BBBdec — Effective Bialynicki-Birula-Brosnan motivic decompositions

BBBdec is a SageMath script computing Bialynicki-Birula-Brosnan motivic decompositions of projective homogeneous varieties for absolutely simple algebraic groups of inner type, following Brosnan's On motivic decompositions arising from the method of Bialynicki-Birula (Thm. 7.4).

Given an absolutely simple group G with Tits index (Δ_G, Θ₀) (all groups will be assumed with this assumptions in the sequel) and a projective G-homogeneous variety X_Θ,G, the script computes the decomposition in terms of the underlying combinatorics :

M(X_Θ,G) ≃ ⊕_{w ∈ 𝒲} M(X_Θw,Gan){ℓ(w)}

where the sum runs over the set 𝒲 of minimal length double coset representatives for W_Θ0\W/W_Θ, and G_an is the anisotropic kernel of G.

The script is described in "Effective Bialynicki-Birula-Brosnan motivic decompositions", see [PDF]

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BBBdec.py — SageMath 9.5 script.

Usage: sage BBBdec.py --dynkin TYPE --Sigma0 LIST --J LIST [options]

Main flags:

Note on conventions.

Examples for each Dynkin diagram type

Example 1 — Type A_n
Flags of ideals. Let A be a central simple algebra, say of degree 6 and index 3. Consider the variety X(1,2,3;A) of flags of right ideals in A of reduced dimension 1,2,3.

Result: M(X(1,2,3;A)) ≃ M(X(1,2;D)){0} ⊕ M(X(1,2,4;D))^⊕3{1,2,3} ⊕ M(X(1,4,5;D))^⊕3{4,5,6} ⊕ M(X(4,5;D)){9}
where D is the division algeba Brauer-equivalent to A.

Example 2 — Type B_n
Orthogonal Grassmannians. Let q be a non-degerenate quadratic form of dimension 13, and Witt index 2. Consider the variety X(2,3;q) of flags of isotropic subspaces for q of dimension 2,3.

Result: M(X(2,3;q)) ≃ M(X(3;q_an))^⊕12{0,1,2,8,9,9,10,10,11,17,18,19}⊕M(X(3,4;q_an))^⊕4{2,3,11,12}⊕M(X(4,5;q_an)){6}⊕M(X(4;q_an))^⊕4{4,5,10,11}
where X( — ;q_an) are varieties of flags of isotropic subspaces for the anisotropic part of q.

Example 3 — Type C_n
Flags of isotropic ideals for symplectic involution. Let (A,σ) be central simple algebra of degree 12, index 2, of Witt index 1. Consider the variety X(2;(A,σ)) of isotropic ideals with respect to σ of reduced dimension 2.

Result: X(2;(A,σ)) ≃ ℤ{0} ⊕ M(SB(A) x X_1,Gan){1} ⊕ M(X_2,Gan){4} ⊕ M(SB(A)){9} ⊕ M(SB(A) x X_1,Gan){10} ⊕ ℤ{19}
where SB(A) is the Severi-Brauer variety of A and G_an is the anisotropic kernel of G.

Example 4 — Type D_n
Orthogonal Grassmannians (recovering Brosnan, Ex. 7.6). Let q be a 12-dimensional quadratic form of Witt index 1, and X(2;q) the variety of isotropic planes.

Result: M(X(2;q)) ≃ M(X(1;q_an)){0} ⊕ M(X(2;q_an)){2} ⊕ M(X(1;q_an)){9}
where X( — ;q_an) are orthogonal Grassmanians for q_an.

Example 5 — Type E₆
Let G be a group of type E₆, of Tits index given by Θ₀={2,3,4,5} (anisotropic kernel of type D₄). Consider a projective G-homogeneous variety of type {1}.

Result: M(X_1,G) ≃ ℤ{0} ⊕ M(X_1,Gan){1} ⊕ M(X_3,Gan){5} ⊕ ℤ{8} ⊕ M(X_4,Gan){9} ⊕ ℤ{16}

Example 6 — Type E₇
Let G be a group of type E₇, of Tits index given by Θ₀={2,3,4,5} (anisotropic kernel of type D₄). Consider a projective G-homogeneous variety of type {1}.

Example 7 — Type E₈.
Let G be a group of type E₈, of Tits index given by Θ₀={2,3,4,5,6,7,8} (anisotropic kernel of type D₇). Consider a projective homogeneous variety of type {8}.

Result: M(X_8,G) ≃ M(X_1,Gan){0} ⊕ M(X_6,Gan){7} ⊕ M(X_2,Gan){18} ⊕ M(X_7,Gan){29} ⊕ M(X_1,Gan){45}.

Example 8 — Type F₄.
Let G is a group of type F₄, of Tits index given by Θ₀={1,2,3} (anisotropic kernel of type B₃). Consider a projective G-homogeneous variety of type {2}.

Result: M(X_2,G) ≃ M(X_2,Gan){0} ⊕ M(X_{1,2},Gan){3} ⊕ M(X_{1,3},Gan){6} ⊕ M(X_{1,2},Gan){9} ⊕ M(X_2,Gan){13}.

Example 9 — Type G₂ (split case fork).
Let G be a split group of type G₂. Consider a projective G-homogeneous variety of type {1,2}.

Result: M(X_{1,2},G) ≃ ℤ{0} ⊕ ℤ{1}^⊕2 ⊕ ℤ{2}^⊕2 ⊕ ℤ{3}^⊕2 ⊕ ℤ{4}^⊕2 ⊕ ℤ{5}^⊕2 ⊕ ℤ{6}.

Example 10 — Tate trace mode.
Let G be a group of type E₆, of Tits index given by Θ₀={2,3,4,5} (anisotropic kernel of type D₄).

Result: Tr(X_{1,6},G) = ℤ{0} ⊕ ℤ{8}^⊕2 ⊕ ℤ{16}^⊕2 ⊕ ℤ{24}.