G$, ensuring the complexity scales with the size of the motive rather than the full Weyl group. As applications, we provide new motivic decompositions for some exceptional groups and show how to extend Karpenko's decompositions for classical groups to characteristic $2$.

• The paper introduces a SageMath algorithm that computes Bialynicki-Birula-Brosnan motivic decompositions by enumerating right-minimal Weyl group representatives and filtering for bi-minimality, reducing complexity to the motive’s absolute rank rather than the full Weyl group.
• The implementation handles difficult exceptional cases, including an E8 computation completed in roughly 250 seconds and yielding a new five-term motivic decomposition, while aggregation and rank checks validate the results.
• The paper extends motivic decomposition techniques to characteristic 2, deriving a six-summand formula for isotropic ideals of dimension 2 and identifying limitations involving outer-type groups and the rational-point versus degree-one zero-cycle condition.

Overview

This paper by Charles De Clercq addresses the practical computability of Bialynicki-Birula-Brosnan motivic decompositions. Brosnan's theorem [Theorem 7.4, bros] gives, for a semi-simple group $G$ of inner type over a field $F$ and a projective $G$-homogeneous variety $X_{\Theta,G}$ of type $\Theta$, an explicit decomposition in the category $\mathrm{CM}(F,\mathbb{Z})$ of Chow motives with integral coefficients:

$$M(X_{\Theta,G}) \simeq \bigoplus_{w \in \mathcal{W}} M(X_{\Theta_w,G_{an}})\{\ell(w)\},$$

where $\mathcal{W}$ is the set of minimal length representatives for the double cosets $W_{\Theta_0}\backslash W / W_{\Theta}$ (with $(\Delta_G,\Theta_0)$ the Tits index of $F$0), $F$1 is the anisotropic kernel, and $F$2. While explicit, these decompositions are intractable by hand when the Weyl group is large — notably for exceptional types such as $F$3, whose Weyl group has order $F$4. The paper provides a SageMath implementation (BBBdec.py) whose central algorithmic idea makes the cost scale with the size of the motive rather than with $F$5.

Algorithmic design

The input consists of a Dynkin type, the subset $F$6 encoding the Tits index, and the subset $F$7 specifying the variety type; the output lists all summands aggregated by motivic isomorphism class. The computation proceeds in three stages.

Enumeration of right-minimal representatives. The set $F$8 is computed by depth-first search on the Cayley graph of $F$9, starting from the identity. Since each simple reflection changes length by exactly one, the graph is graded by length; each candidate $G$0 is immediately reduced to its right-minimal representative via a ReduceRight routine that multiplies on the right by generators from $G$1 whenever this decreases length. Correctness follows from the uniqueness of the minimal representative in each right coset: once the queue empties, every element of $G$2 has been discovered.

Bi-minimality filter. The key improvement over naive approaches rests on the factorization property that any double-coset minimal element $G$3 (with $G$4, $G$5, lengths adding) satisfies that $G$6 is right-minimal and $G$7 left-minimal. Hence it suffices to filter $G$8 for the additional left-minimality condition, rather than running bi-minimal reduction on arbitrary elements of $G$9. Two equivalent tests are implemented — a length-based test comparing $X_{\Theta,G}$0 with $X_{\Theta,G}$1, and a root-based test checking $X_{\Theta,G}$2 — neither dominating the other in practice; they can be run in parallel as a cross-validation.

Aggregation and consistency checks. Each surviving representative contributes a summand determined by $X_{\Theta,G}$3, and contributions sharing the same triple are grouped to give multiplicities. The quantity $X_{\Theta,G}$4 equals the absolute rank of $X_{\Theta,G}$5, i.e., the number of Tate motives appearing after scalar extension, and serves as a per-summand size measure. The implementation enforces two checks: agreement between the two minimality tests, and the identity $X_{\Theta,G}$6, computed as a ratio of integers without polynomial division.

For split $X_{\Theta,G}$7, where $X_{\Theta,G}$8 is trivial, the algorithm bypasses Weyl group enumeration entirely and computes the decomposition via the quotient $X_{\Theta,G}$9 of Poincaré polynomials, using the Chevalley-Solomon formula available through SageMath's W.degrees().

Complexity

The enumeration costs $\Theta$0 time, where $\Theta$1 is the rank and $\Theta$2 the Coxeter number, versus $\Theta$3 for naive bi-minimal reduction over all of $\Theta$4 — a saving of a factor $\Theta$5. For type $\Theta$6 with $\Theta$7 of complemented size corresponding to $\Theta$8, this factor is $\Theta$9, nearly three million. More structurally, since $\mathrm{CM}(F,\mathbb{Z})$0 is precisely the absolute rank of $\mathrm{CM}(F,\mathbb{Z})$1, the complexity is naturally calibrated to the size of the motive under study. Memory is dominated by the seen set storing all of $\mathrm{CM}(F,\mathbb{Z})$2; retaining sample reduced words is optional.

Scope restrictions

Two limitations are stated plainly. First, the implementation handles only groups of inner type: although Bialynicki-Birula-Brosnan decompositions exist in outer type, they involve corestrictions over field extensions whose algorithmic encoding the author deems impractical. Second, the Tate trace mode assumes that every projective $\mathrm{CM}(F,\mathbb{Z})$3-homogeneous variety has a rational point if and only if it has a zero-cycle of degree $\mathrm{CM}(F,\mathbb{Z})$4 — a condition known for spinor and orthogonal groups, special and projective linear groups, and in general over number fields, but open for arbitrary fields. Under this hypothesis, an integral version of a lemma of De Clercq–Quéguiner-Mathieu shows that a summand $\mathrm{CM}(F,\mathbb{Z})$5 carries non-trivial integral Tate trace only when $\mathrm{CM}(F,\mathbb{Z})$6, in which case it is itself a Tate motive; the Tate trace mode therefore retains only those representatives satisfying $\mathrm{CM}(F,\mathbb{Z})$7 for all $\mathrm{CM}(F,\mathbb{Z})$8.

Explicit computations

Three examples illustrate the tool. For a $\mathrm{CM}(F,\mathbb{Z})$9-dimensional quadratic form of Witt index $$M(X_{\Theta,G}) \simeq \bigoplus_{w \in \mathcal{W}} M(X_{\Theta_w,G_{an}})\{\ell(w)\},$$0 (type $$M(X_{\Theta,G}) \simeq \bigoplus_{w \in \mathcal{W}} M(X_{\Theta_w,G_{an}})\{\ell(w)\},$$1, reproducing Brosnan's Example 7.6), the decomposition of the orthogonal Grassmannian $$M(X_{\Theta,G}) \simeq \bigoplus_{w \in \mathcal{W}} M(X_{\Theta_w,G_{an}})\{\ell(w)\},$$2 is recovered as $$M(X_{\Theta,G}) \simeq \bigoplus_{w \in \mathcal{W}} M(X_{\Theta_w,G_{an}})\{\ell(w)\},$$3, with $$M(X_{\Theta,G}) \simeq \bigoplus_{w \in \mathcal{W}} M(X_{\Theta_w,G_{an}})\{\ell(w)\},$$4 the anisotropic quadric and $$M(X_{\Theta,G}) \simeq \bigoplus_{w \in \mathcal{W}} M(X_{\Theta_w,G_{an}})\{\ell(w)\},$$5 its variety of isotropic planes. For type $$M(X_{\Theta,G}) \simeq \bigoplus_{w \in \mathcal{W}} M(X_{\Theta_w,G_{an}})\{\ell(w)\},$$6 with anisotropic kernel of type $$M(X_{\Theta,G}) \simeq \bigoplus_{w \in \mathcal{W}} M(X_{\Theta_w,G_{an}})\{\ell(w)\},$$7, the Tate trace mode yields $$M(X_{\Theta,G}) \simeq \bigoplus_{w \in \mathcal{W}} M(X_{\Theta_w,G_{an}})\{\ell(w)\},$$8. Most significantly, for type $$M(X_{\Theta,G}) \simeq \bigoplus_{w \in \mathcal{W}} M(X_{\Theta_w,G_{an}})\{\ell(w)\},$$9 with anisotropic kernel of type $\mathcal{W}$0 and the variety attached to vertex $\mathcal{W}$1, the computation produces a five-term decomposition

$\mathcal{W}$2

in roughly 250 seconds on standard hardware — a computation entirely out of reach of brute-force bi-minimal reduction over $\mathcal{W}$3. These constitute new motivic decompositions for exceptional groups.

Application to characteristic $\mathcal{W}$4

Karpenko's decompositions for classical groups [Corollary 15.14, kar] were established in characteristic different from $\mathcal{W}$5. The paper shows how the algorithm helps extend them to characteristic $\mathcal{W}$6 (with orthogonal involutions replaced by quadratic pairs for type $\mathcal{W}$7). The main result concerns a central-simple algebra $\mathcal{W}$8 with symplectic involution of degree $\mathcal{W}$9, $W_{\Theta_0}\backslash W / W_{\Theta}$0, Brauer-equivalent to a quaternion division algebra $W_{\Theta_0}\backslash W / W_{\Theta}$1, isotropic with orthogonal decomposition $W_{\Theta_0}\backslash W / W_{\Theta}$2 adjoint to $W_{\Theta_0}\backslash W / W_{\Theta}$3. In characteristic $W_{\Theta_0}\backslash W / W_{\Theta}$4, the motive of the variety $W_{\Theta_0}\backslash W / W_{\Theta}$5 of $W_{\Theta_0}\backslash W / W_{\Theta}$6-isotropic ideals of reduced dimension $W_{\Theta_0}\backslash W / W_{\Theta}$7 decomposes as

$W_{\Theta_0}\backslash W / W_{\Theta}$8

The proof identifies four minimal double-coset representatives ($W_{\Theta_0}\backslash W / W_{\Theta}$9, $(\Delta_G,\Theta_0)$0, $(\Delta_G,\Theta_0)$1, and the palindrome $(\Delta_G,\Theta_0)$2) extrapolated from low-rank outputs of the algorithm, verifies their minimality via the root criterion of Proposition 2.3, and obtains the two remaining summands by symmetry via push-forward along the exchange involution. Completeness follows because the absolute ranks of the six summands sum to the rank of the whole motive, a check independent of characteristic. This result is stated as a key tool in forthcoming work on classifying motives of projective homogeneous varieties with finite coefficients.

Limitations and open questions

Beyond the restriction to inner type and the hypothesis underlying the Tate trace mode discussed above, the memory footprint of the seen set may become significant for large groups, though disabling word retention mitigates this. The question whether the point-versus-zero-cycle condition holds over arbitrary fields remains open. The author also notes that the two implementations of the left-minimality test have incomparable performance across group types, so no single optimal choice exists.

Conclusion

The paper converts Brosnan's existence theorem into a practical computational tool, achieving complexity proportional to the absolute rank of the motive rather than the order of the Weyl group, and validates the implementation through new exceptional-type decompositions and a characteristic-$(\Delta_G,\Theta_0)$3 extension of Karpenko's classical-group results.