Структуры комодулей на кольцах Чжоу флаговых многообразий

1.1
Main Results
Comodule Structures
Let G be a connected complex reductive group, and B its Borel subgroup. The homogenous variety G/B is called the flag manifold of G.
Let W = NG(T )/T be the Weyl group and l the standard length function. For elements u, v, w ∈ W , we write the reduced decomposition w = u ⊙ v if w = uv and l(w) = l(u) + l(v).
For an element w ∈ W , we define the lower Schubert variety Σw to be the Zariski closure of BwB/B ⊆ G/B, and the (upper) Schubert variety Σw to be the Zariski closure of B−wB/B ⊆ G/B, where B− is the opposite Borel subgroup. It is well-known [Spr09] that the dimC Σw = codimC Σw = l(w). Furthermore, Σw and Σw are both isomorphic to affine linear spaces.
1.1.2 Let A•(X;R) be the Chow group of an algebra variety X, with coefficients in a commutative ring R. We denote CH•(X; R) the same ring but with degrees doubled. When R = Z, we will simply write CH•(X).
When X is smooth, there is a commutative ring structure over the Chow group. We will call it Chow ring in this case. For a closed subvariety Y of X, we denote [Y] the fundamental class of Y in CH•(X;R). Its degree is 2 times its complex codimension.
By the affine stratification theorem (for example [EH16]), the Chow ring of the flag variety G/B is freely generated by the Schubert classes [Σw] with w going through all elements of Weyl group W.
1.1.3 Let H•(X;R) be the cohomology ring of a space X with coefficients in a commutative ring R. When R = Z, we will simply write H•(X). We will denote the cup product by ⌣.
For any smooth algebraic variety X, there is a natural graded ring homomorphism ηX : CH•(X;R) → H•(X; R) sending the fundamental class [Y ] to the topological fundamental class (see [GH94]).
The affine stratification structure equips G/B a cellular structure thus ηG/B is an isomorphism.
1.1.4 We denote π∗ the map CH•(G/B; R) → CH•(G; R) induced by the natural projection π : G → G/B. By abuse of notation, we use the same name for the cohomology map H•(G; R) → H•(G/B; R).
Let α : G × G/B → G/B be the morphism of left action (g, xB) → gxB. Due to the relative affine stratification theorem, and the universal coefficient theorem, we have
CH•(G × G/B; R) = CH•(G; R) ⊗R CH•(G/B; R) H•(G × G/B; R) = H•(G; R) ⊗R H•(G/B; R).
We denote α∗ the natural map of Chow rings CH•(G/B; R) → CH•(G; R) ⊗R CH•(G/B : R) and the coho- mology map H•(G/B; R) → H•(G; R) ⊗R H•(G/B : R) induced by α.
1.1.1
1.1.5 By our discussion above, we have the following commuting diagram
• ooπ∗ • • • ooα∗ •
CH (G;R) CH (G/B;R) CH (G;R)⊗R CH (G/B;R) CH (G/B;R)
ηG/B ≀ ηG⊗ηG/B ηG/B ≀
ηG
H•(G; R) oo ∗ H•(G/B; R) H•(G; R) ⊗R H•(G/B; R) oo ∗
H•(G/B; R) As a result π∗ : H•(G/B; R) → H∗(G; R) factors through CH•(G; R). The same case for μ∗.
πα
1.1.6 The main result is
Theorem 1.1 The map α∗ induced by the left action α : G × G/B → G/B
w=u⊙v
is given by
The cohomology map α∗ has the same expression.
α∗ : CH•(G/B; R) −→ CH•(G; R) ⊗ CH•(G/B; R) [Σw] −→ π∗[Σu] ⊗ [Σv].
2
1.1.7 Let P be a standard parabolic subgroup P containing B. That is, P is given by a subset Θ of simple reflections of W. Let WP ⊆ W be the Weyl group generated by Θ. Then P = w∈WP BwB by the Tits system (see [Spr09]).
For the longest representative w for a coset wWP ∈ W/WP , we denote ΣwP the Zariski closure of B−wP/P ⊆ G/P. It is well-known the natural map Σw → ΣPw is an isomorphism. Then CH•(G/P) has a natural affine stratification structure, thus is freely generated by Schubert classes [ΣwP ] with w going through the longest representatives of all elements of W/WP . In particular ηG/P is an isomorphism, the natural map CH•(G/P;R)→CH•(G/B;R)inducedbyG/B→G/P maps[ΣwP]to[Σw],thusinjective. Thuswehavethe following
Corollary 1.2 The map αP∗ induced the left action G × G/P → G/P
αP∗ :CH•(G/P;R)−→CH•(G;R)⊗CH•(G/P;R)
is given by the same expression as in theorem 1.1.
1.1.8 When R be a field, it follows from [Gro58] that π∗ induces a surjection onto CH•(G; R), and CH•(G; R) is a subring of H•(G;R).
In this case, we denote μ∗ the natural map CH•(G; R) → CH•(G; R) ⊗R CH•(G; R) induced by the group multiplication μ : (x, y) → xy. We denote ν∗ the natural map CH•(G; R) → CH•(G; R) induced by the inverse μ : x → x−1. These two maps equip CH•(G) a Hopf algebra structure over R.
1.1.9 We can describe the Hopf structure of CH•(G;R) as follows.
Theorem 1.3 When R is a field, the Hopf structure of CH•(G;R) can be described by
and the antipode
μ∗ : π∗[Σw] −→
ν∗ : π∗[Σw] −→ (−1)l(w)π∗[Σw−1 ].
w=u⊙v
π∗[Σu] ⊗ π∗[Σv],
Note that the first assertion follows from Theorem 1.1.
1.1.10 Historically, the determination of the Hopf structure of compact Lie groups is quite tough. The classic approach is to use cohomology operators, see for example [IKT76]. The determination based on computer- assisted computation can be found in [DZ14], [DZ15]. The method using motives can be found in
1.2 Computational Results
1.2.1 Let Z(G) be the center of G. Then the quotient group G/Z(G) is semisimple of adjoint type and has the same flag variety as G’s by the natural morphism, for example [Spr09].
1.2.2 When R = Q, the map π∗ is trivial. That is, π∗ maps elements of positive degrees to 0. This follows from Borel’s presentation that H•(G/B;Q) is generated by the second Chern classes of G-equivariant line bundles. But pull back to G, it has to be trivial.
1.2.3 We will compute the case when G is semisimple, and R = Fp the finite field of p elements where p is a prime number.
We will give an explicit formula for π∗ and α∗ for A-type, C-type. We will develop an algorithm to compute π∗, see Algorithm 5.1.5, so that we can describe π∗ and α∗ for G2, F4 and some partial result for E6 and E7.
TheresultsofthispaperaresummarizedinTable1,wheredegxi =iinCH•(G;Fp).