Motivic decomposition of anisotropic varieties of type F_{4} and generalized Rost motives^{∗}

S.Nikolenko, N.Semenov, K.Zainoulline

February 17, 2005

Abstract

We provide an explicit decomposition of the motives of two anisotropic projective homogeneous varieties of type F_{4}. In particular, we provide an explicit construction of a generalized Rost motive for a norm variety that corresponds to a symbol (3, 3).

We also establish a motivic isomorphism between two projective homogeneous varieties of type F_{4}. We provide an explicit construction of this isomorphism.

All our proofs work for Chow motives with integral coefficients.

1 Introduction

The main motivation for our work was the result of N. Karpenko where he gave a shortened construction of a Rost motive for a norm quadric [Ka98]. In the present paper we provide a shortened and explicit construction of a generalized Rost motive for a norm variety that corresponds to a symbol (3, 3). The latter is given by the Rost–Serre invariant g_{3} for an Albert algebra and the respective norm variety is a projective homogeneous variety of type F_{4}. Namely, we prove the following

1.1 Theorem. Let k be a field of characteristic different from 2 and 3. Let X = G/P be a projective homogeneous variety over k, where G is an anisotropic group of type F_{4} obtained by the first Tits process and P its maximal parabolic subgroup corresponding to the first (last) three vertices of the respective Dynkin diagram. Then

∗MSC2000: 14C15, 57T15 Keywords: anisotropic homogeneous variety, Rost motive, exceptional algebraic group

(i) the (integral) Chow motive of X decomposes as

M(X) ≅ R ⊕ R_{12} ⊕ R(3) ⊕ R^{t} ⊕ R_{12}^{t} ⊕ R^{t}(3),

where the motive R = (X, p) is the (integral) generalized Rost motive, i.e., over the separable closure k′ of k it splits as the direct sum of Lefschetz motives ℤ ⊕ ℤ(4) ⊕ ℤ(8), the motive R^{t} = (X, p^{t}) denotes its transpose and the motive R_{12} is isomorphic over k′ to the direct sum R(1) ⊕ R(2);

(ii) the Chow motive of X with ℤ/3-coefficients decomposes as

M(X, ℤ/3) ≅ ⨁_{i=0}^{3}(R(i) ⊕ R^{t}(i)).

1.2. Observe that the part (ii) of the Theorem is the consequence of Voevodsky’s results [Vo03, § 5], where the generalized Rost motive with ℤ/pcoefficients (p is any prime) was constructed.

By the next result, we provide the first known “purely exceptional” example of two non-isomorphic varieties with isomorphic motives. Recall that the similar result for groups of type G_{2} obtained in [Bo03] provides a motivic isomorphism between quadric and Fano variety.

1.3 Theorem. Under the hypotheses of theorem 1.1 let X_{1} and X_{2} be two projective homogeneous varieties corresponding to the maximal parabolic subgroups generated by the last (first) three vertices of the Dynkin diagram respectively. Then the motives of X_{1} and X_{2} are isomorphic.

1.4. Our proof is quite elementary. It uses well-known facts concerning linear algebraic groups and projective homogeneous varieties, a computer program that computes the Chow ring CH(G/P) for a split group G, Rost Nilpotence Theorem and several simple procedures that allow to produce rational cycles on CH(G/P × G/P). Moreover, the proof works not only for projective homogeneous varieties of type F_{4}. Applying the similar arguments to Pfister quadrics and their maximal neighbors one obtains the well-known decompositions into Rost motives [Ro98]. For exceptional groups of type G_{2} one immediately obtains the motivic decomposition of the variety G_{2}/P_{2} together with the motivic isomorphism found by J.-P. Bonnet [Bo03].

1.5. The paper is organized as follows. In Section 2 we provide background information on Chow motives and rational cycles. Section 3 is devoted to computational matters of Chow rings. Namely, we introduce Pieri and Giambelli formulae and discuss their relationship with Hasse diagrams. In Section 4 we apply the formulae introduced in Section 3 to projective homogeneous varieties X_{1} and X_{2} of type F_{4}. In Section 5 we prove Theorem 1.1. Section 6 is devoted to the proof of Theorem 1.3. In Appendix I we provide the complete multiplication tables for Chow rings of X_{1} and X_{2} obtained by using the algorithm described in Appendix II.

2 Rational cycles on projective homogeneous varieties

2.1. Let k be a field and Var_{k} be a category of smooth projective varieties over k. We define the category of Chow motives (over k) following [Ma68].

First, we define the category of correspondences (over k) denoted by Cor_{k}. Its objects are smooth projective varieties over k. For morphisms, called correspondences, we set Mor(X, Y) := CH^{dim X}(X × Y). For any two correspondences α ∈ CH(X × Y) and β ∈ CH(Y × Z) we define the composition β ∘ α ∈ CH(X × Z)

β ∘ α = pr_{13}_{∗}(pr_{12}^{∗}(α) · pr_{23}^{∗}(β)),  (1)

where pr_{ij} denotes the projection on the i-th and j-th factors of X × Y × Z respectively and pr_{ij}_{∗}, pr_{ij}^{∗} denote the induced push-forwards and pull-backs for Chow groups.

The pseudo-abelian completion of Cor_{k} is called the category of Chow motives and is denoted by M_{k}. The objects of M_{k} are pairs (X, p), where X is a smooth projective variety and p is a projector, that is, p ∘ p = p. The motive (X, id) will be denoted by M(X).

2.2. By the construction M_{k} is a tensor additive category with self-duality. given by the transposition of cycles α ↦ α^{t}. Moreover, the Chow functor CH : Var_{k} → ℤ-Ab (to the category of ℤ-graded abelian groups) factors through M_{k}, that is, one has

\xymatrix{Var_{k} \ar[rr]^{CH} \ar[dr]_{Γ} &  & ℤ-Ab \\  & M_{k} \ar[ur]_{R}}

where Γ : f ↦ Γ_{f} is the graph and R : (X, p) ↦ im(p^{∗}) is the realization.

2.3. Observe that the composition product ∘ induces the ring structure on the abelian group CH^{dim X}(X × X). The unit element of this ring is the class of the diagonal map Δ_{X} such that Δ_{X} ∘ α = α ∘ Δ_{X} = α for all α ∈ CH^{dim X}(X × X).

2.4. Let G be a split linear algebraic group over k. Let X be a projective G-homogeneous variety, i.e., X = G/P, where P is a parabolic subgroup of G. The abelian group structure of CH(X), as well as its ring structure, is well-known. Namely, X has a cellular filtration and the base of this filtration generates the free abelian group CH(X) (see [Ka01]). Note that the product of two projective homogeneous varieties X×Y has a cellular filtration as well, and CH^{∗}(X × Y) ≅ CH^{∗}(X) ⊗ CH^{∗}(Y) as graded rings. The correspondence product of two cycles α = f_{α}×g_{α} ∈ CH(X×Y) and β = f_{β}×g_{β} ∈ CH(Y ×X) is given by (cf. [Bo03, Lem. 5])

(f_{β} × g_{β}) ∘ (f_{α} × g_{α}) = deg(g_{α} · f_{β})(f_{α} × g_{β}),  (2)

where deg : CH(Y) → CH({pt}) = ℤ is the degree map.

2.5. Let X be a projective variety of dimension n over a field k. Let k′ be the separable closure of the field k. Consider the scalar extension X′ = X ×_{k} k′. We say a cycle J ∈ CH(X′) is rational if it lies in the image of the pull-back homomorphism CH(X) → CH(X′). For instance, there is an obvious rational cycle Δ_{X′} on CH^{n}(X′ × X′) that is given by the diagonal class. Clearly, any linear combinations, intersections and correspondence products of rational cycles are rational.

2.6. Several techniques allow to produce rational cycles (cf. [Ka04, Prop. 3.3] for the case of quadrics). We shall use the following:

(i) Consider a variety Y and a morphism X → Y such that X′ = Y′ ×_{Y} X, where Y′ = Y ×_{k} k′. Then any rational cycle on CH(Y′) gives rise to a rational cycle on CH(X′) by the induced pull-back CH(Y′) → CH(X′).

(ii) Consider a variety Y and a projective morphism Y → X such that Y′ = X′ ×_{X} Y. Then any rational cycle on CH(Y′) gives rise to a rational cycle on CH(X′) by the induced push-forward CH(Y′) → CH(X′).

(iii) Let X and Y be projective homogeneous varieties over k that split completely over the function fields k(Y) and k(X) respectively. Consider the following pull-back diagram

\xymatrix{CH^{i}(X × Y) \ar[d]_{f} \ar[r]^{g} & CH^{i}(X′ × Y′) \ar[d]^{f′} \\ CH^{i}(X_{k(Y)}) \ar[r]^{=} & CH^{i}(X′_{k′(Y′)})}

where the vertical arrows are surjective by [IK00, §5]. Now take any cycle α ∈ CH^{i}(X′ × Y′), i ≤ dim X. Let β = g(f^{−1}(f′(α))). Then f′(β) = f′(α) and β is rational. Hence, β = α + J, where J ∈ kerf′, and we conclude that α + J ∈ CH^{i}(X′ × Y′) is rational.

2.7. We will use the following fact (see [CGM, Cor. 8.3]) that follows from the Rost Nilpotence Theorem. Let p′ be a non-trivial rational projector on CH^{n}(X′ × X′), i.e., p′ ∘ p′ = p′. Then there exists a non-trivial projector p on CH^{n}(X × X) such that p ×_{k} k′ = p′. Hence, the existence of a non-trivial rational projector p′ on CH^{n}(X′ × X′) gives rise to the decomposition of the Chow motive of X

M(X) ≅ (X, p) ⊕ (X, 1 − p)

We shall find such a projector in the case of an “exceptional” projective homogeneous variety “of type F_{4}”.

3 Hasse diagrams and Chow rings

3.1. To each projective homogeneous variety X we may associate an oriented labeled graph H called Hasse diagram. It is known that the ring structure of CH(X) is determined by H. In the present section we remind several facts concerning relations between Hasse diagrams and Chow rings. For a precise reference on this account see [De74], [Hi82a] and [Ko91].

3.2. Let G be a split simple algebraic group defined over a field k. We fix a maximal split torus T in G and a Borel subgroup B of G containing T and defined over k. Denote by Φ the root system of G, by Π the set of simple roots of Φ corresponding to B, by W the Weyl group, and by S the corresponding set of fundamental reflections.

Let P = P_{Θ} be a (standard) parabolic subgroup corresponding to a subset Θ ⊂ Π, i.e., P = BW_{Θ}B, where W_{Θ} = ⟨s_{θ} | θ ∈ Θ⟩. Denote

W^{Θ} = {w ∈ W | ∀ s ∈ Θ l(ws) = l(w) + 1},

where l is the length function. The pairing

W^{Θ} × W_{Θ} → W (w, v) ↦ wv

is a bijection and l(wv) = l(w)+l(v). It is easy to see that W^{Θ} consists of all representatives in the cosets W/W_{Θ} which have minimal length. Sometimes it is also convenient to consider the set of all representatives of maximal length. We shall denote this set as ^{Θ}W. Observe that there is a bijection W^{Θ} → ^{Θ}W given by v ↦ vw_{θ}, where w_{θ} is the longest element of W_{Θ}. The longest element of W^{Θ} corresponds to the longest element w_{0} of the Weyl group.

3.3. To a subset Θ of the finite set Π we associate an oriented labeled graph, which we call a Hasse diagram and denote by H_{W} (Θ). This graph is constructed as follows. The vertices of this graph are the elements of W^{Θ}. There is an edge from a vertex w to a vertex w′ with a label i if and only if l(w) < l(w′) and w′ = ws_{i}. The example of such a graph is provided in 4.4. Observe that the diagram H_{W} (∅) coincides with the Cayley graph associated to the pair (W, S).

3.4 Lemma. The assignment H_{W} : Θ ↦ H_{W} (Θ) is a contravariant functor from the category of subsets of the finite set Π (with embeddings as morphisms) to the category of oriented graphs.

Proof. It is enough to embed the diagram H_{W} (Θ) to the diagram H_{W} (∅). We do this as follows. We identify the vertices of H_{W} (Θ) with the subset of vertices of H_{W} (∅) by means of the bijection W^{Θ} → ^{Θ}W. Then the edge from w to w′ of ^{Θ}W ⊂ W has a label i if and only if l(w) < l(w′) and w′ = s_{i}w (as elements of W). Clearly, the obtained graph will coincide with H_{W} (Θ).

3.5. Now consider the Chow ring of a projective homogeneous variety G/P_{Θ}. It is well known that CH(G/P_{Θ}) is a free abelian group with a basis given by the varieties [X_{w}] that correspond to the vertices w of the Hasse diagram H_{W} (Θ). The degree of the basis element [X_{w}] corresponds to the minimal number of edges needed to connect the respective vertex w with w_{θ} (which is the longest one). The multiplicative structure of CH(G/P_{Θ}) depends only on the root system of G and the diagram H_{W} (Θ).

3.6 Lemma. The contravariant functor CH: Θ ↦ CH(G/P_{Θ}) factors through the category of Hasse diagrams H_{W}, i.e., the pull-back (ring inclusion)

CH(G/P_{Θ′}) ↪ CH(G/P_{Θ})

arising from the embedding Θ ⊂ Θ′ is induced by the embedding of the respective Hasse diagrams H_{W} (Θ′) ⊂ H_{W} (Θ).

3.7 Corollary. Let B be a Borel subgroup of G and P its (standard) parabolic subgroup. Then CH(G/P) is a subring of CH(G/B). The generators of CH(G/P) are [X_{w}], where w ∈ ^{Θ}W ⊂ W. The cycle [X_{w}] in CH(G/P) has the codimension l(w_{0}) − l(w).

Proof. Apply the lemma to the case B = P_{∅} and P = P_{Θ′}.

Hence, in order to compute CH(G/P) it is enough to compute CH(X), where X = G/B is the variety of complete flags. The following results provide tools to perform such computations.

3.8. In order to multiply two basis elements h and g of CH(G/P) such that deg h + deg g = dim G/P we use the following formula (see [Ko91, 1.4]):

[X_{w}] · [X_{w′}] = δ_{w,w_{0}w′w_{θ}} · [pt].

3.9 (Pieri formula). In order to multiply two basis elements of CH(X) one of which is of codimension 1 we use the following formula (see [De74, Cor. 2 of 4.4]):

[X_{w_{0}s_{α}}][X_{w}] = ∑_{β∈Φ^{+}, l(ws_{β})=l(w)−1}⟨β^{∨}, \overline{ω}_{α}⟩[X_{ws_{β}}],

where the sum runs through the set of positive roots β ∈ Φ^{+}, s_{α} denotes the simple reflection corresponding to α and ω̄_{α} is the fundamental weight corresponding to α. Here [X_{w_{0}s_{α}}] is the element of codimension 1.

3.10 (Giambelli formula). Let P = P (Φ) be the weight space. We denote as ω̄_{1}, … ω̄_{l} the basis of P consisting of fundamental weights. The symmetric algebra S^{∗}(P) is isomorphic to ℤ[ω̄_{1}, … ω̄_{l}]. The Weyl group W acts on P, hence, on S^{∗}(P). Namely, for a simple root α_{i},

w_{α_{i}}(ω̄_{j}) = \begin{cases}ω̄_{i} − α_{i}, & i = j, \\ ω̄_{j}, & otherwise. \end{cases}

We define a linear map c: S^{∗}(P) → CH^{∗}(G/B) as follows. For a homogeneous u ∈ ℤ[ω̄_{1}, …, ω̄_{l}]

c(u) = ∑_{w∈W, l(w)=deg(u)} Δ_{w}(u)[X_{w_{0}w}],

where for w = w_{α_{1}} … w_{α_{k}} we denote by Δ_{w} the composition of derivations Δ_{α_{1}} ∘ … ∘ Δ_{α_{k}} and the derivation Δ_{α_{i}}: S^{∗}(P) → S^{∗−1}(P) is defined by Δ_{α_{i}}(u) = \frac{u−w_{α_{i}}(u)}{α_{i}}. Then (see [Hi82a, ch. IV, 2.4])

[X_{w}] = c(Δ_{w^{−1}}(\frac{d}{|W |})),

where d is the product of all positive roots in S^{∗}(P). In other words, the element Δ_{w^{−1}}(\frac{d}{|W |}) ∈ c^{−1}([X_{w}]).

Hence, in order to multiply two basis elements h, g ∈ CH(X) take their preimages under the map c and multiply them in S^{∗}(P) = ℤ[ω̄_{1}, … ω̄_{l}]. Then apply c to their product.

4 Projective homogeneous varieties of type F_{4}

4.1. From now on, we assume that the characteristics of the base field k is not equal to 2 or 3. In the present section we remind several well-known facts concerning Albert algebras, groups of type F_{4} and respective projective homogeneous varieties (see [PR94], [Inv], [Gar97]). At the end we provide partial computations of Chow rings of these varieties.

We start with the following observation concerning the Picard group of a projective homogeneous variety of type F_{4} 4.2 Lemma. Let G be an anisotropic group of type F_{4} over a field k and P be its maximal parabolic subgroup. Let X = G/P be a respective projective homogeneous variety and X′ = X ×_{k} k′ be its scalar extension to the separable closure k′. Then the Picard group Pic(X′) is a free abelian group of rank 1 with a rational generator.

Proof. Since P is maximal, Pic(X′) is a free abelian group of rank 1. We use the following exact sequence (see [Ar82] and [MT95, 2.3]):

0 ⟶ Pic X ⟶ (Pic X′)^{Γ} ⟶^{α_{X}} Br(k),

where Γ = Gal(k′/k) is the absolute Galois group and Br(k) the Brauer group of k. The map α_{X} is explicitly described in [MT95] in terms of Tits classes. Since groups of type F_{4} are adjoint and simply-connected, their Tits classes are trivial and so is α_{X}. Since Γ acts trivially on Pic(X′) and the image of α_{X} is trivial, we have Pic(X) ≃ Pic(X′).

4.3. It is well known that the classification of algebraic groups of type F_{4} is equivalent to the classification of Albert algebras (those are 27-dimensional exceptional simple Jordan algebras). All Albert algebras can be obtained from one of the two Tits constructions.

An Albert algebra A obtained by the first Tits construction is produced from a central simple algebra of degree 3. By using the Rost-Serre invariant g_{3} (if the input central simple algebra is split, then g_{3} = 0) one can show that for the respective group G = Aut(A) only two Tits diagrams ([Ti66, Table II]) are allowed, namely the completely split case and the anisotropic case. This means that

(i) anisotropic G splits completely by a cubic field extension;

(ii) for each maximal parabolic subgroup P_{i}, i = 1, 2, 3, 4, the respective projective homogeneous varieties X = G/P_{i} split completely over the function field k(G/P_{j}), j = 1, 2, 3, 4.

4.4. From this point on we consider a split group G of type F_{4} obtained by the first Tits process. Let X_{1} = G/P_{1} and X_{2} = G/P_{4} be projective homogeneous varieties, corresponding to maximal parabolic subgroups P_{1} and P_{4} generated by the last {2, 3, 4} and the first {1, 2, 3} three vertices of the Dynkin diagram

∘_{1}  ∘_{2} ^{>} ∘_{3}  ∘_{4}

Varieties X_{1} and X_{2} are not isomorphic and have the dimension 15. We provide the Hasse diagrams (graphs) for X_{1}:

[figure: ∘; 1; ∘; 2; ∘; ∘; 2; ∘; ∘; ∘; 1; ∘; 2; ∘; 3; ∘; 2; ∘; ∘; 2; ∘; 3; ∘; ∘; 3; ∘; 2; ∘; 1; ∘; ∘; 1; ∘; 3; ∘; 4; ∘; 2; ∘; 4; ∘; 1]

and X_{2}:

[figure: ∘; 4; ∘; 3; ∘; ∘; 3; ∘; ∘; ∘; 4; ∘; 3; ∘; 2; ∘; 3; ∘; ∘; 3; ∘; 2; ∘; ∘; 2; ∘; 3; ∘; 4; ∘; ∘; 4; ∘; 2; ∘; 1; ∘; 3; ∘; 1; ∘; 4]

We draw the diagrams in such a way that the labels on the opposite sides of a parallelogram are equal and in that case we omit all labels but one.

Recall that (see 3.5) the vertices of this graph correspond to the basis elements of the Chow group CH(X_{1}). The leftmost vertex is the unit class 1 = [X_{w_{0}}] and the rightmost one is the class of a 0-cycle of degree 1.

4.5. We denote the basis elements of the respective Chow groups as follows

CH^{i}(X_{1}) = \begin{cases}⟨h_{1}^{i}⟩, & i = 0 … 3, 12 … 15, \\ ⟨h_{1}^{i}, g_{1}^{i}⟩, & i = 4 … 11. \end{cases}

CH^{i}(X_{2}) = \begin{cases}⟨h_{2}^{i}⟩, & i = 0 … 3, 12 … 15, \\ ⟨h_{2}^{i}, g_{2}^{i}⟩, & i = 4 … 11. \end{cases}

The generators h_{i} correspond to the upper vertices of the respective Hasse diagrams, and g_{i} to the lower ones (if the corresponding rank is 2).

4.6. Applying 3.8 we immediately obtain the following partial multiplication table

h_{k}^{s}g_{k}^{15−s} = 0, h_{k}^{s}h_{k}^{15−s} = g_{k}^{s}g_{k}^{15−s} = h_{k}^{15},

where k = 1, 2, for all s.

4.7. By Pieri formula 3.9 we obtain the following partial multiplication tables for CH(X_{1}):

h_{1}^{1}h_{1}^{1} = h_{1}^{2},  h_{1}^{1}h_{1}^{2} = 2h_{1}^{3},  h_{1}^{1}h_{1}^{3} = 2h_{1}^{4} + g_{1}^{4},  h_{1}^{1}h_{1}^{4} = h_{1}^{5},
h_{1}^{1}g_{1}^{4} = 2h_{1}^{5} + g_{1}^{5}, h_{1}^{1}h_{1}^{5} = 2h_{1}^{6} + g_{1}^{6}, h_{1}^{1}g_{1}^{5} = 2g_{1}^{6},  h_{1}^{1}h_{1}^{6} = h_{1}^{7} + g_{1}^{7},
h_{1}^{1}g_{1}^{6} = 2g_{1}^{7}, h_{1}^{1}h_{1}^{7} = 2h_{1}^{8} + g_{1}^{8}, h_{1}^{1}g_{1}^{7} = h_{1}^{8} + 2g_{1}^{8}, h_{1}^{1}h_{1}^{8} = h_{1}^{9},
h_{1}^{1}g_{1}^{8} = h_{1}^{9} + 2g_{1}^{9}, h_{1}^{1}h_{1}^{9} = 2h_{1}^{10},  h_{1}^{1}g_{1}^{9} = h_{1}^{10} + 2g_{1}^{10}, h_{1}^{1}h_{1}^{10} = h_{1}^{11} + 2g_{1}^{11},
h_{1}^{1}g_{1}^{10} = g_{1}^{11}, h_{1}^{1}h_{1}^{11} = 2h_{1}^{12}, h_{1}^{1}g_{1}^{11} = h_{1}^{12}, h_{1}^{1}h_{1}^{12} = 2h_{1}^{13},
h_{1}^{1}h_{1}^{13} = h_{1}^{14},  h_{1}^{1}h_{1}^{14} = h_{1}^{15}.

for CH(X_{2}):

h_{2}^{1}h_{2}^{1} = h_{2}^{2},  h_{2}^{1}h_{2}^{2} = h_{2}^{3},  h_{2}^{1}h_{2}^{3} = h_{2}^{4} + g_{2}^{4},  h_{2}^{1}h_{2}^{4} = h_{2}^{5},
h_{2}^{1}g_{2}^{4} = h_{2}^{5} + g_{2}^{5}, h_{2}^{1}h_{2}^{5} = h_{2}^{6} + g_{2}^{6}, h_{2}^{1}g_{2}^{5} = g_{2}^{6},  h_{2}^{1}h_{2}^{6} = h_{2}^{7} + g_{2}^{7},
h_{2}^{1}g_{2}^{6} = g_{2}^{7},  h_{2}^{1}h_{2}^{7} = 2h_{2}^{8} + g_{2}^{8}, h_{2}^{1}g_{2}^{7} = h_{2}^{8} + 2g_{2}^{8}, h_{2}^{1}h_{2}^{8} = h_{2}^{9},
h_{2}^{1}g_{2}^{8} = h_{2}^{9} + g_{2}^{9}, h_{2}^{1}h_{2}^{9} = h_{2}^{10},  h_{2}^{1}g_{2}^{9} = h_{2}^{10} + g_{2}^{10}, h_{2}^{1}h_{2}^{10} = h_{2}^{11} + g_{2}^{11},
h_{2}^{1}g_{2}^{10} = g_{2}^{11}, h_{2}^{1}h_{2}^{11} = h_{2}^{12}, h_{2}^{1}g_{2}^{11} = h_{2}^{12}, h_{2}^{1}h_{2}^{12} = h_{2}^{13},
h_{2}^{1}h_{2}^{13} = h_{2}^{14},  h_{2}^{1}h_{2}^{14} = h_{2}^{15}.

4.8. Observe that the multiplication tables 4.7 can be visualized by means of the Hasse diagrams. Namely, for the variety X_{1} consider the following graph which is obtained from the respective Hasse diagram by adding a few more edges and erasing all the labels:

[figure: ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘]

and for X_{2}:

[figure: ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘]

The multiplication rules can be restored from this graph as follows: for a vertex u (that corresponds to a basis element of the Chow group) we set

h_{i}^{1}u = ∑_{u→v} v,

where the sum runs through all the edges going from u to the right (cf. [Hi82b, Cor. 3.3]), i = 1, 2.

4.9. Applying Giambelli formula 3.10 we obtain the following products which will be essentially used in the next section

g_{1}^{4}g_{1}^{4} = 6h_{1}^{8} + 8g_{1}^{8}, g_{2}^{4}g_{2}^{4} = 3h_{2}^{8} + 4g_{2}^{8}.

5 Construction of rational projectors

The goal of the present section is to prove Theorem 1.1.

5.1. According to 2.7 in order to decompose the motive M(X) it is enough to construct rational projectors on CH^{15}(X′ × X′), where X′ = X ×_{k} k′. Observe that our anisotropic group G of type F_{4} is obtained by the first Tits construction, so by 4.3.(i) there is a splitting field extension of degree 3. Therefore, for any basis element h of CH(X′) the cycle 3h is rational (it follows immediately by transfer arguments). Hence, in order to construct rational cycles in CH(X′ ×X′) it is enough to work modulo 3. We shall write x =_{3} y iff x − y = 3z for some cycle z. Please note that all results hold for Chow groups with integral coefficients.

5.2. Recall that the cycles h_{1}^{1} and h_{2}^{1} are rational by Lemma 4.2. Hence, their powers (h_{1}^{1})^{i} and (h_{2}^{1})^{i}, i = 2 … 7, are rational as well. Using multiplication tables 4.7 or their graph interpretation 4.8 we immediately obtain the following rational cycles: in codimensions 2 through 7 for CH(X′_{1}):

h_{1}^{2}, h_{1}^{3}, h_{1}^{4} − g_{1}^{4}, h_{1}^{5} + g_{1}^{5}, h_{1}^{6}, h_{1}^{7} + g_{1}^{7},

and for CH(X′_{2}):

h_{2}^{2}, h_{2}^{3}, h_{2}^{4} + g_{2}^{4}, g_{2}^{5} − h_{2}^{5}, h_{2}^{6}, h_{2}^{7} + g_{2}^{7}.

5.3. Apply the arguments of 2.6.(iii) to CH^{4}(X′_{2} × X′_{1}) (this can be done because all the properties for X′_{1} and X′_{2} hold by 4.3.(ii)). There exists a rational cycle α_{1} ∈ CH^{4}(X′_{2} × X′_{1}) such that f′(α_{1}) = g_{2}^{4} × 1. This cycle must have the following form:

α_{1} = g_{2}^{4} × 1 + a_{1}h_{2}^{3} × h_{1}^{1} + a_{2}h_{2}^{2} × h_{1}^{2} + a_{3}h_{2}^{1} × h_{1}^{3} + a_{4}1 × h_{1}^{4} + a′1 × g_{1}^{4}, where a_{i}, a′ ∈ {−1, 0, 1}. We may reduce α_{1} by adding cycles that are known to be rational (by 5.2) to

α_{1} = (g_{2}^{4} × 1) + a′(1 × g_{1}^{4}).

Repeating the same procedure for a rational cycle α_{2} ∈ CH^{4}(X′_{2} × X′_{1}) such that f′(α_{2}) = 1 × g_{1}^{4} and reducing it we obtain the rational cycle

α_{2} = b(g_{2}^{4} × 1) + (1 × g_{1}^{4}),

where b ∈ {−1, 0, 1}. Hence, there is a rational cycle of the form

r = g_{2}^{4} × 1 − a(1 × g_{1}^{4}),

where a ∈ {−1, 1}. 5.4. To obtain a rational projector p_{1} we proceed as follows. First, we obtain the following rational cycles in CH(X′_{2} × X′_{1}) modulo 3.

r^{2} = (g_{2}^{4} × 1 − a · 1 × g_{1}^{4})^{2} =_{3} g_{2}^{8} × 1 + a(g_{2}^{4} × g_{1}^{4}) − 1 × g_{1}^{8}, r_{1} = (1 × (h_{1}^{7} + g_{1}^{7}))r^{2} =_{3} g_{2}^{8} × (h_{1}^{7} + g_{1}^{7}) − 1 × h_{1}^{15} − a(g_{2}^{4} × (g_{1}^{11} + h_{1}^{11})), r_{2} = ((h_{2}^{7} + g_{2}^{7}) × 1)r^{2} =_{3} −(h_{2}^{7} + g_{2}^{7}) × g_{1}^{8} + h_{2}^{15} × 1 + a((g_{2}^{11} − h_{2}^{11}) × g_{1}^{4}), r_{3} = (h_{2}^{1} × h_{1}^{6})r^{2} =_{3} −h_{2}^{1} × h_{1}^{14} + (g_{2}^{9} + h_{2}^{9}) × h_{1}^{6} + a((h_{2}^{5} + g_{2}^{5}) × (g_{1}^{10} − h_{1}^{10})), r_{4} = (h_{2}^{6} × h_{1}^{1})r^{2} =_{3} h_{2}^{6} × (g_{1}^{9} − h_{1}^{9}) + h_{2}^{14} × h_{1}^{1} + a((h_{2}^{10} + g_{2}^{10}) × (h_{1}^{5} − g_{1}^{5})), r_{5} = (h_{2}^{2} ×(h_{1}^{5} +g_{1}^{5}))r^{2} =_{3} h_{2}^{2} ×h_{1}^{13} +(g_{2}^{10} −h_{2}^{10})×(h_{1}^{5} +g_{1}^{5})+a((h_{2}^{6} −g_{2}^{6})×g_{1}^{9}), r_{6} = ((g_{2}^{5}−h_{2}^{5})×h_{1}^{2}))r^{2} =_{3} −h_{2}^{13}×h_{1}^{2}−(g_{2}^{5}−h_{2}^{5})×(h_{1}^{10}+g_{1}^{10})+a(g_{2}^{9}×(h_{1}^{6}+g_{1}^{6})), r_{7} = (h_{2}^{3} × (h_{1}^{4} − g_{1}^{4}))r^{2} =_{3} h_{2}^{3} × h_{1}^{12} − h_{2}^{11} × (h_{1}^{4} − g_{1}^{4}) + a(h_{2}^{7} × (g_{1}^{8} − h_{1}^{8})), r_{8} = ((h_{2}^{4} + g_{2}^{4}) × h_{1}^{3})r^{2} =_{3} −h_{2}^{12} × h_{1}^{3} + (h_{2}^{4} + g_{2}^{4}) × h_{1}^{11} + a((h_{2}^{8} − g_{2}^{8}) × h_{1}^{7}).

5.5. To obtain motivic decompositions, we need to find rational projectors in CH(X′_{1} × X′_{1}) and CH(X′_{2} × X′_{2}). By compositioning (still modulo 3), we obtain the following rational cycles:

p_{0} = −(r_{2} ∘ r_{1}^{t}) =_{3} h_{1}^{15} × 1 + (g_{1}^{11} + h_{1}^{11}) × g_{1}^{4} + (g_{1}^{7} + h_{1}^{7}) × g_{1}^{8},

p_{1} = −(r_{4} ∘ r_{3}^{t}) =_{3} h_{1}^{14} × h_{1}^{1} + (h_{1}^{10} − g_{1}^{10}) × (g_{1}^{5} − h_{1}^{5}) + h_{1}^{6} × (h_{1}^{9} − g_{1}^{9}),

p_{2} = −(r_{6} ∘ r_{5}^{t}) =_{3} h_{1}^{13} × h_{1}^{2} − (h_{1}^{5} + g_{1}^{5}) × (h_{1}^{10} + g_{1}^{10}) + g_{1}^{9} × (h_{1}^{6} + g_{1}^{6}),

p_{3} = −(r_{8} ∘r_{7}^{t}) =_{3} h_{1}^{12} ×h_{1}^{3} +(h_{1}^{8} −g_{1}^{8})×h_{1}^{7} +(h_{1}^{4} −g_{1}^{4})×h_{1}^{11} ∈ CH^{15}(X′_{1} ×X′_{1});

q_{0} = −(r_{1}^{t} ∘ r_{2}) =_{3} h_{2}^{15} × 1 + (g_{2}^{11} − h_{2}^{11}) × g_{2}^{4} + (g_{2}^{7} + h_{2}^{7}) × g_{2}^{8},

q_{1} = −(r_{4}^{t} ∘ r_{3}) =_{3} h_{2}^{1} × h_{2}^{14} + (g_{2}^{9} + h_{2}^{9}) × h_{2}^{6} − (h_{2}^{5} + g_{2}^{5}) × (h_{2}^{10} + g_{2}^{10}),

q_{2} = −(r_{6}^{t} ∘ r_{5}) =_{3} h_{2}^{2} × h_{2}^{13} − (g_{2}^{10} − h_{2}^{10}) × (g_{2}^{5} − h_{2}^{5}) + (g_{2}^{6} − h_{2}^{6}) × g_{2}^{9},

q_{3} = −(r_{7}^{t} ∘r_{8}) =_{3} h_{2}^{12} ×h_{2}^{3} +(h_{2}^{4} +g_{2}^{4})×h_{2}^{11} +(h_{2}^{8} −g_{2}^{8})×h_{2}^{7} ∈ CH^{15}(X′_{2} ×X′_{2}).

It remains to note that

p_{0} ∘ p_{0} = p_{0}, p_{3} ∘ p_{3} = p_{3} in CH^{15}(X′_{1} × X′_{1}),
q_{0} ∘ q_{0} = q_{0}, q_{3} ∘ q_{3} = q_{3} in CH^{15}(X′_{2} × X′_{2}),

where the equalities hold with integral coefficients (not just modulo 3).

Note that p_{1}, p_{2}, q_{1}, and q_{2} are not projectors with integral coefficients, but the sums

p_{12} = p_{1} + p_{2}^{t} + 3(h_{1}^{10} × h_{1}^{5} + g_{1}^{10} × g_{1}^{5}),
q_{12} = q_{1} + q_{2}^{t} + 3(h_{2}^{5} × h_{2}^{10} + g_{2}^{5} × g_{2}^{10})

are rational projectors. In fact, p_{1}, p_{2}, q_{1}, and q_{2} are projectors modulo 3, that is, p_{1} ∘ p_{1} =_{3} p_{1} etc.

Also note that

p_{0} + p_{12} + p_{3} + p_{0}^{t} + p_{12}^{t} + p_{3}^{t} = Δ_{X′_{1}},
q_{0} + q_{12} + q_{3} + q_{0}^{t} + q_{12}^{t} + q_{3}^{t} = Δ_{X′_{2}},

where Δ_{X′_{i}} are the diagonal cycles, and all these projectors are orthogonal to each other. Hence, by 2.7 we obtain the decomposition of the motive of X_{1} and X_{2}:

M(X_{1}) = (X_{1}, p_{0}) ⊕ (X_{1}, p_{12}) ⊕ (X_{1}, p_{3}) ⊕ (X_{1}, p_{0}^{t}) ⊕ (X_{1}, p_{12}^{t}) ⊕ (X_{1}, p_{3}^{t}), M(X_{2}) = (X_{2}, q_{0}) ⊕ (X_{2}, q_{12}) ⊕ (X_{2}, q_{3}) ⊕ (X_{2}, q_{0}^{t}) ⊕ (X_{2}, q_{12}^{t}) ⊕ (X_{2}, q_{3}^{t}),

It is also easy to see that over the separable closure k′ the motives (X_{1}, p_{0}), (X_{1}, p_{3}), (X_{2}, q_{0}), and (X_{2}, q_{3}) split as the direct sums of Lefschetz motives ℤ ⊕ ℤ(4) ⊕ ℤ(8). Indeed, the images of the realizations of p_{0}, p_{3}, q_{0}, and q_{3} are free abelian groups ⟨1, g_{1}^{4}, g_{1}^{8}⟩, ⟨h_{1}^{3}, h_{1}^{7}, h_{1}^{11}⟩, ⟨1, g_{2}^{4}, g_{2}^{8}⟩, and ⟨h_{2}^{3}, h_{2}^{7}, h_{2}^{11}⟩ respectively.

5.6. Since p_{1}, p_{2}, q_{1}, and q_{2} are projectors modulo 3, we obtain here a consequence of Voevodsky’s theorem [Vo03] for the case of the symbol (3, 3). These projectors are the generalized Rost projectors in this case.

6 Motivic isomorphism between M(X_{1}) and M(X_{2})

The goal of this section is to prove Theorem 1.3.

6.1. We continue to use the notation from the previous section. Consider the following cycle in CH^{15}(X′_{2} × X′_{1})

J = g_{2}^{8} × g_{1}^{7} − 1 × h_{1}^{15} − ag_{2}^{4} × g_{1}^{11} − ah_{2}^{12} × h_{1}^{3} + ah_{2}^{4} × h_{1}^{11} + h_{2}^{8} × h_{1}^{7} − g_{2}^{7} × g_{1}^{8} + h_{2}^{15} × 1 + ag_{2}^{11} × g_{1}^{4} + ah_{2}^{3} × h_{1}^{12} − ah_{2}^{11} × h_{1}^{4} − h_{2}^{7} × h_{1}^{8} − h_{2}^{1} × h_{1}^{14} + h_{2}^{9} × h_{1}^{6} + ah_{2}^{5} × h_{1}^{10} − ag_{2}^{5} × g_{1}^{10} + ah_{2}^{13} × h_{1}^{2} − g_{2}^{9} × g_{1}^{6} − h_{2}^{6} × h_{1}^{9} + h_{2}^{14} × h_{1}^{1} − ah_{2}^{2} × h_{1}^{13} + g_{2}^{6} × g_{1}^{9} + ag_{2}^{10} × g_{1}^{5} − ah_{2}^{10} × h_{1}^{5}.

Observe that

J ∘ (−J^{t}) = Δ_{X′_{1}}, (−J^{t}) ∘ J = Δ_{X′_{2}}.

In other words, the correspondence J provides a motivic isomorphism between X′_{2} and X′_{1} with the inverse (−J)^{t}. From the other hand,

J =_{3} (r_{1} + ar_{8}) + (r_{2} + ar_{7}) + (r_{3} − ar_{6}) + (r_{4} − ar_{5})

and, hence, is rational. Theorem 1.3 is proved.

7 Further work

We plan to use our methods to obtain similar decompositions for other projective homogeneous varieties. Our closest plans include decomposing the motives of other projective homogeneous varieties of type F_{4} and of other types, e.g., E_{8}.

Another interesting question is to determine whether the motive R_{12} obtained above is decomposable.

Acknowledgements

These notes appeared as a result of the seminar organized at the University of Bielefeld in the winter of 2004-2005. We give our thanks to the participants of the seminar: V. Petrov, O. Röndigs and others for their remarks. We are very grateful to N. Karpenko and A. Vishik for the comments concerning rational cycles and norm varieties.

The first two authors are sincerely grateful to Bielefeld University and personally to Anthony Bak for hospitality and the opportunity for us to work together. We are also grateful to the DAAD foundation for support. The second author appreciates the support of the INTAS foundation. The last author is grateful to Alexander von Humboldt foundation, RFFI, and RTN projects for support.

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Appendix I

The multiplication table for X_{1}:

h^{4}h^{4} = h^{8} + g^{8},  h^{4}h^{5} = 2h^{9} + 2g^{9},  h^{4}h^{6} = 2h^{10} + g^{10}, h^{4}h^{7} = h^{11} + 2g^{11},
h^{4}h^{8} = h^{12},  h^{4}h^{9} = 2h^{13},  h^{4}h^{10} = h^{14},  h^{4}g^{4} = 2h^{8} + 3g^{8},
h^{4}g^{5} = h^{9} + 2g^{9},  h^{4}g^{6} = 2h^{10} + 2g^{10}, h^{4}g^{7} = h^{11} + 3g^{11}, h^{4}g^{8} = 2h^{12},
h^{4}g^{9} = h^{13}, h^{4}g^{10} = 0, g^{4}g^{4} = 6h^{8} + 8g^{8}, g^{4}g^{5} = 4h^{9} + 4g^{9},
g^{4}g^{6} = 6h^{10} + 4g^{10}, g^{4}g^{7} = 3h^{11} + 8g^{11}, g^{4}g^{8} = 6h^{12},  g^{4}g^{9} = 4h^{13},
g^{4}g^{10} = h^{14},  g^{4}h^{5} = 5h^{9} + 6g^{9},  g^{4}h^{6} = 5h^{10} + 4g^{10}, g^{4}h^{7} = 2h^{11} + 6g^{11},
g^{4}h^{8} = 2h^{12},  g^{4}h^{9} = 4h^{13},  g^{4}h^{10} = 2h^{14},  h^{5}h^{5} = 6h^{10} + 4g^{10},
h^{5}h^{6} = 2h^{11} + 5g^{11}, h^{5}h^{7} = 4h^{12},  h^{5}h^{8} = 2h^{13},  h^{5}h^{9} = 2h^{14},
h^{5}g^{5} = 4h^{10} + 4g^{10}, h^{5}g^{6} = 2h^{11} + 6g^{11}, h^{5}g^{7} = 5h^{12},  h^{5}g^{8} = 4h^{13},
h^{5}g^{9} = h^{14},  g^{5}g^{5} = 4h^{10},  g^{5}g^{6} = 2h^{11} + 4g^{11}, g^{5}g^{7} = 4h^{12},
g^{5}g^{8} = 4h^{13},  g^{5}g^{9} = 2h^{14},  g^{5}h^{6} = h^{11} + 4g^{11}, g^{5}h^{7} = 2h^{12},
g^{5}h^{8} = 0,  g^{5}h^{9} = 0,  h^{6}h^{6} = 3h^{12},  h^{6}h^{7} = 3h^{13},
h^{6}h^{8} = h^{14},  h^{6}g^{6} = 3h^{12},  h^{6}g^{7} = 3h^{13},  h^{6}g^{8} = h^{14},
g^{6}g^{6} = 4h^{12},  g^{6}g^{7} = 4h^{13},  g^{6}g^{8} = 2h^{14},  g^{6}h^{7} = 2h^{13},
g^{6}h^{8} = 0,  h^{7}h^{7} = 2h^{14},  h^{7}g^{7} = h^{14},  g^{7}g^{7} = 2h^{14}.

The multiplication table for X_{2}:

h^{4}h^{4} = 2h^{8} + 2g^{8}, h^{4}h^{5} = 4h^{9} + 2g^{9},  h^{4}h^{6} = 4h^{10} + g^{10}, h^{4}h^{7} = 2h^{11} + 2g^{11},
h^{4}h^{8} = h^{12}, h^{4}h^{9} = h^{13}, h^{4}h^{10} = h^{14}, h^{4}g^{4} = 2h^{8} + 3g^{8},
h^{4}g^{5} = h^{9} + g^{9},  h^{4}g^{6} = 2h^{10} + g^{10}, h^{4}g^{7} = 2h^{11} + 3g^{11}, h^{4}g^{8} = 2h^{12},
h^{4}g^{9} = h^{13}, h^{4}g^{10} = 0, g^{4}g^{4} = 3h^{8} + 4g^{8}, g^{4}g^{5} = 2h^{9} + g^{9},
g^{4}g^{6} = 3h^{10} + g^{10}, g^{4}g^{7} = 3h^{11} + 4g^{11}, g^{4}g^{8} = 3h^{12},  g^{4}g^{9} = 2h^{13},
g^{4}g^{10} = h^{14},  g^{4}h^{5} = 5h^{9} + 3g^{9},  g^{4}h^{6} = 5h^{10} + 2g^{10}, g^{4}h^{7} = 2h^{11} + 3g^{11},
g^{4}h^{8} = h^{12},  g^{4}h^{9} = h^{13},  g^{4}h^{10} = h^{14},  h^{5}h^{5} = 6h^{10} + 2g^{10},
h^{5}h^{6} = 4h^{11} + 5g^{11}, h^{5}h^{7} = 4h^{12},  h^{5}h^{8} = h^{13},  h^{5}h^{9} = h^{14},
h^{5}g^{5} = 2h^{10} + g^{10}, h^{5}g^{6} = 2h^{11} + 3g^{11}, h^{5}g^{7} = 5h^{12},  h^{5}g^{8} = 2h^{13},
h^{5}g^{9} = h^{14},  g^{5}g^{5} = h^{10},  g^{5}g^{6} = h^{11} + g^{11},  g^{5}g^{7} = 2h^{12},
g^{5}g^{8} = h^{13},  g^{5}g^{9} = h^{14},  g^{5}h^{6} = h^{11} + 2g^{11}, g^{5}h^{7} = h^{12},
g^{5}h^{8} = 0,  g^{5}h^{9} = 0,  h^{6}h^{6} = 6h^{12},  h^{6}h^{7} = 3h^{13},
h^{6}h^{8} = h^{14},  h^{6}g^{6} = 3h^{12},  h^{6}g^{7} = 3h^{13},  h^{6}g^{8} = h^{14},
g^{6}g^{6} = 2h^{12},  g^{6}g^{7} = 2h^{13},  g^{6}g^{8} = h^{14},  g^{6}h^{7} = h^{13},
g^{6}h^{8} = 0,  h^{7}h^{7} = 2h^{14},  h^{7}g^{7} = h^{14},  g^{7}g^{7} = 2h^{14}.

Appendix II

In this appendix we describe how we obtained the necessary multiplication tables. Our root enumeration follows Bourbaki ([Bou]). We fix an orthonormal base {e_{1}, e_{2}, e_{3}, e_{4}} in ℝ^{4}. F_{4} has the following simple roots:

α_{1} = e_{3} − e_{2}, α_{2} = e_{2} − e_{1},
α_{3} = e_{1},  α_{4} = −\frac{1}{2}e_{1} − \frac{1}{2}e_{2} − \frac{1}{2}e_{3} + \frac{1}{2}e_{4}.

The set of fundamental weights:

ω̄_{1} = e_{3} + e_{4},  ω̄_{2} = e_{2} + e_{3} + 2e_{4},
ω̄_{3} = \frac{1}{2}e_{1} + \frac{1}{2}e_{2} + \frac{1}{2}e_{3} + \frac{3}{2}e_{4}, ω̄_{4} = e_{4}.

For the expressions of other positive roots by the base roots we refer to [Bou]. We list the expressions of these roots in the basis of fundamental weights:

−ω̄_{3} + 2ω̄_{4}, − ω̄_{2} + 2ω̄_{3} − ω̄_{4}, −ω̄_{1} + 2ω̄_{2} − 2ω̄_{3}, 2ω̄_{1} − ω̄_{2},
−ω̄_{2} + ω̄_{3} + ω̄_{4}, − ω̄_{1} + ω̄_{2} − ω̄_{4}, −ω̄_{1} + 2ω̄_{3} − 2ω̄_{4}, ω̄_{1} + ω̄_{2} − 2ω̄_{3},
−ω̄_{1} + ω̄_{2} − ω̄_{3} + ω̄_{4}, ω̄_{1} − ω̄_{4},  −ω̄_{1} + 2ω̄_{4}, ω̄_{1} − ω̄_{2} + 2ω̄_{3} − 2ω̄_{4},
−ω̄_{1} + ω̄_{3}, ω̄_{1} − ω̄_{3} + ω̄_{4},  ω̄_{1} − ω̄_{2} + 2ω̄_{4}, ω̄_{2} − 2ω̄_{4},
ω̄_{1} − ω̄_{2} + ω̄_{3}, ω̄_{2} − 2ω̄_{3} + 2ω̄_{4},  ω̄_{2} − ω̄_{3}, − ω̄_{2} + 2ω̄_{3},
ω̄_{3} − ω̄_{4}, − ω̄_{1} + ω̄_{2},  ω̄_{4}, ω̄_{1}.

We denote by d the product of all positive roots.

Using the Giambelli formula, we obtain the preimages of g_{i}^{4} in S^{∗}(P). Here is the list:

g_{1}^{4} = c(\frac{1}{3}ω̄_{4}^{4} − \frac{2}{3}ω̄_{3}ω̄_{4}^{3} + ω̄_{3}^{2}ω̄_{4}^{2} − \frac{4}{3}ω̄_{2}ω̄_{3}ω̄_{4}^{2} − \frac{2}{3}ω̄_{1}ω̄_{2}ω̄_{3}^{2} + \frac{2}{3}ω̄_{2}^{2}ω̄_{4}^{2}+

\frac{2}{3}ω̄_{1}^{2}ω̄_{4}^{2} + ω̄_{1}ω̄_{3}ω̄_{4}^{2} + \frac{2}{3}ω̄_{1}ω̄_{2}ω̄_{3}ω̄_{4} − ω̄_{1}ω̄_{3}^{2}ω̄_{4} − \frac{2}{3}ω̄_{1}^{2}ω̄_{3}ω̄_{4} + \frac{4}{3}ω̄_{2}ω̄_{3}^{2}ω̄_{4}−

\frac{2}{3}ω̄_{3}^{3}ω̄_{4} − \frac{2}{3}ω̄_{2}^{2}ω̄_{3}ω̄_{4} − \frac{1}{2}ω̄_{1}^{3}ω̄_{2} − \frac{2}{3}ω̄_{2}ω̄_{3}^{3} − \frac{2}{3}ω̄_{1}^{2}ω̄_{2}ω̄_{3} + \frac{1}{3}ω̄_{2}^{2}ω̄_{3}^{2}+

\frac{1}{2}ω̄_{1}^{4} + \frac{1}{3}ω̄_{3}^{4} + \frac{2}{3}ω̄_{1}^{2}ω̄_{3}^{2} + \frac{1}{2}ω̄_{1}^{2}ω̄_{2}^{2} − \frac{1}{3}ω̄_{1}ω̄_{2}^{2}ω̄_{3} + \frac{1}{3}ω̄_{1}ω̄_{2}ω̄_{3}^{2}),

g_{2}^{4} = c(\frac{11}{6}ω̄_{4}^{4} − \frac{7}{6}ω̄_{4}^{3}ω̄_{3} + \frac{11}{12}ω̄_{4}^{2}ω̄_{1}^{2} + \frac{3}{2}ω̄_{4}^{2}ω̄_{3}^{2} − \frac{11}{6}ω̄_{3}ω̄_{4}^{2}ω̄_{2} + \frac{11}{12}ω̄_{4}^{2}ω̄_{2}^{2}

− \frac{11}{12}ω̄_{4}^{2}ω̄_{1}ω̄_{2} − \frac{2}{3}ω̄_{4}ω̄_{3}^{3} − \frac{1}{2}ω̄_{4}ω̄_{1}^{2}ω̄_{2} + \frac{1}{3}ω̄_{4}ω̄_{3}ω̄_{1}^{2} + \frac{4}{3}ω̄_{3}^{2}ω̄_{4}ω̄_{2} + \frac{1}{2}ω̄_{4}ω̄_{1}ω̄_{2}^{2}−

\frac{2}{3}ω̄_{3}ω̄_{4}ω̄_{2}^{2} − \frac{1}{3}ω̄_{4}ω̄_{3}ω̄_{1}ω̄_{2} + \frac{1}{3}ω̄_{3}^{4} − \frac{1}{3}ω̄_{3}ω̄_{1}ω̄_{2}^{2} + \frac{1}{3}ω̄_{3}^{2}ω̄_{1}ω̄_{2} − \frac{1}{3}ω̄_{1}^{2}ω̄_{3}^{2}+

\frac{1}{3}ω̄_{3}ω̄_{1}^{2}ω̄_{2} + \frac{1}{3}ω̄_{3}^{2}ω̄_{2}^{2} − \frac{2}{3}ω̄_{3}^{3}ω̄_{2}),

Multiplying the correspondent polynomials and taking the c function, we find the products.