TRIALITY AND ÉTALE ALGEBRAS

MAX-ALBERT KNUS AND JEAN-PIERRE TIGNOL

Dedicated with great friendship to R. Parimala at the occasion of her 60^{th} birthday

Abstract. Trialitarian automorphisms are related to automorphisms of order 3 of the Dynkin diagram of type D_{4}. Octic étale algebras with trivial discriminant, containing quartic subalgebras, are classified by Galois cohomology with value in the Weyl group of type D_{4}. This paper discusses triality for such étale extensions.

1. Introduction

All Dynkin diagrams but one admit at most automorphisms of order two, which are related to duality in algebra and geometry. The Dynkin diagram of D_{4}

\begin{matrix}α_{3} \\ α_{4} \end{matrix} [figure: α_{1}; α_{2}]

is special, in the sense that it admits automorphisms of order 3. Algebraic and geometric objects related to D_{4} are of particular interest as they also usually admit exceptional automorphisms of order 3, which are called trialitarian. For example the special projective orthogonal group PGO_{8}^{+} or the simply connected group Spin_{8} admit outer automorphisms of order 3. As already observed by E. Cartan, [5], the Weyl group W (D_{4}) = 𝔖_{2}^{3} ⋊ 𝔖_{4} of Spin_{8} or of PGO_{8}^{+} similarly admits trialitarian automorphisms. Let F be a field and let F_{s} be a separable closure of F. The Galois cohomology set H^{1}(Γ, W (D_{4})), where Γ is the absolute Galois group Gal(F_{s}/F), classifies isomorphism classes of étale extensions S/S_{0} where S has dimension 8, S_{0} dimension 4 and S has trivial discriminant (see §3). There is an induced trialitarian action on H^{1}(Γ, W (D_{4})), which associates to the isomorphism class of an extension S/S_{0} as above, two extensions S′/S′_{0} and S′′/S′′_{0}, of the same kind, so that the triple (S/S_{0}, S′/S′_{0}, S′′/S′′_{0}) is cyclically permuted by triality. This paper is devoted to the study of such triples of étale algebras. It grew out of a study, in the spirit of [16], of Severi-Brauer varieties over the “field of one element”, [13], which is in preparation (see also [17] and [18]). In Part 2 we describe some basic constructions on finite Γ-sets and étale algebras. Some results are well-known, others were taken from [13], like the Clifford construction. In Section 3 we recall how Γ-sets and étale algebras are related to Galois cohomology. Section 4 is devoted to triality in connection with Γ-sets and in Section 5 we discuss trialitarian automorphisms of the Weyl group W (D_{4}). In Section 6 we consider triality at the level of étale algebras. We give in Table 1 a list of isomorphism classes of étale algebras corresponding to the conjugacy classes of subgroups of W (D_{4}), together with a description of the triality action. We also consider étale algebras associated to subgroups of W (D_{4}) which are fixed under triality. We then view in Section 7 triality as a way to create resolvents and give explicit formulae for polynomials defining étale algebras. Finally we give in the last section results of Serre on Witt invariants of W (D_{4}). We are grateful to Parimala for her unshakable interest in triality, in particular for many discussions at earlier stages of this work and we specially thank J-P. Serre for communicating to us his results on Witt and cohomological invariants of the group W (D_{4}). We also thank Emmanuel Kowalski who introduced us to Magma [2] with much patience, Jean Barge for his help with Galois cohomology and J. E. Humphreys and B. Mühlherr for the reference to the paper [8]. The paper [10] on octic fields was a very useful source of inspiration. Finally we are highly thankful to the referee for many improvements.

The second author is supported in part by the F.R.S.–FNRS (Belgium).

2. Étale algebras and Γ-sets

Throughout most of this work, F is an arbitrary field. We denote by F_{s} a separable closure of F and by Γ the absolute Galois group Γ = Gal(F_{s}/F), which is a profinite group. A finite-dimensional commutative F -algebra S is called étale (over F) if S ⊗_{F} F_{s} is isomorphic to the F_{s}-algebra F_{s}^{n} = F_{s} × ⋯ × F_{s} (n factors) for some n ≥ 1. Étale F -algebras are the direct products of finite separable field extensions of F. We refer to [11, §18.A] for various equivalent characterizations. Étale algebras (with F -algebra homomorphisms) form a category Ét_{F} in which finite direct products and finite direct sums (= tensor products) are defined. Finite sets with a continuous left action of Γ (for the discrete topology) are called (finite) Γ-sets. They form a category Set_{Γ} whose morphisms are the Γ-equivariant maps. Finite direct products and direct sums (= disjoint unions) are defined in this category. We denote by |X| the cardinality of any finite set X. For any étale F -algebra S of dimension n, the set of F -algebra homomorphisms

X(S) = Hom_{F -alg}(S, F_{s})

is a Γ-set of n elements since Γ acts on F_{s}. Conversely, if X is a Γ-set of n elements, the F -algebra M(X) of Γ-equivariant maps X → F_{s} is an étale F -algebra of dimension n,

M(X) = {f: X → F_{s} | γ(f(x)) = f(^{γ}x) for γ ∈ Γ, x ∈ X}.

As first observed by Grothendieck, there are canonical isomorphisms

M(X(S)) ≅ S, X(M(X)) ≅ X,

so that the functors M and X define an anti-equivalence of categories

(2.1)  Set_{Γ} ≡ Ét_{F}

(see [6, Proposition (4.3), p. 25] or [11, (18.4)]). Under this anti-equivalence, the cardinality of Γ-sets corresponds to the dimension of étale F -algebras, the disjoint union ⊔ in Set_{Γ} corresponds to the product × in Ét_{F}, and the product × in Set_{Γ} to the tensor product ⊗ in Ét_{F}. For any integer n ≥ 1, we let Ét_{F}^{n} denote the groupoid^{1} whose objects are n-dimensional étale F -algebras and whose morphisms are F -algebra isomorphisms, and Set_{Γ}^{n} the groupoid of Γ-sets with n elements. The anti-equivalence (2.1) restricts to an anti-equivalence Set_{Γ}^{n} ≡ Ét_{F}^{n}. The split étale algebra F^{n} corresponds to the Γ-set n of n elements with trivial Γ-action. Étale algebras of dimension 2 are also called quadratic étale algebras. A morphism^{2} of Γ-sets Y ⟵^{π} Z is called a Γ-covering if the number of elements in each fiber y^{π^{−1}} ⊂ Z does not depend on y ∈ Y. This number is called the degree of the covering. For n, d ≥ 1 we let Cov_{Γ}^{d/n} denote the groupoid whose objects are coverings of degree d of a Γ-set of n elements and whose morphisms are isomorphisms of Γ-coverings. A homomorphism S ⟶^{ε} T of étale F -algebras is said to be an extension of degree d of étale algebras if ε endows T with a structure of a free S-module of rank d. This corresponds under the anti-equivalence (2.1) to a covering of degree d:

1A groupoid is a category in which all morphisms are isomorphisms.

X(S) ⟵^{X(ε)} X(T)

(see [13]). Let Étex_{F}^{d/n} denote the groupoid of étale extensions S ⟶^{ε} T of degree d of F -algebras with dim_{F} S = n (hence dim_{F} T = dn). From (2.1) we obtain an anti-equivalence of groupoids

Étex_{F}^{d/n} ≡ Cov_{Γ}^{d/n}.

The Γ-covering with trivial Γ-action

(2.2)  d/n : n ⟵^{p_{1}} n × d

where p_{1} is the first projection corresponds to the extension F^{n} → (F^{d})^{n}. Of particular importance in the sequel are coverings of degree 2, which are also called double coverings. Each such covering Y ⟵^{π} Z defines a canonical automorphism Z ⟵^{σ} Z of order 2, which interchanges the elements in each fiber of π. Clearly, this automorphism has no fixed points. Conversely, if Z is any Γ-set and Z ⟵^{σ} Z is an automorphism of order 2 without fixed points, the set of orbits

Z/σ = {{z, z^{σ}} | z ∈ Z}

is a Γ-set and the canonical map (Z/σ) ← Z is a double covering. An involution of a Γ-set with an even number of elements is any automorphism of order 2 without fixed points. Let σ: S → S be an automorphism of order 2 of an étale F -algebra S, and let S^{σ} ⊂ S denote the F -sub-algebra of fixed elements, which is necessarily étale. The following conditions are equivalent (see [13]):

(a) the inclusion S^{σ} → S is a quadratic étale extension of F -algebras;

(b) the automorphism X(σ) is an involution on X(S).

We say under these equivalent conditions that the automorphism σ is an involution of the étale F -algebra S.

2We let morphisms of Γ-sets act on the right of the arguments (with the exponential notation)

and use the usual function notation for morphisms in the anti-equivalent category of étale algebras.

Basic constructions on Γ-sets. We recall from [11, §18] and [12, §2.1] the construction of the discriminant Δ(X) of a Γ-set X with |X| = n ≥ 2. Consider the set of n-tuples of elements in X:

Σ_{n}(X) = {(x_{1}, …, x_{n}) | X = {x_{1}, …, x_{n}}}.

This Γ-set carries an obvious transitive (right) action of the symmetric group 𝔖_{n}. The discriminant Δ(X) is the set of orbits under the alternating group 𝔄_{n}:

Δ(X) = Σ_{n}(X)/𝔄_{n}.

It is a Γ-set of two elements, so Δ is a functor

Δ: Set_{Γ}^{n} → Set_{Γ}^{2}.

For any covering Z_{0} ⟵^{π} Z of degree 2 with |Z_{0}| = n, (hence |Z| = 2n), we consider the set of (not necessarily Γ-equivariant) sections of π:

C(Z/Z_{0}) = {{z_{1}, …, z_{n}} ⊂ Z | {z_{1}^{π}, …, z_{n}^{π}} = Z_{0}}.

It is a Γ-set with 2^{n} elements, so C is a functor

C: Cov_{Γ}^{2/n} → Set_{Γ}^{2^{n}},

called the Clifford functor (see [13]). The Γ-set C(Z/Z_{0}) is equipped with a canonical surjective morphism

(2.3)  Δ(Z) ⟵^{δ} C(Z/Z_{0}),

which is defined in [12, §2.2] as follows: let σ: Z → Z be the involution canonically associated to the double covering Z_{0} ⟵^{π} Z, so the fiber of z^{π} is {z, z^{σ}} for each z ∈ Z; then δ maps each section {z_{1}, …, z_{n}} to the 𝔄_{2n}-orbit of the 2n-tuple (z_{1}, …, z_{n}, z_{1}^{σ}, …, z_{n}^{σ}),

{z_{1}, …, z_{n}}^{δ} = (z_{1}, …, z_{n}, z_{1}^{σ}, …, z_{n}^{σ})^{𝔄_{2n}}.

Note that the canonical involution σ induces an involution \underline{σ} on C(Z/Z_{0}), which maps each section ω to its complement Z\ω. We may view C(Z/Z_{0}) as a covering of degree 2 of the set of orbits C(Z/Z_{0})/\underline{σ}, and thus consider the Clifford construction as a functor

(2.4)  C: Cov_{Γ}^{2/n} → Cov_{Γ}^{2/2^{n−1}}.

Proposition 2.5. For sections ω, ω′ ∈ C(Z/Z_{0}), we have ω^{δ} = (ω′)^{δ} if and only if |ω ∩ ω′| ≡ n mod 2. Moreover, denoting by ι the nontrivial automorphism of Δ(Z), we have

\underline{σ} ∘ δ = \begin{cases}δ & if n is even, \\ δ ∘ ι & if n is odd. \end{cases}

Proof. Let ω = {z_{1}, …, z_{n}} and ω′ = {z_{1}, …, z_{r}, z_{r+1}^{σ}, …, z_{n}^{σ}}, so r = |ω ∩ ω′|,

ω^{δ} = (z_{1}, …, z_{n}, z_{1}^{σ}, …, z_{n}^{σ})^{𝔄_{2n}}

and

(ω′)^{δ} = (z_{1}, …, z_{r}, z_{r+1}^{σ}, …, z_{n}^{σ}, z_{1}^{σ}, …, z_{r}^{σ}, z_{r+1}, …, z_{n})^{𝔄_{2n}}.

The permutation σ′ that interchanges z_{i} and z_{i}^{σ} for i = r + 1, …, n satisfies

(z_{1}, …, z_{n}, z_{1}^{σ}, …, z_{n}^{σ})^{σ′} = (z_{1}, …, z_{r}, z_{r+1}^{σ}, …, z_{n}^{σ}, z_{1}^{σ}, …, z_{r}^{σ}, z_{r+1}, …, z_{n}); it is in 𝔄_{2n} if and only if n − r is even, which means |ω ∩ ω′| ≡ n mod 2. For ω′ = ω\underline{σ} the complement of ω we have |ω ∩ ω\underline{σ}| = 0, hence ω^{\underline{σ}δ} = ω^{δ} if and only if n ≡ 0 mod 2. □

Oriented Γ-sets. An oriented Γ-set is a pair (Z, ∂_{Z}) where Z is a Γ-set and ∂_{Z} is a fixed isomorphism of Γ-sets 2 − ←^{∼} Δ(Z). In particular the Γ-action on Δ(Z) is trivial. There are two possible choices for ∂_{Z}. A choice is an orientation of Z. Oriented Γ-sets with n elements form a groupoid (Set_{Γ}^{n})^{+} whose morphisms are isomorphisms Z_{2} ⟵^{f} Z_{1} such that Δ(f) ∘ ∂_{Z_{2}} = ∂_{Z_{1}}. Similarly oriented coverings are pairs (Z/Z_{0}, ∂_{Z}) where Z_{0} ← Z is a Γ-covering and ∂_{Z} is an orientation of Z. We denote by (Cov_{Γ}^{d/n})^{+} the groupoid of oriented coverings of degree d of Γ-sets with n elements. Changing the orientation through the twist 2 ⟵^{ι} 2 defines an involutive functor

κ: (Cov_{Γ}^{d/n})^{+} → (Cov_{Γ}^{d/n})^{+}.

Proposition 2.6. If n is even the functor C: Cov_{Γ}^{2/n} → Cov_{Γ}^{2/2^{n−1}} of (2.4) restricts to a pair of functors

C_{1}, C_{2} : (Cov_{Γ}^{2/n})^{+} → Cov_{Γ}^{2/2^{n−2}}.

Moreover two sections ω and ω′ of the oriented Γ-covering (Z/Z_{0}, ∂_{Z}) lie in the same set C_{1}(Z/Z_{0}, ∂_{Z}) or C_{2}(Z/Z_{0}, ∂_{Z}) if and only if |ω ∩ ω′| ≡ 0 mod 2.

Proof. Let Z/Z_{0} be a 2/n-covering. Proposition 2.5 implies that the covering Δ(Z) ⟵^{δ} C(Z/Z_{0}) factors through C(Z/Z_{0})/\underline{σ}, where \underline{σ} is the canonical involution of C(Z/Z_{0}):

Δ(Z) ← C(Z/Z_{0})/\underline{σ} ← C(Z/Z_{0}).

Thus, if Z/Z_{0} is oriented, we may use the given isomorphism 2 ⟵^{∂_{Z}} Δ(Z) to define the Γ-sets

C_{1}(Z/Z_{0}, ∂_{Z}) = {ω ∈ C(Z/Z_{0}) | ω^{δ∂_{Z}} = 1}

and

C_{2}(Z/Z_{0}, ∂_{Z}) = {ω ∈ C(Z/Z_{0}) | ω^{δ∂_{Z}} = 2}.

Obviously, we have C(Z/Z_{0}) = C_{1}(Z/Z_{0}, ∂_{Z}) ⊔ C_{2}(Z/Z_{0}, ∂_{Z}), and Proposition 2.5 shows that \underline{σ} restricts to involutions on C_{1}(Z/Z_{0}, ∂_{Z}) and C_{2}(Z/Z_{0}, ∂_{Z}). The last claim also follows from Proposition 2.5. □

We call the two functors C_{1} and C_{2} the spinor functors. Note that when n is even an orientation ∂_{Z} on Z/Z_{0} can also be defined by specifying whether a given section ω ∈ C(Z/Z_{0}) lies in C_{1}(Z/Z_{0}, ∂_{Z}) or C_{2}(Z/Z_{0}, ∂_{Z}). Indeed, ω ∈ C_{1}(Z/Z_{0}, ∂_{Z}) if and only if ω^{δ} ∈ Δ(Z) is mapped to 1, which determines ∂_{Z} uniquely. We shall avail ourselves of this possibility to define orientations on coverings in Cov_{Γ}^{2/4} in §4.

Basic constructions on étale algebras. We now consider analogues of the functors Δ and C for étale algebras and étale extensions. For S an étale F -algebra of dimension n ≥ 2, the discriminant Δ(S) is a quadratic étale F -algebra such that

X(Δ(S)) = Δ(X(S)).

We thus have a functor

Δ: Ét_{F}^{n} → Ét_{F}^{2} for n ≥ 2.

If the field F has characteristic different from 2, it is usual to represent Δ(S) as F [x]/(x^{2} − Disc(S)), Disc(S) ∈ F^{×}, and the class of Disc(S) in F^{×}/(F^{×})^{2} is the usual discriminant. We refer to [11, p. 291–293] and [12, §3.1] for details. Let S ⟶^{ε} T be an étale extension of degree 2 of (étale) F -algebras, with dim_{F} S = n, dim_{F} T = 2n. In [12, §3.2]^{3} we define an étale F -algebra C(T/S) such that

X(C(T/S)) = C(X(T)/ X(S)).

Example 2.7. If dim_{F} T = 2 and S = F, we have C(T/S) = T. For S_{1}, S_{2} étale algebras of arbitrary dimension, and for arbitrary étale extensions T_{1}/S_{1} and T_{2}/S_{2} of degree 2, there is a canonical isomorphism

P: C((T_{1} × T_{2})/(S_{1} × S_{2})) ⟶^{∼} C(T_{1}/S_{1}) ⊗ C(T_{2}/S_{2}).

We call the 2^{n}-dimensional algebra C(T/S) the Clifford algebra of T/S. It admits a canonical involution \underline{σ}. If dim_{F} S is even \underline{σ} is the identity on Δ(T). The canonical morphism δ of (2.3)

Δ(X(T)) ⟵^{δ} C(X(T)/ X(S))

yields a canonical F -algebra homomorphism which we again denote by δ,

Δ(T) ⟶^{δ} C(T/S),

so that C(T/S) is an étale extension of degree 2^{n−1} of a quadratic étale F -algebra.

Oriented étale algebras. As for oriented Γ-sets we define oriented étale algebras as pairs (S, ∂_{S}) where S is an étale algebra and ∂_{S}: Δ(S) ⟶^{∼} F × F is an isomorphism of F -algebras. Oriented extensions of étale algebras are pairs (S/S_{0}, ∂_{S}) where S/S_{0} is an extension of étale algebras and ∂_{S}: Δ(S) ⟶^{∼} F × F is an isomor-

phism of F -algebras. We have corresponding groupoids (Ét_{F}^{n})^{+}, (Étex_{F}^{d/n})^{+} and anti-equivalences

(Set_{Γ}^{n})^{+} ≡ (Ét_{F}^{n})^{+} and (Cov_{Γ}^{d/n})^{+} ≡ (Étex_{F}^{d/n})^{+}.

Switching the orientation induces an involutive functor κ on these groupoids. The Clifford functor C restricts to a pair of spinor functors

(2.8)  C_{1}, C_{2} : (Étex_{F}^{2/n})^{+} → Étex_{F}^{2/2^{n−2}}

if n is even.

Remark 2.9. The terminology used above owes its origin to the fact that the Clifford functor is related to the theory of Clifford algebras in the framework of quadratic forms and central simple algebras with involution. We refer to [13] for details and more properties of the Clifford construction.

3The notation Ω is used for C in [12].

3. Cohomology

For any integer n ≥ 1, we consider the Γ-set n = {1, …, n} with the trivial Γ-action and let 𝔖_{n} denote the symmetric group on n, i.e., the automorphism group of n,

𝔖_{n} = Aut(n).

Recall from [11, §28.A] that the cohomology set H^{1}(Γ, 𝔖_{n}) (for the trivial action of Γ on 𝔖_{n}) is the set of continuous group homomorphisms Γ → 𝔖_{n} (“cocycles”) up to conjugation. Letting Iso(Set_{Γ}^{n}) denote the set of isomorphism classes in Set_{Γ}^{n}, we have a canonical bijection of pointed sets

(3.1)  Iso(Set_{Γ}^{n}) ⟶^{∼} H^{1}(Γ, 𝔖_{n}).

Cohomology sets can also be used to describe isomorphism classes of Γ-coverings: for any integers n, d ≥ 1, the group of automorphisms of the Γ-covering with trivial Γ-action d/n is the wreath product (of order (d!)^{n}n!)

Aut(d/n) = 𝔖_{d} ≀ 𝔖_{n} (= 𝔖_{d}^{n} ⋊ 𝔖_{n}).

The same construction as above yields a canonical bijection

(3.2)  Iso(Cov_{Γ}^{d/n}) ⟶^{∼} H^{1}(Γ, 𝔖_{d} ≀ 𝔖_{n}),

where the Γ-action on 𝔖_{d} ≀ 𝔖_{n} is trivial; see [12, §4.2]. The automorphism group of the oriented Γ-covering (d/n, ∂_{n×d}) is the group

(𝔖_{d} ≀ 𝔖_{n})^{+} = (𝔖_{d} ≀ 𝔖_{n}) ∩ 𝔄_{dn}

so that

(3.3)  Iso ((Cov_{Γ}^{d/n})^{+}) ⟶^{∼} H^{1}(Γ, (𝔖_{d} ≀ 𝔖_{n})^{+}).

We now assume that Γ is the absolute Galois group Γ = Gal(F_{s}/F) of a field F and use the notation H^{1}(F, 𝔖_{n}) for H^{1}(Γ, 𝔖_{n}). The anti-equivalence Set_{Γ}^{n} ≡ Ét_{F}^{n} and the bijection (3.1) induce canonical bijections

Iso(Ét_{F}^{n}) ≅ Iso(Set_{Γ}^{n}) ≅ H^{1}(F, 𝔖_{n})

The bijection Iso(Ét_{F}^{n}) ≅ H^{1}(F, 𝔖_{n}) may of course also be defined directly since

Aut_{F -alg}(F^{n}) ≅ 𝔖_{n},

see [11, (29.9)]. Similarly, it follows from (3.2), (3.3), and the anti-equivalence

of groupoids Étex_{F}^{d/n} ≡ Cov_{Γ}^{d/n}, (Étex_{F}^{d/n})^{+} ≡ (Cov_{Γ}^{d/n})^{+}, that we have canonical bijections of pointed sets:

(3.4)  Iso(Étex_{Γ}^{d/n}) ≅ Iso(Cov_{Γ}^{d/n}) ≅ H^{1}(F, 𝔖_{d} ≀ 𝔖_{n})

and

(3.5)  Iso ((Étex_{Γ}^{d/n})^{+}) ≅ Iso ((Cov_{Γ}^{d/n})^{+}) ≅ H^{1}(F, (𝔖_{d} ≀ 𝔖_{n})^{+}).

Remark 3.6. Any group homomorphism φ: G → H, where G and H are automorphism groups of finite sets or of finite double coverings, induces a map on the level of cocycles φ_{∗}: (γ: Γ → G) ↦ (φ ∘ γ: Γ → H). Thus φ associates in a “canonical way” an étale algebra (or an étale algebra with involution) E_{φ}, whose isomorphism class belongs to H^{1}(F, H), to an étale algebra E (or an étale algebra E with involution), whose class belongs to H^{1}(F, G). We say that the algebra E_{φ} is a resolvent of E. For example the discriminant Δ(E) is the resolvent of E associated to the parity map 𝔖_{n} → 𝔖_{2}. Other examples of resolvents will be discussed in relation with triality.

4. Triality and Γ-coverings

Recall the functor C, which associates to any double covering its set of sections. For oriented 2/4-coverings of Γ-sets, it leads to two functors

C_{1}, C_{2} : (Cov_{Γ}^{2/4})^{+} → Cov_{Γ}^{2/4},

see Proposition 2.6. The functors C_{1} and C_{2} together with the functor κ, which changes the orientation, give an explicit description of an action of the group 𝔖_{3} on the pointed set Iso((Cov_{Γ}^{2/4})^{+}).

Theorem 4.1. The functors C_{1}, C_{2} : (Cov_{Γ}^{2/4})^{+} → Cov_{Γ}^{2/4} factor through the forgetful functor F: (Cov_{Γ}^{2/4})^{+} → Cov_{Γ}^{2/4}, i.e., there are functors

C_{1}^{+}, C_{2}^{+} : (Cov_{Γ}^{2/4})^{+} → (Cov_{Γ}^{2/4})^{+}

such that F ∘ C_{i}^{+} = C_{i} for i = 1, 2. These functors satisfy natural equivalences:

(C_{1}^{+})^{3} = Id,  (C_{1}^{+})^{2} = C_{2},  C_{1}^{+}κ = κC_{2}^{+}.

Proof. Let (Z/Z_{0}, ∂) be an object in (Cov_{Γ}^{2/4})^{+} and let σ denote the involution of Z/Z_{0}. Consider a real vector space V with basis (e_{1}, e_{2}, e_{3}, e_{4}). Fixing a bijection φ between a section ω ∈ C_{1}(Z/Z_{0}, ∂) and {e_{1}, …, e_{4}}, we identify Z with a subset of V by

z ↦ \begin{cases}z^{φ} & if z ∈ ω, \\ −z^{σφ} & if z ∉ ω. \end{cases}

Thus, Z = {±e_{1}, ±e_{2}, ±e_{3}, ±e_{4}} and σ acts on Z by mapping each element to its opposite. The action of Γ on Z extends to a linear action on V since it commutes with σ. We also identify C(Z/Z_{0}) with a subset of V by the map

ω′ ↦ \frac{1}{2} ∑_{z∈ω′} z.

The set C(Z/Z_{0}) then consists of the following vectors and their opposite:

f_{1} = \frac{1}{2}(e_{1} + e_{2} + e_{3} + e_{4})  g_{1} = \frac{1}{2}(e_{1} + e_{2} + e_{3} − e_{4}),
f_{2} = \frac{1}{2}(e_{1} + e_{2} − e_{3} − e_{4})  g_{2} = \frac{1}{2}(e_{1} + e_{2} − e_{3} + e_{4}),
f_{3} = \frac{1}{2}(e_{1} − e_{2} + e_{3} − e_{4})  g_{3} = \frac{1}{2}(e_{1} − e_{2} + e_{3} + e_{4}),
f_{4} = \frac{1}{2}(−e_{1} + e_{2} + e_{3} − e_{4})  g_{4} = \frac{1}{2}(e_{1} − e_{2} − e_{3} − e_{4}).

Since ω ∈ C_{1}(Z/Z_{0}, ∂), we have (see Proposition 2.6

C_{1}(Z/Z_{0}, ∂) = {±f_{1}, ±f_{2}, ±f_{3}, ±f_{4}}

and

C_{2}(Z/Z_{0}, ∂) = {±g_{1}, ±g_{2}, ±g_{3}, ±g_{4}}.

The canonical involutions on C_{1}(Z/Z_{0}, ∂) and C_{2}(Z/Z_{0}, ∂) map each vector to its opposite. Note that these identifications are independent of the choice of the section ω in C_{1}(Z/Z_{0}, ∂) and of the bijection φ.

Let ∂_{1} be the orientation of C_{1}(Z/Z_{0}, ∂) such that C_{1}(C_{1}(Z/Z_{0}, ∂), ∂_{1}) contains the section {f_{1}, f_{2}, f_{3}, f_{4}}. Thus, by Proposition 2.6, C_{1}(C_{1}(Z/Z_{0}, ∂), ∂_{1}) consists of the following sections:

±{f_{1}, f_{2}, f_{3}, f_{4}}, ±{f_{1}, f_{2}, −f_{3}, −f_{4}}, ±{f_{1}, −f_{2}, f_{3}, −f_{4}}, ±{−f_{1}, f_{2}, f_{3}, −f_{4}}.

They are characterized by the property that for each i = 1, …, 4 they contain a section ±f_{j} containing e_{i} and a section ±f_{k} containing −e_{i}. Identifying these sections to vectors in V as above, we obtain

C_{1}(C_{1}(Z/Z_{0}, ∂), ∂_{1}) = \{±\frac{1}{2}(f_{1} + f_{2} + f_{3} + f_{4}), ±\frac{1}{2}(f_{1} + f_{2} − f_{3} − f_{4}),

± \frac{1}{2}(f_{1} − f_{2} + f_{3} − f_{4}), ±\frac{1}{2}(−f_{1} + f_{2} + f_{3} − f_{4})\}.

Likewise, let ∂_{2} be the orientation of C_{2}(Z/Z_{0}, ∂) such that

C_{1}(C_{2}(Z/Z_{0}, ∂), ∂_{2}) = \{±\frac{1}{2}(g_{1} + g_{2} + g_{3} + g_{4}), ±\frac{1}{2}(g_{1} + g_{2} − g_{3} − g_{4}),

± \frac{1}{2}(g_{1} − g_{2} + g_{3} − g_{4}), ±\frac{1}{2}(−g_{1} + g_{2} + g_{3} − g_{4})\}.

Define the functors C_{1}^{+}, C_{2}^{+} : (Cov_{Γ}^{2/4})^{+} → (Cov_{Γ}^{2/4})^{+} by

C_{1}^{+}(Z/Z_{0}, ∂) = (C_{1}(Z/Z_{0}), ∂_{1})  and  C_{2}^{+}(Z/Z_{0}, ∂) = (C_{2}(Z/Z_{0}), ∂_{2}).

By definition, it is clear that F ∘ C_{i}^{+} = C_{i} for i = 1, 2. To establish the natural equivalences, consider the linear map μ: V → V defined by μ(e_{i}) = f_{i} for i = 1, …, 4. Using this map, we may rephrase the definition of C_{1}^{+} as follows: for Z = {±e_{1}, ±e_{2}, ±e_{3}, ±e_{4}} with the orientation ∂ such that C_{1}(Z/Z_{0}, ∂) ∋ μ(e_{1}), we have

C_{1}^{+}(Z/Z_{0}, ∂) = {±μ(e_{1}), ±μ(e_{2}), ±μ(e_{3}), ±μ(e_{4})}

with the orientation such that

C_{1}(C_{1}^{+}(Z/Z_{0}, ∂)) ∋ \frac{1}{2}(μ(e_{1}) + μ(e_{2}) + μ(e_{3}) + μ(e_{4})).

Note that \frac{1}{2}(μ(e_{1})+μ(e_{2})+μ(e_{3})+μ(e_{4})) = μ(f_{1}) = μ^{2}(e_{1}). Therefore, substituting μ(e_{i}) for e_{i}, for i = 1, …, 4, we obtain

(C_{1}^{+})^{2}(Z/Z_{0}, ∂) = {±μ^{2}(e_{1}), ±μ^{2}(e_{2}), ±μ^{2}(e_{3}), ±μ^{2}(e_{4})},

endowed with an orientation such that

C_{1}((C_{1}^{+})^{2}(Z/Z_{0}, ∂)) ∋ μ^{3}(e_{1}).

Computation shows that μ^{2}(e_{i}) = g_{i} for i = 1, …, 4, and μ^{3} = Id. Since \frac{1}{2}(g_{1} + g_{2} + g_{3} + g_{4}) = e_{1}, it follows that (C_{1}^{+})^{2}(Z/Z_{0}, ∂) = C_{2}^{+}(Z/Z_{0}, ∂). Similarly, we have

(C_{1}^{+})^{3}(Z/Z_{0}, ∂) = {±μ^{3}(e_{1}), ±μ^{3}(e_{2}), ±μ^{3}(e_{3}), ±μ^{3}(e_{4})} = Z,

endowed with an orientation such that

C_{1}((C_{1}^{+})^{3}(Z/Z_{0}, ∂)) ∋ μ^{4}(e_{1}) = f_{1},

hence (C_{1}^{+})^{3}(Z/Z_{0}, ∂) = (Z/Z_{0}, ∂). Finally, we have

κ(Z/Z_{0}, ∂) = {±e_{1}, ±e_{2}, ±e_{3}, ±e_{4}}

with an orientation such that C_{1}κ(Z/Z_{0}, ∂) ∋ μ^{2}(e_{1}), hence

C_{1}^{+}κ(Z/Z_{0}, ∂) = {±μ^{2}(e_{1}), ±μ^{2}(e_{2}), ±μ^{2}(e_{3}), ±μ^{2}(e_{4})}

endowed with an orientation such that

C_{1}(C_{1}^{+}κ(Z/Z_{0}, ∂)) ∋ μ^{4}(e_{1}) = f_{1}.

Therefore, C_{1}^{+}κ(Z/Z_{0}, ∂) = κC_{2}^{+}(Z/Z_{0}, ∂). □

Remark 4.2. The decomposition C(Z/Z_{0}) = C_{1}(Z/Z_{0}, ∂) ⊔ C_{2}(Z/Z_{0}, ∂) can also be viewed geometrically on a hypercube: suppose V = ℝ^{4} and let (e_{1}, e_{2}, e_{3}, e_{4}) be the standard basis. The set

C(Z/Z_{0}) = \{\frac{1}{2}(±e_{1} ± e_{2} ± e_{3} ± e_{4})\}

is the set of vertices of a hypercube K (see Figure 1), and the set

Z = {±e_{1}, ±e_{2}, ±e_{3}, ±e_{4}}

is in bijection with the set of 3-dimensional cells of K.

Figure 1

We identify

Z = {A, Ā, B, B̄, C, C̄, D, D̄}

where A, …, C̄ are as in Figure 1, D is the big cell and D̄ the small cell inside. The involution permutes a cell with its opposite cell and the set Z_{0} is obtained by identifying pairs of opposite cells

Z_{0} = {{A, Ā}, {B, B̄}, {C, C̄}, {D, D̄}}.

To obtain a corresponding identification of C(Z/Z_{0}) with the set of vertices of K, observe that a section of Z/Z_{0} consists of a set of four cells which are pairwise not opposite. Four such cells intersect in exactly one vertex and conversely each vertex lies in four cells. With the notation in Figure 1 we have the following identification:

\begin{matrix}1 = \{A, B̄, C, D\} & ¯1 = \{Ā, B, C̄, D̄\} & 2 = \{A, B̄, C̄, D\} & ¯2 = \{Ā B, C, D̄\} \\ 3 = \{A, B, C̄, D\} & ¯3 = \{Ā, B̄, C, D̄\} & 4 = \{A, B, C, D\} & ¯4 = \{Ā, B̄, C̄, D̄\} \\ 5 = \{Ā, B, C, D\} & ¯5 = \{A, B̄, C̄, D̄\} & 6 = \{Ā, B, C̄, D\} & ¯6 = \{A, B̄, C, D̄\} \\ 7 = \{Ā, B̄, C̄, D\} & ¯7 = \{A, B, C, D̄\} & 8 = \{Ā, B̄, C, D\} & ¯8 = \{A, B, C̄, D̄\} \end{matrix}

This set of vertices decomposes into two classes, two vertices being in the same class if the number of edges in any path connecting them is even. One class is

X = {1, ¯1, 3, ¯3, 5, ¯5, 7, ¯7}

and the other

Y = {2, ¯2, 4, ¯4, 6, ¯6, 8, ¯8}.

We get coverings X/X_{0} and Y/Y_{0} by identifying opposite vertices v and v̄. If Δ(Z) ≃ 2, the decomposition of C(Z/Z_{0}) as the disjoint union X/X_{0} ⊔ Y/Y_{0} is Γ-compatible; the functors C_{1} and C_{2} are given (up to a possible permutation) by the rule

C_{1}(Z/Z_{0}, ∂) = X/X_{0} and C_{2}(Z/Z_{0}, ∂) = Y/Y_{0}.

A section of X/X_{0} is a set of four vertices in X which are pairwise not opposite. Four such vertices either lie on a 3-dimensional cell or are adjacent to exactly one vertex in the complementary set Y. A similar claim holds for a section of Y/Y_{0}. This leads to identifying:

(4.3)  \begin{matrix}A = \{1, 3, ¯5, ¯7\} = \{2, 4, ¯6, ¯8\} & Ā = \{¯1, ¯3, 5, 7\} = \{¯2, ¯4, 6, 8\} \\ B = \{¯1, 3, 5, ¯7\} = \{¯2, 4, 6, ¯8\} & B̄ = \{1, ¯3, ¯5, 7\} = \{2, ¯4, ¯6, 8\} \\ C = \{1, ¯3, 5, ¯7\} = \{¯2, 4, ¯6, 8\} & C̄ = \{¯1, 3, ¯5, 7\} = \{2, ¯4, 6, ¯8\} \\ D = \{1, 3, 5, 7\} = \{2, 4, 6, 8\} & D̄ = \{¯1, ¯3, ¯5, ¯7\} = \{¯2, ¯4, ¯6, ¯8\} \end{matrix}

and

(4.4)  \begin{matrix}1 = \{A, B̄, C, D\} = \{2, 4, ¯6, 8\} & ¯1 = \{Ā, B, C̄, D̄\} = \{¯2, ¯4, 6, ¯8\} \\ 3 = \{A, B, C̄, D\} = \{2, 4, 6, ¯8\} & ¯3 = \{Ā, B̄, C, D̄\} = \{¯2, ¯4, ¯6, 8\} \\ 5 = \{Ā, B, C, D\} = \{¯2, 4, 6, 8\} & ¯5 = \{A, B̄, C̄, D̄\} = \{2, ¯4, ¯6, ¯8\} \\ 7 = \{Ā, B̄, C̄, D\} = \{2, ¯4, 6, 8\} & ¯7 = \{A, B, C, D̄\} = \{¯2, 4, ¯6, ¯8\} \\ 2 = \{A, B̄, C̄, D\} = \{1, 3, ¯5, 7\} & ¯2 = \{Ā, B, C, D̄\} = \{¯1, ¯3, 5, ¯7\} \\ 4 = \{A, B, C, D\} = \{1, 3, 5, ¯7\} & ¯4 = \{Ā, B̄, C̄, D̄\} = \{¯1, ¯3, ¯5, 7\} \\ 6 = \{Ā, B, C̄, D\} = \{¯1, 3, 5, 7\} & ¯6 = \{A, B̄, C, D̄\} = \{1, ¯3, ¯5, ¯7\} \\ 8 = \{Ā, B̄, C, D\} = \{1, ¯3, 5, 7\} & ¯8 = \{A, B, C̄, D̄\} = \{¯1, 3, ¯5, ¯7\}, \end{matrix}

hence the existence of decompositions C(X/X_{0}) = Y/Y_{0} ⊔ Z/Z_{0} and C(Y/Y_{0}) = Z/Z_{0} ⊔ X/X_{0} which, in fact, are decompositions as Γ-sets.

Remark 4.5. In the proof of Theorem 4.1, μ is not the unique linear map that can be used to describe the C_{1}^{+} and the C_{2}^{+} construction. An alternative description uses Hurwitz’ quaternions. Choosing for V the skew field of real quaternions ℍ and for (e_{1}, e_{2}, e_{3}, e_{4}) the standard basis (1, i, j, k), we have

Z = \{±1, ±i, ±j, ±k\}, C(Z/Z_{0}) = \{\frac{1}{2}(±1 ± i ± j ± k)\},

so the union Z ∪ C(Z/Z_{0}) ⊂ ℍ is the group ℍ^{1} of Hurwitz integral quaternions of norm 1. The element

ρ = −\frac{1}{2}(1 + i + j + k)

is of order 3 in ℍ^{1} and conjugation by ρ permutes i, j and k cyclically. The set Z is in fact the underlying set of the quaternionic group 𝔔_{8} and

ℍ^{1} = 𝔔_{8} ⋊ ℭ_{3}

where the cyclic group of three elements ℭ_{3} operates on 𝔔_{8} via conjugation with ρ. If ∂ is the orientation of Z/Z_{0} such that ρ ∈ C_{1}(Z/Z_{0}, ∂), we have C_{1}^{+}(Z/Z_{0}, ∂) = ρ · Z with the orientation such that ρ^{2} ∈ C_{1}(C_{1}^{+}(Z/Z_{0}, ∂)), and C_{2}^{+}(Z/Z_{0}, ∂) = (C_{1}^{+})^{2}(Z/Z_{0}, ∂) = ρ^{2} ·Z with the orientation such that 1 ∈ C_{1}(C_{2}^{+}(Z/Z_{0}, ∂)). Note that, with respect to the standard basis, multiplication by ρ is given by the matrix

(4.6)  ρ = \frac{1}{2} (\begin{matrix}−1 & 1 & 1 & 1 \\ −1 & −1 & 1 & −1 \\ −1 & −1 & −1 & 1 \\ −1 & 1 & −1 & −1 \end{matrix})

whereas the matrix of μ is

(4.7)  μ = \frac{1}{2} (\begin{matrix}1 & 1 & 1 & −1 \\ 1 & 1 & −1 & 1 \\ 1 & −1 & 1 & 1 \\ 1 & −1 & −1 & −1 \end{matrix}) .

5. The Weyl group of D_{4}

The Dynkin diagram D_{4}

(5.1)  \begin{matrix}α_{3} \\ α_{4} \end{matrix} [figure: α_{1}; α_{2}]

has the permutation group 𝔖_{3} as a group of automorphisms. The vertices of the diagram are labeled by the simple roots of the Lie algebra of type D_{4}. Let (e_{1}, e_{2}, e_{3}, e_{4}) be the standard orthonormal basis of the Euclidean space ℝ^{4}. The simple roots are

α_{1} = e_{1} − e_{2}, α_{2} = e_{2} − e_{3}, α_{3} = e_{3} − e_{4} and α_{4} = e_{3} + e_{4}

(see [3]). The permutation α_{1} ↦ α_{4}, α_{4} ↦ α_{3}, α_{3} ↦ α_{1}, α_{2} ↦ α_{2} is an automorphism of order 3 of the Dynkin diagram. Its extension to a linear automorphism of ℝ^{4} is given by the orthogonal matrix μ of (4.7). The matrix

(5.2)  ν = (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix})

extends the automorphism α_{1} ↦ α_{1}, α_{4} ↦ α_{3}, α_{3} ↦ α_{4}, α_{2} ↦ α_{2}. The set {μ, ν} generates a subgroup of O_{4} isomorphic to 𝔖_{3}, which restricts to the automorphism group of the Dynkin diagram. The group

W (D_{4}) = (𝔖_{2} ≀ 𝔖_{4})^{+} = 𝔖_{2}^{3} ⋊ 𝔖_{4}

is the Weyl group of the split adjoint algebraic group PGO_{8}^{+}, which is of type D_{4}. The group W (D_{4}) = 𝔖_{2}^{3} ⋊ 𝔖_{4}, as a subgroup of the orthogonal group O_{4}, is generated by the reflections with respect to the roots of the Lie algebra of PGO_{8}^{+}. Elements of 𝔖_{2} ≀ 𝔖_{4} can be written as matrix products

(5.3)  w = D · P (π),

where D is the diagonal matrix Diag(ε_{1}, ε_{2}, ε_{3}, ε_{4}), ε_{i} = ±1, and P (π) is the permutation matrix of π ∈ 𝔖_{4}. The group 𝔖_{2} ≀ 𝔖_{4} fits into the exact sequence

(5.4)  1 → 𝔖_{2}^{4} → 𝔖_{2} ≀ 𝔖_{4} →^{β} 𝔖_{4} → 1

where β maps w = D·P (π) to π. Elements of W (D_{4}) have a similar representation, with the supplementary condition ∏_{i} ε_{i} = 1. In relation with the geometric description of C_{1} and C_{2} at the end of §4, note that the group 𝔖_{2} ≀ 𝔖_{4} = 𝔖_{2}^{4} ⋊ 𝔖_{4} is the group of automorphisms of the hypercube K. The subgroup W (D_{4}) = (𝔖_{2} ≀ 𝔖_{4})^{+} = 𝔖_{2}^{3} ⋊ 𝔖_{4} consists of the automorphisms of K respecting the decomposition of the set of vertices as X ⊔ Y, i.e., automorphisms of the half-hypercube.

Automorphisms of W (D_{4}). We view W (D_{4}) as a subgroup of O_{4} as in (5.3). Conjugation x ↦ μxμ^{−1} with the matrices μ and ν on O_{4} induce by restriction outer automorphisms μ̃ and ν̃ of W (D_{4}). The set {μ̃, ν̃} generates a group of automorphisms of W (D_{4}) isomorphic to 𝔖_{3} (see already [5, p. 368]). The center of W (D_{4}) is isomorphic to 𝔖_{2} and is generated by

(5.5)  w_{0} = Diag(−1, −1, −1, −1) = −1.

Thus W (D_{4})/⟨w_{0}⟩ acts on W (D_{4}) as the group of inner automorphisms. Let ψ be the automorphism of order 2 of W (D_{4}) given by

ψ: D · P (π) ↦ D · P (π) · (w_{0})^{sgn(π)},

or equivalently by x ↦ x det(x), x ∈ W (D_{4}) ⊂ O_{4}. A proof of the following result can be found in [8, Theorem 31,(5)] or in [7, Prop. 2.8,(e)]:

Proposition 5.6.

Aut (W (D_{4})) ≃ ((W (D_{4})/⟨w_{0}⟩) ⋊ 𝔖_{3}) × ⟨ψ⟩.

For any w ∈ W (D_{4}), we let Int(w): x ↦ wxw^{−1} be the inner automorphism of W (D_{4}) defined by conjugation by w, and by Int(W (D_{4})) the group of inner automorphisms of W (D_{4}). As an immediate consequence of Proposition 5.6, we have

Corollary 5.7.

Aut(W (D_{4}))/ Int(W (D_{4})) ≃ 𝔖_{3} × ⟨ψ⟩.

We call trialitarian the outer automorphisms of order 3 of W (D_{4}). As observed above, the automorphisms μ̃ and μ̃^{2} are trialitarian. Conjugation by the matrix ρ of (4.6) also yields a trialitarian automorphism ρ̃: indeed, we have ρ^{3} = 1 and

ρμ^{−1} = −1 · (\begin{matrix}0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{matrix}) ∈ W (D_{4});

hence, letting w be the matrix on the right side, we have ρ̃ = Int(w) ∘ μ̃.

Proposition 5.8. Any trialitarian automorphism of W (D_{4}) is conjugate in the group Aut (W (D_{4})) to either μ̃ or ρ̃.

Proof. Explicit computation (with the help of the algebra computational system Magma [2]) shows that the conjugation class of ρ̃ contains 16 elements and the conjugation class of μ̃ contains 32 elements. In view of Proposition 5.6 any trialitarian automorphism of W (D_{4}) is the restriction to W (D_{4}) of conjugation by an element u ∈ O_{4} of the form u = μ · w or u = μ^{2} · w, with w ∈ W (D_{4}) and u^{3} = 1. There are 48 elements u of this form, hence the claim. □

Corollary 5.9. There are up to isomorphism two types of subgroups of fixed points of trialitarian automorphisms of W (D_{4}), those isomorphic to Fix(μ̃) and those isomorphic to Fix(μ̃) .

Proof. Trialitarian automorphisms which are conjugate in Aut (W (D_{4})) have isomorphic groups of fixed points. □

Proposition 5.10. 1) The 2-dimensional subspace of ℝ^{4} generated by the set of elements {e_{1} + e_{3}, e_{2} − e_{3}} is fixed under μ.

2) The set {e_{1} +e_{3}, e_{2} −e_{3}} generates a root system of type G_{2} and the group Fix(μ̃) is the corresponding Weyl group, which is the dihedral group 𝔇_{6} of order 12.

Proof. By explicit computation. □

Proposition 5.11. The group Fix(ρ̃) is isomorphic to the group of order 24 of Hurwitz quaternions ℍ^{1} = 𝔔_{8} ⋊ℭ_{3}. This group is isomorphic to the double covering 𝔄̃_{4} of 𝔄_{4}.

Proof. Recall that the matrix ρ is obtained by choosing (1, i, j, k) as basis of ℝ^{4} and letting −\frac{1}{2}(1 + i + j + k) operate by left multiplication in ℍ. The group ℍ^{1} has a representation in W (D_{4}) by right multiplication which obviously commutes with the action of ρ. Hence Fix(ρ̃) contains a copy of ℍ^{1}. The claim then follows from the fact that Fix(ρ̃) has 24 elements. □

Cohomology with W (D_{4}) coefficients. Each automorphism α ∈ Aut(W (D_{4})) acts on H^{1}(Γ, W (D_{4})) by

α_{∗}: [φ] ↦ [α ∘ φ],

where φ: Γ → W (D_{4}) is a cocycle with values in W (D_{4}). If α′ = Int(w) ∘ α for some w ∈ W (D_{4}), then for all cocycles φ: Γ → W (D_{4}) we have

w · α(φ(γ)) · w^{−1} = α′(φ(γ)) for all γ ∈ Γ,

hence [α ∘ φ] = [α′ ∘ φ] and therefore α_{∗} = α′_{∗}. Thus, the action of Aut(W (D_{4})) on H^{1}(Γ, W (D_{4})) factors through Aut(W (D_{4}))/ Int(W (D_{4})) ≃ 𝔖_{3} × ⟨ψ⟩. In particular the symmetric group 𝔖_{3} acts on H^{1}(Γ, W (D_{4})). Under the bijections (3.5), the

symmetric group 𝔖_{3} also acts on Iso ((Cov_{Γ}^{2/4})^{+}) and Iso((Étex_{Γ}^{2/4})^{+}). The action of the outer automorphism ν̃ associates to the oriented 2/4-covering (Z/Z_{0}, ∂_{Z}) the oriented covering κ(Z/Z_{0}, ∂) = (Z/Z_{0}, ∂_{Z} ∘ ι) where 2 ⟵^{ι} 2 twists the orientation. The proof of Theorem 4.1 shows that the action of μ̃ maps the class of an oriented covering (Z/Z_{0}, ∂_{Z}) to the class of C_{1}^{+}(Z/Z_{0}, ∂_{Z}).

6. Triality and étale algebras

We next investigate triality on isomorphism classes of étale algebras using Galois cohomology. Oriented extensions of étale algebras S/S_{0} with dim_{F} S = 8 and dim_{F} S_{0} = 4 correspond to cocycles, i.e., continuous homomorphisms Γ → W (D_{4}), and isomorphism classes of such algebras correspond to cocycles up to conjugation. If the cocycle factors through a subgroup G of W (D_{4}), the conjugacy class of G in W (D_{4}) is determined by the isomorphism class of the algebra. Thus it makes sense to classify isomorphism classes of algebras according to the conjugacy classes of the subgroups G of W (D_{4}). We give in Table 1 a list of all conjugacy classes of subgroups of W (D_{4}). We still consider W (D_{4}) as a subgroup of O_{4} (see (5.3)) and use the following notation. The group W (D_{4}) fits into the split exact sequence:

(6.1)  1 → 𝔖_{2}^{3} → W (D_{4}) →^{β} 𝔖_{4} → 1.

where β is as in (5.4). For each subgroup G of W (D_{4}) we denote by G_{1} the restriction G ∩ 𝔖_{2}^{3} and by G_{0} the projection β(G). The center of W (D_{4}), generated by w_{0} = Diag(−1, −1, −1, −1) = −1 is denoted by C and we set w_{1} = Diag(1, −1, 1, −1), w_{2} = Diag(1, −1, −1, 1) and w_{3} = Diag(−1, −1, 1, 1) for special elements of the subgroup 𝔖_{2}^{3} ⊂ W (D_{4}) given by diagonal matrices. We denote by 𝔖_{n} the permutation group of n elements, 𝔄_{n} is the alternating subgroup, ℭ_{n} is cyclic of order n, 𝔇_{n} is the dihedral group of order 2n, 𝔙_{4} is the Klein 4-group, and 𝔔_{8} is the quaternionic group with eight elements. In Column S we summarize the various possibilities for étale algebras of dimension 8 associated to the class of a cocycle α: Γ → W which factors through G and in Column S_{0} étale algebras of dimension 4 associated to the class of the induced cocycle β ∘ α: Γ → 𝔖_{4} which factors through G_{0}. The entry K in one of the columns S or S_{0} denotes a quadratic separable field extension. We use symbols E, respectively E_{0} for separable field extensions whose Galois closures have Galois groups G, respectively G_{0}. The symbol R(E) stands for the cubic resolvent of E if E is a quartic separable field extension and λ^{2}E stands for the second lambda power of E (see [12], where it is denoted Λ_{2}(E) or [9], where it is denoted E(2)^{4}). If dim_{F} E = 4, λ^{2}E admits an involution σ and for any quadratic étale algebra K with involution ι we set K ∗ λ^{2}E = (K ⊗_{F} λ^{2}E)^{ι ⊗ σ}. We write \overline{E}_{0} for the Galois closure of E_{0}. The symbol ℓ gives the number of subgroups in the conjugacy class of G, MS refers to the maximal subgroups of G and in column T we give the two conjugacy classes which are the images of the class of G under the trialitarian automorphisms μ̃ and μ̃^{2}. Entries N, |G|, ℓ and MS in the table were generated with the help of the Magma algebra software [2]. The computation of the entry T, the explicit representation of the group G as an exact sequence and the decomposition of the étale algebras as products of fields were checked case by case. Explicit computations of trialitarian triples were made in [1] and [19] using the description of the trialitarian action given in the proof of Theorem 4.1.

4The corresponding Γ-set λ^{2}X is obtained by removing the diagonal from X × X and dividing by the involution (x, y) ↦ (y, x).

16 MAX-ALBERT KNUS AND JEAN-PIERRE TIGNOL

\begin{matrix}N & S_{0} & S & G_{1} → G → G_{0} & |G| & ℓ & MS & T \\ 1 & F^{4} & F^{8} & 1 → 1 → 1 & 1 & 1 & 1 & 1, 1 \\ 2 & F^{4} & K^{4} & C → 𝔖_{2} → 1 & 2 & 6 & 1 & 2, 2 \\ 3 & K^{2} & K^{4} & 1 → 𝔖_{2} → 𝔖_{2} ⊂ 𝔙_{4} & 2 & 6 & 1 & 3, 5 \\ 4 & K^{2} & K^{4} & 1 → 𝔖_{2} → 𝔖_{2} ⊂ 𝔙_{4} & 2 & 6 & 1 & 4, 3 \\ 5 & F^{4} & F^{2} × K^{2} & ⟨w_{1}⟩ → 𝔖_{2} → 1 & 2 & 6 & 1 & 5, 4 \\ 6 & F^{2} × K & F^{2} × K^{2} & 1 → 𝔖_{2} → 𝔖_{2} ⊄ 𝔙_{4} & 2 & 12 & 1 & 6, 6 \\ 7 & F^{2} × K & F^{2} × K^{2} & 1 → 𝔖_{2} → 𝔖_{2} ⊄ 𝔙_{4} & 2 & 12 & 1 & 7, 7 \\ 8 & F × E_{0} & F^{2} × S_{0}^{2} & 1 → ℭ_{3} → ℭ_{3} & 3 & 16 & 1 & 8, 8 \\ 9 & F^{4} & K_{1}^{2} × K_{2}^{2} & ⟨C, w_{1}⟩ → 𝔖_{2}^{2} → 1 & 4 & 3 & 2 5 & 11, 10 \\ 10 & K^{2} & K ⊗ K_{1}^{2} & C → 𝔖_{2}^{2} → 𝔖_{2} ⊂ 𝔙_{4} & 4 & 3 & 2 3 & 9, 11 \\ 11 & K^{2} & K ⊗ K_{1}^{2} & C → 𝔖_{2}^{2} → 𝔖_{2} ⊂ 𝔙_{4} & 4 & 3 & 2 4 & 10, 9 \\ 12 & K_{1} ⊗ K_{2} & K_{1} ⊗ K_{2}^{2} & 1 → 𝔖_{2}^{2} → 𝔙_{4} & 4 & 4 & 4 & 14, 13 \\ 13 & F^{4} & K_{1}^{2} × K_{2}^{2} & ⟨w_{1}, w_{2}⟩ → 𝔖_{2}^{2} → 1 & 4 & 4 & 5 & 12, 14 \\ 14 & K_{1} ⊗ K_{2} & K_{1} ⊗ K_{2}^{2} & 1 → 𝔖_{2}^{2} → 𝔙_{4} & 4 & 4 & 3 & 13, 12 \\ 15 & F^{2} × K & F^{4} × K_{1} ⊗ K & ⟨w_{1}⟩ → 𝔖_{2}^{2} → 𝔖_{2} ⊄ 𝔙_{4} & 4 & 6 & 5 6 & 18, 17 \\ 16 & K_{1} × K_{2} & K_{1} ⊗ K_{2}^{2} & 1 → 𝔖_{2}^{2} → 𝔖_{2}^{2} & 4 & 6 & 4 7 & 21, 19 \\ 17 & K_{1} × K_{2} & K_{1}^{2} × K_{2}^{2} & 1 → 𝔖_{2}^{2} → 𝔖_{2}^{2} & 4 & 6 & 3 6 & 15, 18 \\ 18 & K_{1} × K_{2} & K_{1}^{2} × K_{2}^{2} & 1 → 𝔖_{2}^{2} → 𝔖_{2}^{2} & 4 & 6 & 4 6 & 17, 15 \\ 19 & F^{2} × K & K_{1}^{2} × K_{1} × K & ⟨w_{1}⟩ → 𝔖_{2}^{2} → 𝔖_{2} ⊄ 𝔙_{4} & 4 & 6 & 5 7 & 16, 21 \\ 20 & K^{2} & E^{2} & C → ℭ_{4} → 𝔖_{2} ⊂ 𝔙_{4} & 4 & 2 & 2 & 20, 20 \\ 21 & K_{1} × K_{2} & K_{1} ⊗ K_{2}^{2} & 1 → 𝔖_{2}^{2} → 𝔖_{2}^{2} & 4 & 6 & 3 7 & 19, 16 \\ 22 & K_{1} × K_{2} & K_{1}^{2} × K_{1} ⊗ K_{2} & 1 → 𝔖_{2}^{2} → 𝔖_{2}^{2} & 4 & 12 & 4 6 7 & 23, 27 \\ 23 & K_{1} × K_{2} & K_{1}^{2} × K_{1} ⊗ K_{2} & 1 → 𝔖_{2}^{2} → 𝔖_{2}^{2} & 4 & 12 & 3 6 7 & 27, 22 \\ 24 & K^{2} & K^{2} × K ⊗ K_{1} & ⟨w_{1}⟩ → 𝔖_{2}^{2} → 𝔖_{2} ⊂ 𝔙_{4} & 4 & 12 & 3 4 5 & 24, 24 \\ 25 & F^{2} × K & F^{4} × E & ⟨w_{1}⟩ → ℭ_{4} → 𝔖_{2} ⊄ 𝔙_{4} & 4 & 12 & 5 & 26, 28 \\ 26 & E_{0} & E_{0}^{2} & 1 → ℭ_{4} → ℭ_{4} & 4 & 12 & 4 & 28, 25 \\ 27 & F^{2} × K & K_{1}^{2} × K_{1} ⊗ K & ⟨−w_{1}⟩ → 𝔖_{2}^{2} → 𝔖_{2} ⊄ 𝔙_{4} & 4 & 12 & 5 6 7 & 22, 23 \\ 28 & E_{0} & E_{0} × E_{0} & 1 → ℭ_{4} → ℭ_{4} & 4 & 12 & 3 & 25, 26 \\ 29 & F^{2} × K & K^{2} × K ⊗ K_{1} & C → 𝔖_{2}^{2} → 𝔖_{2} ⊄ 𝔙_{4} & 4 & 12 & 2 6 7 & 29, 29 \\ 30 & F × E_{0} & F^{2} × E_{0} ⊗ Δ(E_{0}) & 1 → 𝔖_{3} → 𝔖_{3} & 6 & 16 & 7 8 & 30, 30 \\ 31 & F × E_{0} & F^{2} × E & C → ℭ_{6} → ℭ_{3} & 6 & 16 & 2 8 & 31, 31 \\ 32 & F × E_{0} & F^{2} × E_{0}^{2} & 1 → 𝔖_{3} → 𝔖_{3} & 6 & 16 & 6 8 & 32, 32 \\ 33 & E_{0} & E & C → 𝔖_{2}^{3} → 𝔙_{4} & 8 & 1 & 11 12 & 34, 35 \\ 34 & E_{0} & E & C → 𝔖_{2}^{3} → 𝔙_{4} & 8 & 1 & 10 14 & 35, 33 \\ 35 & F^{4} & K_{1} × K_{2} × K_{3} × K_{123} & 𝔖_{2}^{3} → 𝔖_{2}^{3} → 1 & 8 & 1 & 9 13 & 33, 34 \\ 36 & E_{0} & E & C → 𝔔_{8} → 𝔙_{4} & 8 & 2 & 20 & 36, 36 \\ 37 & K^{2} & E_{1} × E_{2} & ⟨C, w_{1}⟩ → 𝔖_{2} × ℭ_{4} → 𝔖_{2} ⊂ 𝔙_{4} & 8 & 3 & 3 20 & 38, 40 \\ 38 & E_{0} & E_{0} ⊗ K & C → 𝔖_{2} × ℭ_{4} → ℭ_{4} & 8 & 3 & 11 20 & 40, 37 \\ 39 & K^{2} & K ⊗ K_{1} × K ⊗ K_{2} & ⟨C, w_{1}⟩ → 𝔖_{2}^{3} → 𝔖_{2} ⊂ 𝔙_{4} & 8 & 3 & 9 10 11 24 & 39, 39 \\ 40 & E_{0} & E_{0} ⊗ K & C → 𝔖_{2} × ℭ_{4} → ℭ_{4} & 8 & 3 & 10 20 & 37, 38 \\ 41 & K^{2}K & E^{2} & ⟨C, w_{2}⟩ → 𝔇_{4} → 𝔖_{2} ⊂ 𝔙_{4} & 8 & 6 & 9 11 20 & 49, 45 \\ 42 & F^{2} × K & K_{1} × E & ⟨C, w_{1}⟩ → 𝔖_{2} × ℭ_{4} → 𝔖_{2} ⊄ 𝔙_{4} & 8 & 6 & 9 25 & 44, 47 \\ 43 & K_{1} × K_{2} & K ⊗ K_{1} × K ⊗ K_{2} & C → 𝔖_{2}^{3} → 𝔖_{2}^{2} & 8 & 6 & 10 17 21 23 29 & 48, 46 \\ 44 & E_{0} & E & C → 𝔖_{2} × ℭ_{4} → ℭ_{4} & 8 & 6 & 11 26 & 47, 42 \\ 45 & K^{2} & E^{2} & ⟨C, w_{2}⟩ → 𝔇_{4} → 𝔖_{2} ⊂ 𝔙_{4} & 8 & 6 & 9 10 20 & 41, 49 \\ 46 & K_{1} × K_{2} & K ⊗ K_{1} × K ⊗ K_{2} & C → 𝔖_{2}^{3} → 𝔖_{2}^{2} & 8 & 6 & 11 16 18 22 29 & 43, 48 \\ 47 & E_{0} & E_{0} ⊗ K & C → 𝔖_{2} × ℭ_{4} → ℭ_{4} & 8 & 6 & 10 28 & 42, 44 \\ 48 & F^{2} × K & K_{1}^{2} × K ⊗ K_{2} & ⟨C, w_{1}⟩ → 𝔖_{2}^{3} → 𝔖_{2} ⊄ 𝔙_{4} & 8 & 6 & 9 15 19 27 29 & 46, 43 \\ 49 & E_{0} & E & C → 𝔇_{4} → 𝔙_{4} & 8 & 6 & 10 11 20 & 45, 41 \end{matrix} Table 1

\begin{matrix}N & S_{0} & S & G_{1} → G → G_{0} & |G| & ℓ & MS & T \\ 50 & F^{2} × K & K^{2} × E & ⟨w_{1}, w_{2}⟩ → 𝔇_{4} → 𝔖_{2} ⊄ 𝔙_{4} & 8 & 6 & 13 19 25 & 55, 51 \\ 51 & E_{0} & E & 1 → 𝔇_{4} → 𝔇_{4} & 8 & 12 & 14 21 28 & 50, 55 \\ 52 & E_{0} & E_{0}^{2} & 1 → 𝔇_{4} → 𝔇_{4} & 8 & 12 & 14 17 28 & 54, 57 \\ 53 & K_{1} × K_{2} & K ⊗ K_{1} × K ⊗ K_{2} & ⟨w_{1}⟩ → 𝔖_{2}^{3} → 𝔖_{2}^{2} & 8 & 12 & 16 19 21 22 23 24 27 & 53, 53 \\ 54 & F^{2} × K & F^{2} × K × E & ⟨w_{1}, w_{2}⟩ → 𝔇_{4} → 𝔖_{2} ⊄ 𝔙_{4} & 8 & 12 & 13 15 25 & 57, 52 \\ 55 & E_{0} & Ē_{0} & 1 → 𝔇_{4} → 𝔇_{4} & 8 & 12 & 12 16 26 & 51, 50 \\ 56 & K_{1} × K_{2} & K_{1}^{2} × E & ⟨w_{1}⟩ → 𝔖_{2}^{3} → 𝔖_{2}^{2} & 8 & 12 & 15 17 18 22 23 24 27 & 56, 56 \\ 57 & E_{0} & E_{0}^{2} & 1 → 𝔇_{4} → 𝔇_{4} & 8 & 12 & 12 18 26 & 52, 54 \\ 58 & E_{0} & E_{0}^{2} & 1 → 𝔄_{4} → 𝔄_{4} & 12 & 4 & 8 14 & 59, 60 \\ 59 & F × E_{0} & F^{2} × E_{0}^{2} & ⟨w_{1}, w_{2}⟩ → 𝔖_{2}^{2} ⋊ ℭ_{3} → ℭ_{3} & 12 & 4 & 8 13 & 60, 58 \\ 60 & E_{0} & E_{0}^{2} & 1 → 𝔄_{4} → 𝔄_{4} & 12 & 4 & 8 12 & 58, 59 \\ 61 & F × E_{0} & K × K ⊗ E_{0} & C → 𝔖_{2} × 𝔖_{3} → 𝔖_{3} & 12 & 4 & 29, 30, 31, 32 & 61, 61 \\ 62 & K_{1} × K_{2} & E^{2} & ⟨C, w_{1}⟩ → [2^{2}]4 → 𝔖_{2}^{2} & 16 & 3 & 39 42 & 66, 63 \\ 63 & E_{0} & E & ⟨C, w_{1}⟩ → [2^{2}]4 → ℭ_{4} & 16 & 3 & 39 47 & 62, 66 \\ 64 & K^{2} & E_{1} × E_{2} & 𝔖_{2}^{3} → 𝔖_{2} × 𝔇_{4} → 𝔖_{2} ⊂ 𝔙_{4} & 16 & 3 & 35 37 39 41 45 & 68, 67 \\ 65 & K_{1} × K_{2} & K_{1} ⊗ K_{3} × K_{2} ⊗ K_{4} & ⟨C, w_{1}⟩ → 𝔖_{2}^{4} → 𝔖_{2}^{2} & 16 & 3 & 39 43 46 48 53 56 & 65, 65 \\ 66 & E_{0} & E & ⟨C, w_{1}⟩ → [2^{2}]4 → ℭ_{4} & 16 & 3 & 39 44 & 63, 62 \\ 67 & E_{0} & E & ⟨C, w_{1}⟩ → 𝔖_{2} × 𝔇_{4} → 𝔙_{4} & 16 & 3 & 34 39 40 45 49 & 64, 68 \\ 68 & E_{0} & E & ⟨C, w_{2}⟩ → 𝔖_{2} × 𝔇_{4} → 𝔙_{4} & 16 & 3 & 33 38 39 41 49 & 67, 64 \\ 69 & K_{1} × K_{2} & E^{2} & ⟨C, w_{1}⟩ → [2^{2}]4 → 𝔖_{2}^{2} & 16 & 6 & 37 42 48 & 73, 72 \\ 70 & E_{0} & E & ⟨C, w_{1}⟩ → 𝔔_{8} : 2 → 𝔙_{4} & 16 & 6 & 36 37 38 40 41 45 49 & 70, 70 \\ 71 & F^{2} × K & K_{1}^{2} × E & 𝔖_{2}^{3} → 𝔖_{2} × 𝔇_{4} → 𝔖_{2} ⊄ 𝔙_{4} & 16 & 6 & 35 42 48 50 54 & 75, 74 \\ 72 & E_{0} & E & ⟨C, w_{1}⟩ → [2^{2}]4 → 𝔙_{4} & 16 & 6 & 40 43 47 & 69, 73 \\ 73 & E_{0} & E & C → [2^{2}]4 → 𝔇_{4} & 16 & 6 & 38 44 46 & 72, 69 \\ 74 & E_{0} & E_{0} ⊗ K & C → 𝔖_{2} × 𝔇_{4} → 𝔇_{4} & 16 & 6 & 34 43 47 51 52 & 71, 75 \\ 75 & E_{0} & E_{0} ⊗ K & C → 𝔖_{2} × 𝔇_{4} → 𝔇_{4} & 16 & 6 & 33 44 46 55 57 & 74, 71 \\ 76 & E_{0} & E_{0}^{2} & 1 → 𝔖_{4} → 𝔖_{4} & 24 & 4 & 32 52 58 & 79, 81 \\ 77 & F × R(E) & Δ(E) × λ^{2}E & ⟨w_{1}, w_{3}⟩ → 𝔖_{2}^{2} ⋊ 𝔖_{3} → 𝔖_{3} & 24 & 4 & 30 50 59 & 84, 82 \\ 78 & F × E_{0} & Δ(E) × λ^{2}E & 𝔖_{2}^{3} → 𝔖_{2}^{3} ⋊ ℭ_{3} → ℭ_{3} & 24 & 4 & 31 35 59 & 83, 80 \\ 79 & F × R(E) & F^{2} × λ^{2}E & ⟨w_{1}, w_{3}⟩ → 𝔖_{2}^{2} ⋊ 𝔖_{3} → 𝔖_{3} & 24 & 4 & 32 54 59 & 76, 81 \\ 80 & E_{0} & E_{0} ⊗ K & C → 𝔖_{2} × 𝔄_{4} → 𝔄_{4} & 24 & 4 & 31 34 58 & 78, 83 \\ 81 & E_{0} & E_{0}^{2} & 1 → 𝔖_{4} → 𝔖_{4} & 24 & 4 & 32 57 60 & 76, 79 \\ 82 & E_{0} & E_{0} ⊗ Δ(E_{0}) & 1 → 𝔖_{4} → 𝔖_{4} & 24 & 4 & 30 51 58 & 77, 84 \\ 83 & E_{0} & E & C → 𝔖_{2} × 𝔄_{4} → 𝔄_{4} & 24 & 4 & 31 33 60 & 78, 80 \\ 84 & E_{0} & E_{0} ⊗ Δ(E_{0}) & 1 → 𝔖_{4} → 𝔖_{4} & 24 & 4 & 30 55 66 & 82, 77 \\ 85 & E_{0} & E & 𝔖_{2} → 𝔄̃_{4} → 𝔄_{4} & 24 & 4 & 31 36 & 85, 85 \\ 86 & E_{0} & E & 𝔖_{2}^{3} → 𝔖_{2}^{3} ⋊ 𝔙_{4} → 𝔙_{4} & 32 & 1 & 64 67 68 70 & 86, 86 \\ 87 & E_{0} & E & ⟨C, w_{1}⟩ → 𝔖_{2}^{2} ⋊ 𝔇_{4} → 𝔇_{4} & 32 & 3 & 63 65 67 72 74 & 92, 90 \\ 88 & E_{0} & E & 𝔖_{2}^{3} → 𝔖_{2}^{3} ⋊ ℭ_{4} → ℭ_{4} & 32 & 3 & 63 64 66 & 91, 89 \\ 89 & E_{0} & E & ⟨C, w_{1}⟩ → 𝔖_{2}^{3} ⋊ ℭ_{4} → 𝔇_{4} & 32 & 3 & 62 66 67 & 88, 91 \\ 90 & E_{0} & E & ⟨C, w_{1}⟩ → 𝔖_{2}^{2} ⋊ 𝔇_{4} → 𝔇_{4} & 32 & 3 & 65 66 68 73 75 & 92, 87 \\ 91 & E_{0} & E & ⟨C, w_{1}⟩ → 𝔖_{2}^{3} ⋊ ℭ_{4} → 𝔇_{4} & 32 & 3 & 62 63 68 & 89, 88 \\ 92 & K_{1} × K_{2} & E_{1} × E_{2} & 𝔖_{2}^{3} → 𝔖_{2}^{2} ⋊ 𝔇_{4} → 𝔖_{2}^{2} & 32 & 3 & 62 64 65 69 71 & 87, 90 \\ 93 & F × R(E) & Δ(E) × K ∗ λ^{2}E & 𝔖_{2}^{3} → 𝔖_{2} × 𝔖_{4} → 𝔖_{3} & 48 & 4 & 61 71 77 78 79 & 94, 95 \\ 94 & E_{0} & E_{0} ⊗ K & C → 𝔖_{2} × 𝔖_{4} → 𝔖_{4} & 48 & 4 & 61 75 81 83 84 & 95, 93 \\ 95 & E_{0} & E_{0} ⊗ K & C → 𝔖_{2} × 𝔖_{4} → 𝔖_{4} & 48 & 4 & 61 74 76 80 82 & 93, 94 \\ 96 & E_{0} & E & 𝔖_{2}^{3} → 𝔖_{2}^{3} ⋊ 𝔇_{4} → 𝔇_{4} & 64 & 3 & 86 87 88 89 90 91 92 & 96, 96 \\ 97 & E_{0} & E & 𝔖_{2}^{3} → 𝔖_{2}^{3} ⋊ 𝔄_{4} → 𝔄_{4} & 96 & 1 & 78 80 83 85 86 & 97, 97 \\ 98 & E_{0} & E & 𝔖_{2}^{3} → 𝔖_{2}^{3} ⋊ 𝔖_{4} → 𝔖_{4} & 192 & 1 & 93 94 95 96 97 & 98, 98 \end{matrix}

Table 1

Trialitarian triples and fixed points. Let α be any trialitarian automorphism of W (D_{4}). The subset H^{1}(Γ, W (D_{4}))^{C_{3}} of cohomology classes that are fixed under α_{∗} is independent of the particular choice of α. Clearly, the image in H^{1}(Γ, W (D_{4})) of any cohomology class in H^{1}(Γ, Fix(α)) is fixed under α_{∗}, hence this image lies in H^{1}(Γ, W (D_{4}))^{C_{3}}. Thus, we have a canonical map

(6.2)  H^{1}(F, Fix(α)) → H^{1}(F, W (D_{4}))^{C_{3}}

for any trialitarian automorphism α of W (D_{4}).

Theorem 6.3. Any class in H^{1}(F, W (D_{4}))^{C_{3}} lies in the image of the map (6.2) for α = μ̃ or α = ρ̃ as in (5.8).

Proof. If the class [φ] of a cocycle φ: Γ → W (D_{4}) belongs to H^{1}(Γ, W (D_{4}))^{C_{3}}, the cocycle factors through a subgroup G whose conjugacy class is invariant under μ̃. We get from Columns N and T of Table 1 a list of all triples of conjugate classes of subgroups W (D_{4}) which are permuted by a trialitarian automorphism of W (D_{4}) . A triple consists of three identic labels (for example (70, 70, 70)) if there exists a ∈ W (D_{4}) such that

μGμ^{−1} = aGa^{−1},

where μ is as in (4.7). A cocycle factoring through such a group G does not necessarily correspond to a triple of isomorphism classes of étale algebras fixed under triality. For a triple consisting of isomorphic algebras we must have a ∈ W (D_{4}) such that

(6.4)  μxμ^{−1} = axa^{−1}

for all x ∈ G, since isomorphic classes of algebras are given by homomorphisms γ: Γ → G up to conjugation. Thus a necessary condition to get a triple of fixed isomorphism classes of algebras is that the conjugacy classes of all subgroups H of G are invariant under triality. Thus the conjugacy classes N = 24, 39, 53, 56, 65, 70, 86, 96, 97 and 98 do not correspond to triples of isomorphism classes of algebras invariant under triality. The conjugacy classes left over in Table are N= 1, 2, 6, 7, 8, 20, 29, 30, 31, 32, 36, 61 and 85. They give rise to triples of isomorphism classes of étale algebras fixed under triality, since they correspond to subgroups contained in Fix(μ̃) (N = 61) or contained in Fix(ρ̃) (N=85). □

Observe that fixed étale algebras in class N = 61 are of the form K × (E_{0} ⊗ K), where K is quadratic and E_{0} is cubic. Hence they are not fields over F, in contrast to algebras in class N = 85.

7. Trialitarian resolvents

Trialitarian triples of étale algebras can be viewed as one étale algebra with two attached resolvents (see Remark 3.6). For example, let E be a quartic separable field with Galois group 𝔖_{4}. The field E ⊗ Δ(E) is octic with the same Galois group 𝔖_{4} and the extension E ⊗ Δ(E)/E corresponds to Class N = 82 in Table 1. Class N = 77 in the same trialitarian triple corresponds to the extension

(Δ(E) × λ^{2}E)/(F × R(E)),

where Δ(E) is the discriminant, R(E) is the cubic resolvent of E and λ^{2}E is the second lambda power of E, as defined in [12]. In this section we consider the situation where one étale algebra in the triple is given by a separable polynomial and compute polynomials for the two other étale algebras. We assume that the base field F is infinite and has characteristic different from 2.

Proposition 7.1. Let S/S_{0} be an étale algebra with involution of dimension 2n over F.

1) There exists an invertible element x ∈ S such that x generates S and x^{2} generates S_{0}.

2) There exists a polynomial

f_{n}(x) = x^{n} + a_{n−1}x^{n−1} + ⋯ + a_{1}x + a_{0}.

with coefficients in F such that S_{0} ⟶^{∼} F [x]/(f_{n}(x)) and S ⟶^{∼} F [x]/(f_{2n}(x)), where f_{2n}(x) = f_{n}(x^{2}).

3) The algebra S has trivial discriminant if and only if (−1)^{n}a_{0} is a square in F.

Proof. To prove 1) we are looking for invertible elements x of S such that Tr_{S/S_{0}}(x) = 0 and such that the discriminant of the characteristic polynomial of x^{2} is not zero. Any such element generates S and x^{2} ∈ S_{0} generates S_{0}. These elements form an Zariski open subset of the space of trace zero elements. One checks that this open subspace is not empty by going to an algebraic closure of F.

2) follows from 1) and 3) follows from a discriminant formula (see [4, p. 51]) for the discriminant D(f_{2n}) of f_{2n}:

D(f_{2n}) = (−1)^{n}a_{0} · (2^{n}D(f_{n}))^{2}

(recall that Δ(S) ⟶^{∼} F [x]/(x^{2} − D(f_{2n}))). □

Theorem 7.2. Let S/S_{0}, dim_{F} S= 8, with trivial discriminant, be given as in Proposition 7.1, by a polynomial

(7.3)  f_{4}(x) = x^{4} + ax^{3} + bx^{2} + cx + e^{2}.

The polynomials

f′_{4}(x) = x^{4} + ax^{3} + (\frac{3}{8}a^{2} − \frac{1}{2}b + 3e)x^{2} +
(\frac{1}{16}a^{3} − \frac{1}{4}ab + c + \frac{1}{2}ae)x + (\frac{1}{16}a^{2} − \frac{1}{4}b − \frac{1}{2}e)^{2} and
f′′_{4}(x) = x^{4} + ax^{3} + (\frac{3}{8}a^{2} − \frac{1}{2}b − 3e)x^{2} +
(\frac{1}{16}a^{3} − \frac{1}{4}ab + c − \frac{1}{2}ae)x + (\frac{1}{16}a^{2} − \frac{1}{4}b + \frac{1}{2}e)^{2}

define extensions of étale algebras S′/S′_{0}, S′′/S′′_{0} such that the isomorphism classes of S/S_{0}, S′/S′_{0} and S′′/S′′_{0} are in triality.

Proof. Let {y_{1}, y_{2}, y_{3}, y_{4}} be the set of zeroes of f_{4} in a separable closure F_{s} of F. The set \{±x_{i} = ±\sqrt{y}_{i}, i = 1, … 4\} is the set of zeroes of f_{8}. Let ξ be the column vector [x_{1}, x_{2}, x_{3}, x_{4}]^{T}. If φ : Γ → W (D_{4}) ⊂ O_{4} is the cocycle corresponding to S/S_{0}, the group φ(Γ) permutes the elements ±x_{i} through left matrix multiplication on ξ. The cocycle corresponding to S′/S′_{0} is given by

φ′ : γ ↦ μφ(γ)μ^{−1}, γ ∈ Γ,

where μ is as in 4.7. Thus φ′(Γ) permutes the components of ±ξ′ = ±μξ = ±[x′_{1}, x′_{2}, x′_{3}, x′_{4}]^{T} and {±x′_{i}, i = 1, …, 4} is the set of zeroes of f′_{8}. It follows that

f′_{8}(x) = f′_{4}(x^{2}) = ∏_{i}(x − x′_{i})(x + x′_{i}) = ∏_{i}(x^{2} − x′_{i}^{2}).

The x′_{i} are the components of ξ′ = μξ. Thus the coefficients of f′_{8}(x) can be expressed as functions of the x_{i}. Using that the symmetric functions in the x_{i} can be expressed as functions of the coefficients of f_{8} one gets (for example with Magma [2]) the expression given in Proposition 7.2 for f′_{i}. Similar computations with μ^{2} instead of μ lead to the formula for f′′_{4}. □

Remark 7.4. Observe that we move from f′_{8} to f′′_{8} by replacing e by −e, as it should be.

8. Triality and Witt invariants of étale algebras

The results of this section were communicated to us by J-P. Serre, [15]. They are based on results of [14] and [9]. Similar results can be obtained for cohomological invariants of étale algebras instead of Witt invariants. Let k be a fixed base field of characteristic not 2 and F/k be a field extension. Let W Gr(F) be the Witt-Grothendieck ring and W (F) the Witt ring of F, viewed as functors of F. We recall that elements of W Gr(F) are formal differences q − q′ of isomorphism classes of nonsingular quadratic forms over F and that the sum and product are those induced by the orthogonal sum and the tensor product of quadratic forms. The Witt ring W (F) is the quotient of W Gr(F) by the ideal consisting of integral multiples of the 2-dimensional diagonal form ⟨1, −1⟩. Some of the following considerations hold for oriented quadratic extensions S/S_{0} of étale algebras of arbitrary dimension. To simplify notation we assume from now on that dim_{F} S = 8. Let (Étex^{2/4})^{+} be the functor which associates to F the set (Étex_{F}^{2/4})^{+} of isomorphism classes of oriented quadratic extensions S/S_{0} of étale algebras over F such

that dim_{F} S = 8. A Witt invariant on (Étex^{2/4})^{+}, more precisely on W (D_{4}), is a map

H^{1}(F, W (D_{4})) → W (F)

for each F/k, subject to compatibility and specialization conditions (see [9]). The set of Witt invariants

Inv(W (D_{4}), W ) = Inv((Étex^{2/4})^{+}, W )

is a module over W (k). The aim of this section is to describe this module and how triality acts on it. A main tool is the following splitting principle, which is a special case of a variant of the splitting principle for étale algebras (see [9, Theorem 24.9]) and which can be proved following the same lines.

Theorem 8.1. If a ∈ Inv ((Étex^{2/4})^{+}, W ) satisfies a(S/S_{0}) = 0 for every product of two biquadratic algebras

S = F (\sqrt{x}, \sqrt{y}) × F (\sqrt{z}, \sqrt{t}), S_{0} = F (\sqrt{xy}) × F (\sqrt{zt}).

over every extension F of k, then a = 0.

Let G be an elementary abelian subgroup of W (D_{4}) of type (2, 2, 2, 2). It belongs to the conjugacy class N = 65 in Table 1. Theorem 8.1 can be restated in the following form:

Theorem 8.2. The restriction map

Res: Inv(W (D_{4}), W ) → Inv(G, W )

is injective.

Proof. G-torsors correspond to products of two biquadratic algebras. □

A construction of Witt invariants is through trace forms. Let S/S_{0} ∈ (Étex^{2/4})^{+} and let σ be the involution of S. We may associate two nonsingular quadratic trace forms to the extension S/S_{0}:

\begin{aligned}Q(x) & = & Q_{S}(x) & = & \frac{1}{8} Tr_{S/F} (x^{2}) \\ Q′(x) & = & Q′_{S}(x) & = & \frac{1}{8} Tr_{S/F} (xσ(x)), x ∈ S \end{aligned}

The decomposition

S = Sym(S, σ) ⊕ Skew(S, σ)

leads to orthogonal decompositions

Q = Q^{+} ⊥ Q^{−}, Q′ = Q^{+} ⊥ −Q^{−},

hence the forms Q^{+} and Q^{−} define two Witt invariants attached to S/S_{0}. The étale algebras associated to S/S_{0} by triality lead to corresponding invariants. We introduce following notations: S/S_{0} = S_{1}/S_{0,1} and S_{i}/S_{0,i}, i = 2, 3, for the associated étale algebras. We denote the corresponding Witt invariants by Q_{i}^{+} = Q_{S_{i}}^{+} and Q_{i}^{−} = Q_{S_{i}}^{−}, i = 1, 2 and 3. Another construction of Witt invariants is through orthogonal representations. Let O_{n} be the orthogonal group of the n-dimensional form ⟨1, …, 1⟩. Quadratic forms over F of dimension n are classified by the cohomology set H^{1}(F, O_{n}). Thus any group homomorphism W (D_{4}) → O_{n} gives rise to a Witt invariant. In particular we get a Witt invariant q associated with the orthogonal representation W (D_{4}) → O_{4} described in (5.3). Moreover the group W (D_{4}) has three normal subgroups H_{i} of type (2, 2, 2) (i.e., isomorphic to 𝔖_{2}^{3}), corresponding to the classes N = 33, 34, 35 of Table 1. Since the factor groups are isomorphic to 𝔖_{4}, the canonical representation 𝔖_{4} → O_{4} through permutation matrices leads to three Witt invariants q_{1}, q_{2}, q_{3}.

Proposition 8.3. 1) The Witt invariant q is invariant under triality and coincides with Q_{i}^{−}, i = 1, 2, 3.

2) We have q_{i} = Q_{i}^{+}, i = 1, 2, 3, and the three invariants q_{1}, q_{2}, q_{3} are permuted by triality.

Proof. The fact that q is invariant under triality follows from the fact that triality acts on W (D_{4}) by an inner automorphism of O_{4}. Moreover the trialitarian action on W (D_{4}) permutes the normal subgroups H_{i}, hence the invariants q_{i}. For the other claims we may assume by the splitting principle that S_{1} is a product of two biquadratic algebras

(8.4)  S_{1} = F (\sqrt{x}, \sqrt{y}) × F (\sqrt{z}, \sqrt{t}), S_{0,1} = F (\sqrt{xy}) × F (\sqrt{zt}).

An explicit computation, using for example the description of triality given in the proof of Theorem 4.1 (see [1] and [19]) shows that one can make the following identifications

S_{2} = F (\sqrt{x}, \sqrt{z}) × F (\sqrt{y}, \sqrt{t}), S_{0,2} = F (\sqrt{xz}) × F (\sqrt{yt})

and

S_{3} = F (\sqrt{x}, \sqrt{t}) × F (\sqrt{y}, \sqrt{z}), S_{0,3} = F (\sqrt{xt}) × F (\sqrt{yz}).

Observe that, with this identification, the 3-cycle (y, z, t) permutes cyclically the algebras S_{i}/S_{0,i}. We get

Q_{i}^{−} = ⟨x, y, z, t⟩

for i = 1, 2, 3, and

Q_{1}^{+} = ⟨1, 1, xy, zt⟩, Q_{2}^{+} = ⟨1, 1, xz, yt⟩, Q_{3}^{+} = ⟨1, 1, xt, yz⟩.

The equalities q = Q_{i}^{−} and q_{i} = Q_{i}^{+} follow from the fact that the corresponding cocycles are conjugate in O_{4}. □

Further basic invariants are the constant invariant ⟨1⟩ and the discriminant

⟨d⟩ = Disc(q) = Disc(q_{i}), i = 1, 2, 3,

which corresponds to the 1-dimensional representation det: W (D_{4}) → O_{1} = ±1. Since Q_{i}^{+}(1) = 1, the quadratic forms q_{i} = Q_{i}^{+} represent 1 and one can replace them by 3-dimensional invariants ℓ_{i} = (1)^{⊥} ⊂ q_{i}, i = 1, 2, 3. In the following result λ^{2}q denotes the second exterior power of the quadratic form q (see [9]). If q = ⟨α_{1}, …, α_{n}⟩ is diagonal, then λ^{2}q is the n(n − 1)/2-dimensional form λ^{2}q = ⟨α_{1}α_{2}, …, α_{n−1}α_{n}⟩.

Theorem 8.5. 1) The W (k)-module Inv(W (D_{4}), W ) = Inv((Étex^{2/4})^{+}, W ) is free over W (k) with basis

(8.6)  (⟨1⟩, ⟨d⟩, q, ⟨d⟩ · q, ℓ_{1}, ℓ_{2}, ℓ_{3}).

2) The elements ⟨1⟩, ⟨d⟩ , q, ⟨d⟩·q are fixed under triality and the elements ℓ_{1}, ℓ_{2} and ℓ_{3} are permuted.

3) The following nonlinear relations hold among elements of (8.6):

⟨d⟩  = Disc(q) = Disc(q_{i}),
λ^{2}q  = ℓ_{1} + ℓ_{2} + ℓ_{3} − ⟨1, 1, 1⟩
⟨1, d⟩ · q = q · (ℓ_{i} − ⟨1⟩), i = 1, 2, 3.

Proof. 1) The proof follows the pattern of the proof of [9, Theorem 29.2]. Let G be an elementary subgroup of W (D_{4}) of type (2, 2, 2, 2), i.e. isomorphic to 𝔖_{2}^{4}. An arbitrary element of H^{1}(F, G) is given by a 4-tuple (α_{1}, α_{2}, α_{3}, α_{4}) ∈ (F^{×}/F^{×2})^{4}. For I a subset of [1, 4] = {1, 2, 3, 4}, we write α_{I} for the product of the α_{i} for i ∈ I. By [9, Theorem 27.15] the set Inv(G, W) is a free W (k)-module with basis (α_{I})_{I⊂[1,4]}. It then follows from Theorem 8.2 that the family of elements given in

(8.6) is linearly independent over W (k). Let a be an element of Inv((Étex^{2/4})^{+}, W ) and let S_{α} be the algebra (8.4) for α_{1} = x, α_{2} = y, α_{3} = z, α_{4} = t. The map α ↦ a(S_{α}) is a Witt invariant of G, hence by [9, Theorem 27.15] can be uniquely written as a linear combination

(8.7)  ∑ c_{I} · ⟨α_{I}⟩ with c_{I} ∈ W (k).

The claim will follow if we show that the invariant α is in fact a linear combination of the elements given in (8.6). By [9, Prop. 13.2], the image of the restriction map Inv(W (D_{4}), W ) → Inv(G, W) is contained in the W (k)-submodule of Inv(G, W) fixed by the normalizer N = 𝔖_{2}^{3} ⋊ 𝔇_{4} of G in W (D_{4}) (conjugacy class N = 96 in Table 1). The group N acts on the set of isomorphisms classes of algebras S_{i} by acting in the obvious way on the symbols ±\sqrt{x}, ±\sqrt{y}, ±\sqrt{z}, ±\sqrt{t}. It follows that N acts trivially on this set of isomorphisms classes. This shows that only linear combinations of elements in the family

\begin{aligned}B = & \{⟨1⟩, ⟨x⟩ + ⟨y⟩ + ⟨z⟩ + ⟨t⟩, ⟨xy⟩ + ⟨zt⟩, ⟨xz⟩ + ⟨yt⟩, ⟨xt⟩ + ⟨yz⟩, \\ & ⟨xyz⟩ + ⟨xyt⟩ + ⟨xzt⟩ + ⟨yzt⟩, ⟨xyzt⟩\} \end{aligned}

can occur in the sum (8.7). The family B and the family given in 8.6 are equivalent bases. This implies the first claim of Theorem 8.5. Claim 2) follows from Proposition 8.3 and 3) is easy to check for a product of biquadratic extensions. □

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Department Mathematik, ETH Zentrum, CH-8092 Zürich, Switzerland

E-mail address: knus@math.ethz.ch

Institut de Mathématique Pure et Appliquée, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium

E-mail address: jean-pierre.tignol@uclouvain.be