GEOMETRIES, THE PRINCIPLE OF DUALITY, AND ALGEBRAIC GROUPS

MICHAEL CARR AND SKIP GARIBALDI

Abstract. J. Tits gave a general recipe for producing an abstract geometry from a semisimple algebraic group. This expository paper describes a uniform method for giving a concrete realization of Tits’s geometry and works through several examples. We also give a criterion for recognizing the automorphism of the geometry induced by an automorphism of the group. The E_{6} geometry is studied in depth.

Contents

1. Tits’s geometry Γ_{P} 2 2. A concrete geometry Γ_{V}, part I 4 3. A concrete geometry Γ_{V}, part II 5 4. Example: type A (projective geometry) 7 5. Strategy 8 6. Example: type D (orthogonal geometry) 9 7. Example: type E_{6} 12 8. Interlude: more on E_{6} 18 9. Example: type E_{7} 19 10. Loose ends 20 11. Outer automorphisms 23 12. Example: type D (orthogonal duality) 25 13. Example: type D_{4} (triality) 27 14. Example: type E_{6} (duality) 30 References 35

J. Tits’s theory of buildings associated with semisimple algebraic groups gives a unified method of extracting a geometry from a group. For example, the group SL_{n} gives rise to (n − 1)-dimensional projective space. Tits’s geometry however is very abstract. Speaking precisely, one obtains an incidence geometry, which consists of an abstract set of objects each with a given type, and a reflexive, symmetric binary relation on the set of objects called incidence. We find it more palatable to think of projective space in a concrete way, as the collection of subspaces of some explicit vector space. In Section 2 we give an explicit recipe for concretizing Tits’s incidence geometry.

Date: February 25, 2005.

Corresponding author: Garibaldi.

The midsection of this paper consists of explicit descriptions of the concrete realizations of the geometries for groups of type A, D, E_{6}, E_{7}, F_{4}, and G_{2}. (Readers should have little trouble filling in the missing types B and C. We do not know a good description for the geometry of type E_{8}, but we make a few comments in 10.3.) Such descriptions may be found in a variety of places in the literature, e.g., [Coh] or [FF]. The main innovation here is that our recipe produces a realization of the geometry by a largely deterministic process beginning from the root system of the group and a fundamental representation, whereas approaches in the literature have the appearance of being ad hoc. Our principal tool is the representation theory of semisimple groups; we only use the most elementary results, but we exploit those ruthlessly. Consequently, throughout this paper, our base field k is assumed to have characteristic zero.

Élie Cartan was already aware (see [Ca, p. 362]) that the “outer” automorphism g ↦ (g^{−1})^{t} (where t denotes transposition) of SL_{3} gives rise to a polarity in the projective plane and so to the principle of duality. In general, an outer automorphism of a semisimple group gives an automorphism ψ of the corresponding geometry, hence also a “principle of duality” (or “triality” or ...). In Sections 11–14 below, we give a criterion for recognizing such an automorphism ψ and apply our criterion to get an explicit description of ψ in all cases.

We hope that readers who are not familiar with, say, exceptional groups will find the presentation here unusually accessible because of the uniform treatment in the common language of representation theory. Experts will note that the treatment of the E_{6} geometry in §7 largely avoids any discussion of Jordan algebras. Similarly, we do not need to mention octonions at all in our discussion of triality in §13.

Our hypothetical reader is moderately familiar with the theory of linear algebraic groups as in [Bor], [Hu 81], [Sp], or [St] and the classification of irreducible modules via highest weight vectors from [Hu 80, Ch. VI], [FH, §14], [GW, §5.1], or [Va].

1. Tits’s geometry Γ_{P}

In this section, we describe Tits’s recipe for producing an incidence geometry from a certain kind of algebraic group. An incidence geometry is a set of objects, each of some type (e.g., point, line), together with a symmetric binary relation known as incidence. There is just one further axiom: objects x, y of the same type are incident if and only if they are equal.

Remark. Modern formulations of Tits’s recipe take a group and construct a building rather than an incidence geometry. From our perspective, a building is an incidence geometry with extra structure that we do not really need. So we deal only with the much-simpler-to-define incidence geometries as in [T 56]. For a presentation in terms of buildings, see [T 74] or [Br, Ch. V].

We start with a root system Φ in the sense of [Hu 81, §9.2] or [St, §1], a “reduced root system” in the language of [Bou 4–6]. There is a simply connected subgroup G of GL_{n}(k) (for some n) corresponding to Φ, and it is uniquely determined up to isomorphism. Taking Φ of type A_{n}, we obtain a group G isomorphic to SL_{n+1}. When k is algebraically closed, every semisimple algebraic group is obtained in this fashion, or is a quotient of a group obtained in this fashion. For example, the group SO_{2n}(ℂ) is a quotient of Spin_{2n}(ℂ), which is constructed from the root system D_{n}. For general k, we obtain the “split” semisimple groups.

1.1. Parabolic subgroups. A Borel subgroup is a maximal closed, connected, solvable subgroup of G. Fix one and call it B; it (combined with a split maximal torus T contained in it) determines a set of simple roots Δ in Φ. We abuse notation by writing Δ also for the associated Dynkin diagram.

A closed subgroup of G is called parabolic if it contains a Borel subgroup, and we call a parabolic subgroup standard if it contains the Borel B. There is an inclusion-reversing bijection

standard parabolic subgroups of G ↔ subsets of Δ
B ↔ Δ
G ↔ ∅

The maximal proper standard parabolics are in one-to-one correspondence with the elements of Δ. We write P_{δ} for the standard parabolic corresponding to δ ∈ Δ.

1.2. Example (Parabolics in SL_{4}). As an illustration, the Dynkin diagram of SL_{4} is

[figure: α_{1}; α_{2}; α_{3}]

Here we have labeled the vertices as in [Bou 4–6]. The upper triangular matrices are a Borel subgroup, which we take to be B. The maximal proper standard parabolics are

(1.3)  P_{α_{1}} = ( \begin{matrix}∗ & ∗ & ∗ & ∗ \\ 0 & ∗ & ∗ & ∗ \\ 0 & ∗ & ∗ & ∗ \\ 0 & ∗ & ∗ & ∗ \end{matrix} ) , P_{α_{2}} = ( \begin{matrix}∗ & ∗ & ∗ & ∗ \\ ∗ & ∗ & ∗ & ∗ \\ 0 & 0 & ∗ & ∗ \\ 0 & 0 & ∗ & ∗ \end{matrix} ) , P_{α_{3}} = ( \begin{matrix}∗ & ∗ & ∗ & ∗ \\ ∗ & ∗ & ∗ & ∗ \\ ∗ & ∗ & ∗ & ∗ \\ 0 & 0 & 0 & ∗ \end{matrix} ) .

1.4. Tits’s geometry Γ_{P}. Tits defines the objects of the incidence geometry to be the maximal proper parabolic subgroups of G. Since the Borel subgroups of G are all conjugate, every parabolic is conjugate to a unique standard parabolic [Bor, 21.12]. Therefore, every maximal proper parabolic subgroup corresponds to a unique element δ ∈ Δ; this is the type of the parabolic. Two maximal proper parabolics are said to be incident if their intersection contains a Borel subgroup. We write Γ_{P} for this incidence geometry, where the subscript P is meant to remind the reader that the objects are parabolic subgroups of G.

In the case of SL_{4}, one typically calls the parabolics of type α_{1} “points”, α_{2} “lines”, and α_{3} “planes”. This identifies the geometry Γ_{P} with 3-dimensional projective space.

2. A concrete geometry Γ_{V}, part I

We now take Tits’s incidence geometry Γ_{P} —whose objects are certain parabolic subgroups of G—and produce an isomorphic incidence geometry Γ_{V} whose objects are subspaces of a fixed vector space V. For example, in the case where G is SL_{n}, Γ_{V} consists of the nonzero, proper subspaces of k^{n}.

Fix a representation of G on a finite-dimensional vector space V, i.e., a group homomorphism G → GL(V) that may be expressed in terms of polynomials in the entries of G. We assume that the representation is nontrivial (i.e., the image of G is not just the identity transformation) and irreducible (i.e., there is no nonzero, proper G-invariant subspace). For each vertex δ of the Dynkin diagram, we choose a nonzero, proper subspace V_{δ} of V that is invariant under the parabolic P_{δ}.^{1} Every maximal proper parabolic subgroup P′ is conjugate to a unique standard parabolic subgroup P_{δ}, and we define a subspace V_{P′} via

V_{P′} := gV_{δ} for g ∈ G(k) such that P′ = gP_{δ}g^{−1}.

Of course, g is not uniquely determined, but if h ∈ G(k) also satisfies hP_{δ}h^{−1} = P′, then h^{−1}g normalizes P_{δ}. Since P_{δ} is its own normalizer [Bor, 11.16], g equals hp for some p ∈ P_{δ}(k) and hV_{δ} = gV_{δ}. We remark that the stabilizer of V_{P′} in G is precisely P′. Indeed, the stabilizer of V_{P′} is a closed subgroup of G containing P′, hence must be P′ or G. Since the representation V is irreducible, the stabilizer must be P′.

We define an incidence geometry Γ_{V} whose objects are the subspaces V_{P} of V, as P ranges over the maximal proper parabolic subgroups of G. The map P ↦ V_{P} is surjective by definition, but it is also injective because P is precisely the stabilizer of V_{P}. Therefore, the sets of objects in Γ_{P} and Γ_{V} are isomorphic. Define the notions of type and incidence in Γ_{V} by transporting them from Γ_{P}. Speaking precisely, we say that V_{P} in Γ_{V} has type δ ∈ Δ if the parabolic P is of type δ. We define two objects in Γ_{V} to be incident if and only if the corresponding parabolic subgroups are incident.

In this very simple way, we have obtained a realization of Tits’s abstract geometry Γ_{P} as a collection of subspaces of the concrete vector space V.

1For the moment, we assume that such a subspace exists. The doubting reader may wish to glance ahead at Prop. 3.3.

This recipe begs two obvious questions:

Are we guaranteed that a nonzero, proper P_{δ}-invariant subspace of V exists?

(2.1)

Is there a way to tell if two subspaces of V in Γ_{V} are incident without discussing the corresponding parabolic subgroups?

(2.2)

We will address these two questions in the next section and the examples that follow it. But first, here is an example to illustrate the construction.

2.3. Example (Γ_{V} for SL_{4}). Referring to the description of the parabolic subgroups of SL_{4} from Example 1.2, we see that P_{α_{i}} stabilizes the i-dimensional subspace of k^{4} consisting of vectors whose only nonzero entries are in the first i coordinates. We can take this to be V_{α_{i}}. The objects of Γ_{V} are all proper, nonzero subspaces of V.

We claim that two elements of Γ_{V} are incident if and only if one is contained in the other. Indeed, let W, W′ be proper, nonzero subspaces, stabilized by maximal proper parabolics P, P′ in SL_{4}. If W and W′ are incident, then P ∩ P′ contains a Borel subgroup. After conjugation we may assume that P and P′ are standard, hence appear in (1.3). Then clearly W is contained in W′ or vice versa. Conversely, if W is contained in W′, there is some g ∈ SL_{4}(k) such that gW, gW′ are equal to V_{α_{i}}, V_{α_{i′}} for some i ≤ i′. Then gP g^{−1}, gP′g^{−1} equal P_{α_{i}}, P_{α_{i′}}, hence are incident.

3. A concrete geometry Γ_{V}, part II

We will now make the geometry Γ_{V} from the previous section more concrete by focusing on the case where V is a fundamental irreducible representation of G. We completely answer (2.1) in the affirmative in Prop. 3.3 and we partially answer (2.2) in Prop. 3.4.

We view G as being constructed from the root system Φ by the Chevalley construction as in [St, §6]. That is, it is generated by the images of homomorphisms x_{α} : 𝔾_{a} → G as α ranges over the roots in Φ. Write U_{α} for the image of x_{α}. For each α ∈ Φ, the map t ↦ x_{α}(t)x_{α}(1)^{−1} defines a homomorphism 𝔾_{m} → G, which we denote by h_{α}. The images of the h_{α}’s generate a maximal torus T in G. We fix a set of simple roots Δ in Φ and choose our standard Borel subgroup B to be the one generated by T and the U_{α} for α ∈ Δ.

Fix a root β ∈ Δ and let ω be the corresponding fundamental weight. In this section, V denotes the irreducible representation with highest weight ω with respect to our choice of torus T and Borel B. Fix a highest weight vector v in V. 3.1. Write L_{β} for the image of h_{β} in G and V_{β} for the subspace L_{β} v of V. Note that V_{β} is simply the k-span of the highest weight vector v.

For a simple root δ ∈ Δ \ {β}, we define the δ-component of Δ to be the connected component of β in Δ \ {δ}. We write L_{δ} for the subgroup of G generated by the root subgroups U_{α}, U_{−α} for α in the δ-component. The description of G in terms of generators and relations shows that L_{δ} is a simple group whose Dynkin diagram is the δ-component. (It is the semisimple part of the Levi subgroup of the parabolic corresponding to the complement of the δ-component.) We set V_{δ} to be L_{δ} v. 3.2. For each δ ∈ Δ, the subspace V_{δ} is T -invariant because T -normalizes L_{δ}. Therefore, V_{δ} is a direct sum of weight spaces in V. It consists of the weight spaces with weights of the form ω − α, where α is the sum of simple roots in the δ-component (possibly with repetition and with the understanding that the “β-component” is the empty set).

Suppose for the moment that u ∈ V_{δ} is a weight vector, with weight ω−α. For γ ∈ Δ and t ∈ k, we have

x_{γ}(t)u = u + (vectors of weight ω − α + γ, ω − α + 2γ, etc.).

For γ not in the δ-component, the weights ω − α + γ, ω − α + 2γ, etc., are not weights of V, hence U_{γ} fixes u. That is, V_{δ} is fixed elementwise by U_{γ} for every γ ∈ Δ not in the δ-component.

We also have the following proposition:

3.3. Proposition. For every δ ∈ Δ, the subspace V_{δ} is a nonzero, proper subspace of V stabilized by P_{δ}.

Proof. V_{δ} is clearly nonzero because it contains v. We now show that it is a proper subspace. Let α denote the sum of the simple roots in δ; it is a root since δ is connected [Bou 4–6, VI.1.6, Cor. 3]. In standard root system notation, ⟨ω, α⟩ equals 2/(α, α). This is positive (because the inner product is positive definite) and an integer (because ω is a weight). But the weights of V are a saturated set of weights. In particular, V contains a nonzero vector of weight ω − α. But α involves δ, so ω − α is not a weight of V_{δ}. This shows that V_{δ} is proper.

Finally, we show that V_{δ} is stabilized by P_{δ}. It suffices to check that the subgroups T, U_{γ} for γ ∈ Δ, and U_{−γ} for γ ∈ Δ \ {δ} stabilize V_{δ}, since these subgroups generate P_{δ} by [Bor, 14.18]. For γ in the δ-component, U_{γ} and U_{−γ} are in L_{δ}, hence they stabilize V_{δ}. For γ in the other connected components of Δ \ {δ}, U_{γ} and U_{−γ} commute with L_{δ} and fix v because ⟨ω, γ⟩ is 0, hence they stabilize V_{δ}. Finally, T and U_{δ} fix V_{δ} elementwise by 3.2. Together, we have seen that V_{δ} is invariant under P_{δ}. □

The preceding proposition addressed (2.1). Now we address the second question. We call a subspace X ∈ Γ_{V} of type δ a δ-space.

3.4. Proposition. Let X be a δ-space in Γ_{V} and suppose that the δ-component of Δ is of type A.

(1) Every 1-dimensional subspace of X is a β-space.

(2) If the δ-component contains the δ′-component, we have: X is incident to a δ′-space X′ if and only if X contains X′.

Proof. To prove the proposition, we may conjugate X and so assume that X is actually V_{δ}. Since the δ-component is of type A, the group L_{δ} is isomorphic to SL(V_{δ}).

Every nonzero vector in V_{δ} is in the same SL(V_{δ})-orbit as the highest weight vector. Hence every 1-dimensional subspace of V_{δ} is L_{δ}-conjugate to V_{β}, i.e., is a β-subspace. This proves (1).

Now we prove (2). First suppose that X and X′ are incident. Then after conjugation, we may assume that X is V_{δ} and X′ is V_{δ′}. Since L′_{δ} is contained in L_{δ}, clearly X′ is contained in X.

Conversely, suppose that X′ is contained in X. Since L_{δ} is SL(V_{δ}), all subspaces of V_{δ} with the same dimension are L_{δ}-equivalent. Hence X′ is L_{δ}- equivalent to V_{δ′}. The parabolics P_{δ}, P_{δ′} corresponding to V_{δ}, V_{δ′} contain the standard Borel B, hence are incident. □

Before we leave this section, we observe that we know a lot about the spaces V_{δ}. The case δ = β is trivial, so for the rest of this section we fix a δ ∈ Δ that is not β.

3.5. Proposition. For δ ∈ Δ\{β}, the space V_{δ} is a fundamental irreducible representation of L_{δ}.

Proof. Suppose that x ∈ V_{δ} is fixed by U_{α} for every α in the δ-component. By 3.2, x is fixed by U_{α} for all α ∈ Δ, and since V is an irreducible representation of G, x is in the k-span of v.

The previous paragraph shows that v (and scalar multiples of v) is the only highest weight vector for V_{δ} relative to the maximal torus T_{δ} = T ∩ L_{δ} of L_{δ}. Since V_{δ} is a completely reducible representation of L_{δ}, it must be irreducible.

The highest weight of V_{δ} is the restriction of ω to T_{δ}; we denote it by \overline{ω}. Since ω is a fundamental weight of G, the restriction \overline{ω} is a fundamental weight of L_{δ}. □

The dimension of V_{δ} can be looked up in, e.g., [Bou 7–9, chap. 8, Table 2]. 3.6. We also have finer information about the weights of V_{δ}. Computing relative to L_{δ}, the weights of V_{δ} lie between the highest weight \overline{ω} and the lowest weight w_{0}\overline{ω}, where w_{0} is the longest element in the Weyl group of L_{δ}. From the tables in [Bou 4–6], one can quickly find the nonnegative integers k_{α} such that

w_{0}\overline{ω} = \overline{ω} − ∑ k_{α}α

where α runs over the roots in the δ-component. Considering V_{δ} as a subspace of the representation V of G, we see that the weights of V_{δ} are precisely those weights μ of V such that

ω − ∑ k_{α}α ≤ μ ≤ ω.

4. Example: type A (projective geometry)

In this section, we describe the objects in the geometry Γ_{V} constructed from G := SL_{n} as in §3 when the representation is the standard one on V := k^{n}, corresponding to the simple root β := α_{1}. We “discover” that Γ_{V} is projective (n−1)-space. We could do this explicitly in terms of matrices as in Example 2.3, but such an argument would be hard to generalize to other groups. Instead, we give an algebraic-group- and representation-theoretic argument.

We defined V_{β}, a.k.a. V_{α_{1}}, to be the 1-dimensional subspace of V spanned by the highest weight vector. For α_{i} ∈ Δ with i ≠ 1, V_{α_{i}} is the standard representation of L_{α_{i}}. Therefore, the dimension of V_{α_{i}} is precisely i. We summarize this in the Dynkin diagram, where each vertex is labeled with α_{i} and dim V_{α_{i}}:

[figure: β = α_{1}; α_{2}; α_{3}; α_{n−2}; α_{n−1}; 1; 2; 3; n − 2; n − 1]

Since SL_{n} acts transitively on the i-dimensional subspaces of V for all i, we have: the α_{i}-spaces are the i-dimensional subspaces of V. By Prop. 3.4.2, two subspaces are incident if and only if one contains the other. This is the classical description of (n − 1)-dimensional projective space as consisting of lines through the origin in k^{n}.

5. Strategy

In the next few sections, we will fix a split simply connected group G and give an explicit description of the geometry Γ_{V}. One imagines that the geometry Γ_{V} we have just constructed will be easiest to visualize if the ambient vector space V is small. With that in mind, we will focus on the case where V is the smallest irreducible representation of G. For G of type A, D_{4}, or E_{6}, there are multiple equivalent choices, and we arbitrarily pick one.

(5.1) type of G A_{n}  B_{n} C_{n} D_{n} E_{6} E_{7} F_{4} G_{2} β α_{1} α_{1} α_{1} α_{1} α_{1} α_{7} α_{4} α_{1}

dim V n + 1 2n + 1 2n 2n 27 56 26 7

We number the elements of Δ as in the tables in [Bou 4–6]. We call a representation V as in the table above a standard representation of G. We have omitted type E_{8}, see 10.3 for comments. (We remind the reader that despite our focus on the standard representation of G, the recipe in §3 gives a concrete realization of Γ_{P} for every fundamental representation, and one can compute the dimensions of the δ-spaces using Prop. 3.5.)

Roughly speaking, each example consists of three parts: dimensions and properties, transitivity, and incidence.

In “dimensions and properties”, for each δ ∈ Δ we compute the dimensions of the δ-spaces and some algebraic properties P satisfied by the δspaces. Here we restrict ourselves to the tools of elementary representation theory, which we exploit mercilessly. This has two advantages. First, no special background is required to understand the exceptional groups versus the more-familiar classical groups. Second, we hope the reader will view our descriptions of the δ-spaces as reasonably canonical and not ad hoc.

In “transitivity”, we prove that the group G acts transitively on the subspaces of V satisfying the properties P. Since the collection of δ-spaces is a G-orbit, this proves that the δ-spaces are precisely the subspaces of V satisfying P. In many cases, we will refer to the literature for a proof. The proofs in the literature use various interpretations of the standard representation as the vector space underlying some algebraic structure. For example, in the type A example in §4, we used the fact that SL_{n} acts transitively on the subspaces of k^{n} of a given dimension.

In “incidence”, we give a concrete description of how to tell if a δ- and a δ′- subspace are incident. Our final description is purely in terms of subspaces of V, with no mention of the corresponding parabolic subgroups. In most cases, a δ- and δ′-subspace will be incident if and only if one contains the other. When this occurs, we will say that incidence is the same as inclusion. In this “incidence” portion, we return to the techniques of representation theory and eschew algebraic interpretations of the representation.

6. Example: type D (orthogonal geometry)

Consider the split simply connected group G of type D_{n} with n ≥ 4. This group is sometimes denoted Spin_{2n}. The geometry in this case is more complicated than for type A, apparently because the Dynkin diagram has a fork in it. This case will illustrate the basic principles involved in handling forking diagrams, and we will use them when treating the E-groups later. This geometry will be further investigated in Sections 12 and 13 below.

Dimensions and properties. As in the type A case, this is easy.

[figure: β = α_{1}; α_{2}; α_{3}; α_{n−3}; 1; 2; 3; n − 3; α_{n−2}, n − 2; α_{n−1}, n; α_{n}, n]

The standard representation of G has dimension 2n. Since −w_{0}ω_{1} is ω_{1}, where w_{0} is the longest element of the Weyl group, there is a nondegenerate G-invariant bilinear form b on V, unique up to multiplication by an element of k^{×} [Bou 7–9, 8.7.5, Prop. 12]. Moreover, b is symmetric [Bou 7–9, chap. 8, Table 1]. For v the highest weight vector in V and t an element of the maximal torus T, we have:

b(v, v) = b(t · v, t · v) = b(ω(t)v, ω(t)v) = ω(t)^{2}b(v, v).

Since ω is not the trivial character, b(v, v) is 0. Traditionally, a subspace X is called isotropic if b(X, X) is zero. We have just observed that the α_{1}-spaces are isotropic. By Prop. 3.4.1, the α_{i}-spaces are isotropic for every i.

Enlarging the geometry. We now add a new type of object to the geometries Γ_{P} and Γ_{V} to help us study that α_{n}- and α_{n−1}-spaces.

Let Γ̃_{P} be the geometry whose objects are the objects in Γ_{P} (the maximal proper parabolics in G) together with the parabolics of type {α_{n}, α_{n−1}}. Incidence in Γ̃_{P} is as for Γ_{P}; namely, two parabolics are incident if and only if their intersection contains a Borel subgroup.

6.1. Lemma. (Cf. [T 56, pp. 61–64]) Each parabolic P of type {α_{n}, α_{n−1}} is incident with exactly one parabolic Q_{n} of type α_{n} and one parabolic Q_{n−1} of type α_{n−1}. The parabolic P is the intersection Q_{n} ∩ Q_{n−1}.

Proof. First suppose that P is the standard parabolic of type {α_{n}, α_{n−1}}. Suppose that P is incident with Q, a parabolic of type α_{i} for i = n or n − 1. Then P ∩ Q contains a Borel subgroup; let g ∈ G(k) be such that g(P ∩ Q)g^{−1} contains the standard Borel B. In fact, g lies in P by [BT 65, p. 86, 4.4a]. Hence gP g^{−1} is simply P. Since gQg^{−1} contains B, gQg^{−1} is the standard parabolic P_{α_{i}}. However, P is contained in P_{α_{i}} and we have

Q = g^{−1}P_{α_{i}}g = P_{α_{i}}.

The second claim—for P the standard parabolic—is well-known.

For P a parabolic of type {α_{n}, α_{n−1}}, P is conjugate to the standard parabolic of that type. The lemma now follows from the previous paragraph.

□

Similar to what was done in §3, let L_{{α_{n},α_{n−1}}} be the simple subgroup of G generated by the root subgroups U_{α_{i}}, U_{−α_{i}} for 1 ≤ i ≤ n − 2. Let X be the subspace L_{{α_{n},α_{n−1}}}v of V. As in Prop. 3.3, we find that X is a nonzero, proper subspace of V stabilized by the standard parabolic P of type {α_{n}, α_{n−1}}. As in §2, we find a bijection between parabolics of type {α_{n}, α_{n−1}} and G-conjugates of X. Define the new geometry Γ̃_{V} to have the same objects as Γ_{V} as well as the G-conjugates of X. As in §2, we define objects in Γ̃_{V} to be incident precisely when their corresponding parabolics in Γ̃_{P} are incident. 6.2. For δ = α_{n} or α_{n−1}, the group L_{δ} is of type A and L_{{α_{n},α_{n−1}}}v is a subspace of V_{δ}. The proof of Prop. 3.4.2 shows that an {α_{n}, α_{n−1}}-space and a δ-space are incident if and only if one is contained in the other. The lemma gives: each (n − 1)-dimensional isotropic subspace is the intersection of two uniquely determined and incident n-dimensional subspaces, one of type α_{n} and one of type α_{n−1} (compare [Ch, III.1.11]).

Transitivity. We claim that G acts transitively on the m-dimensional isotropic subspaces of V for m < n. Let X, X′ be isotropic of dimension m. They each lie in a direct sum of m hyperbolic planes in V, and one can easily construct an isometry f of b that sends X to X′. Since V is isomorphic to a direct sum of n hyperbolic planes, there is at least one plane where we may choose f as we please. If f has determinant −1, we modify f by a hyperplane reflection in this “extra” hyperbolic plane so that f has determinant 1. Since k is algebraically closed, f is in the image of the map G(k) → SO(b)(k), which proves the claim. Moreover, we have proved that the α_{i}-spaces are the i-dimensional isotropic subspaces for 1 ≤ i ≤ n − 2.

Next let X, X′ be δ-spaces for δ = α_{n} or α_{n−1}. Fix (n − 1)-dimensional subspaces (i.e., {α_{n}, α_{n−1}}-spaces) U in X and U′ in X′. By the previous paragraph, there is some g ∈ G(k) such that gU = U′. Since X, X′ have the same type, we must have gX = X′ by 6.2. An argument similar to the one in the last paragraph gives that SO(b)(k) has at most two orbits on the n-dimensional subspaces of V. Since the δ-spaces are an orbit for δ = α_{n} and α_{n−1}, we have:

{α_{n}- and α_{n−1}-spaces} = {isotropic subspaces of dimension n}.

Incidence. Consider an α_{i}-space X′ and an α_{j}-space X with i ≤ j. If (i, j) is not (n − 1, n), then Prop. 3.4.2 applies and incidence is the same as inclusion.

Now consider the case (i, j) = (n−1, n). We claim that X, X′ are incident if and only if the dimension of X ∩ X′ is n − 1, i.e., X ∩ X′ is an {α_{n}, α_{n−1}}- space. If the two spaces are incident, then X, X′ are simultaneously Gconjugate to the standard subspaces V_{α_{n}}, V_{α_{n−1}}, and the intersection V_{α_{n}} ∩ V_{α_{n−1}} certainly has dimension n−1. Conversely, if the intersection is (n−1)- dimensional, then the spaces are incident as observed just after the proof of the lemma.

6.3. An alternative view. We now outline the geometry that one obtains from G by considering the fundamental representation with highest weight α_{n} (a “half-spin” representation) instead of the standard representation. We continue with the same definitions of V, V_{α_{i}}, etc., as in the rest of this section. We may identify the half-spin representation S with ∧^{even}V_{α_{n}} as described in [Ch, chap. 3].

For i ≠ n−1, the parabolic P_{α_{i}} stabilizes V_{α_{i}}, hence also the ideal of ∧V_{α_{n}} generated by ∧^{i}V_{α_{i}}, hence also its intersection with S. Following the naive algorithm in §2, we define an α_{i}-space to be an intersection of (∧^{i}X)∧(∧V_{α_{n}}) with S, where X is an α_{i}-space relative to the standard representation. (That is, X is isotropic of dimension i, plus an extra condition when i = n.) Thinking in terms of exterior powers of vector spaces, it is clear that the α_{i}-spaces in the half-spin representation have dimension 2^{n−i−1} for i ≤ n−2; they correspond to the right ideals in the even Clifford algebra constructed in [Ga 99, §1]. The α_{n}-spaces are 1-dimensional and are the “pure spinors” corresponding to even maximal isotropic subspaces, in the language of [Ch, §3.1]. We do not know how to describe the α_{n−1}-spaces in this geometry.

7. Example: type E_{6}

The geometry for the split simply connected group G of type E_{6} exhibits two complexities. We are prepared for the first—the fork in the diagram— thanks to our work in the previous section on groups of type D. The second complication is new: the root δ = α_{6} does not satisfy the hypotheses of our workhorse Prop. 3.4.

This is the last example we will do in detail. In the final section of this paper, we will give an explicit description of the duality in this geometry.

Dimensions and properties. The Dynkin diagram for E_{6}, labeled with the dimensions of the corresponding spaces is:

[figure: β = α_{1}; α_{3}; α_{4}; α_{5}; α_{6}; α_{2}; 1; 2; 3; 5; 10; 6]

In order to find the algebraic properties satisfied by the objects in the geometry, we need to know the weights of the representation V. Figure 7.1 displays the 27 weights of V as a Hasse diagram relative to the usual partial ordering of the weights. The row vectors list the coordinates of the weights with respect to the basis consisting of fundamental weights. An edge joining two weights λ > μ is labeled with i if λ − α_{i} = μ. The lowest weight of V_{α_{i}} (cf. 3.6) is labeled λ_{i}.

The automorphism of order 2 of the Dynkin diagram gives an automorphism of the root system, hence an automorphism ϕ of G of order 2. The fixed subgroup is well-known to be simple and split of type F_{4}, and we denote it simply by F_{4}. Examining the restrictions of the weights of V to F_{4}, we find that V decomposes (as a representation of F_{4}) as a direct sum of a 1-dimensional trivial representation C and the standard representation of F_{4}, which we denote by V_{0}.

7.2. Proposition. There is a bilinear form b on V such that

(7.3)  b(ϕ(g)x, gy) = b(x, y) for all g ∈ G and x, y ∈ V.

It is unique up to multiplication by a scalar. Moreover, it is symmetric and nondegenerate, and b|_{C} is not zero.

Proof. First we construct a bilinear form b on V satisfying (7.3). Write ρ: G → GL(V) for the representation of G on V, and write ρ^{∗}: G → GL(V^{∗}) for the dual representation defined by

(ρ^{∗}(g)f)(x) := f(ρ(g)^{−1}x) for g ∈ G, f ∈ V^{∗}, and x ∈ V.

The representations ρϕ and ρ^{∗} are both irreducible with highest weight ω_{6}, hence they are isomorphic. Fix an isomorphism h : V → V^{∗} such that hρϕ(g)h^{−1} = ρ^{∗}(g) for all g ∈ G. Define b by setting

b(x, y) := h(x)(y).

[figure: (1, 0, 0, 0, 0, 0) = ω_{1} = λ_{1}; 1; (−1, 0, 1, 0, 0, 0) = λ_{3}; 3; (0, 0, −1, 1, 0, 0) = λ_{4}; 4; (0, 1, 0, −1, 1, 0); 2; 5; λ_{5} = (0, −1, 0, 0, 1, 0); 5; (0, 1, 0, 0, −1, 1); 2; 6; (0, −1, 0, 1, −1, 1); 4; 6; (0, 1, 0, 0, 0, −1) = λ_{2}; 2; (0, 0, 1, −1, 0, 1); 3; 6; (0, −1, 0, 1, 0, −1); 4; (1, 0, −1, 0, 0, 1); 1; 6; (0, 0, 1, −1, 1, −1); 5; 3; λ_{6} = (−1, 0, 0, 0, 0, 1); 6; (1, 0, −1, 0, 1, −1); 5; 1; (0, 0, 1, 0, −1, 0); 3; (−1, 0, 0, 0, 1, −1); 5; (1, 0, −1, 1, −1, 0); 4; 1; (−1, 0, 0, 1, −1, 0); 4; (1, 1, 0, −1, 0, 0); 2; 1; (−1, 1, 1, −1, 0, 0); 3; 2; (1, −1, 0, 0, 0, 0); 1; (0, 1, −1, 0, 0, 0); 2; (−1, 1, 1, 0, 0, 0); 3; (0, −1, −1, 1, 0, 0); 4; (0, 0, 0, −1, 1, 0); 5; (0, 0, 0, 0, −1, 1); 6; (0, 0, 0, 0, 0, −1) = −ω_{6}]

Figure 7.1. Hasse diagram of the weights of V, where (c_{1}, c_{2}, …, c_{6}) denotes the weight ∑_{i=1}^{6} c_{i}ω_{i}. The map −ϕ reflects the diagram across its horizontal axis of symmetry.

This b is clearly bilinear and

b(ϕ(g)x, gy) = h(ρϕ(g)x)(gy) = [ρ^{∗}(g)h(x)] (gy) = b(x, y).

We now argue that any bilinear form b satisfying (7.3) is symmetric. Set

b_{ε}(x, y) := b(x, y) + εb(y, x).

Then b_{1} and b_{−1} are bilinear, b_{1} is symmetric, b_{−1} is skew-symmetric, and 2b = b_{1}+b_{−1}. We prove that b_{−1} is identically zero. In any case, b_{−1} satisfies (7.3) (using that ϕ is its own inverse), hence b_{−1} is F_{4}-invariant. But V_{0} does not support a nonzero F_{4}-invariant skew-symmetric form, hence b_{−1} restricts to zero on V_{0}. Fix x ∈ V_{0} a nonzero vector with a nonzero weight λ with respect to F_{4}, and let c be a nonzero vector in C. Since

b_{ε}(c, x) = b_{ε}(tc, tx) = λ(t)b_{ε}(c, x)

for t in the F_{4}-torus, b_{ε}(c, x) is zero. Since V_{0} is an irreducible representation of F_{4}, b_{ε}(c, V_{0}) is zero. But b_{−1} is skew-symmetric, so b_{−1}(c, c) is also zero, and we have proved the claim.

The previous paragraph also gives more. Continue the assumption that b satisfies (7.3) and suppose that b is not identically zero. Then C and V_{0} are orthogonal subspaces. For x ∈ V and r in the radical of b, we have

b(gr, x) = b(r, ϕ(g)^{−1}x) = 0,

hence the radical is G-invariant. Since V is irreducible and b is not identically zero, the radical is zero, i.e., b is nondegenerate. Since C is 1-dimensional and orthogonal to V_{0}, we find that b restricts to be nonzero on C.

We now prove uniqueness. Let b, b′ be bilinear forms on V satisfying (7.3). The representation V_{0} of F_{4} supports a unique symmetric bilinear form up to a scalar multiple, so by modifying b′ by a factor in k^{×}, we may assume that b and b′ have the same restriction to V_{0}. Then b−b′ is a bilinear form on V satisfying (7.3) that restricts to be zero on V_{0}. By the previous paragraph, b − b′ is identically zero, and we have proved uniqueness. □

We can use representation theory to decompose (V^{∗})^{⊗3} (or, if the reader prefers, S^{3}(V^{∗})) as a direct sum of irreducible representations; we find just one trivial, 1-dimensional representation. That is, there is a G-invariant cubic form N on V, uniquely determined up to a factor in k^{×}. We abuse notation and write N also for the trilinear “polarization” of N on V such that N(x, x, x) = 6N(x) for all x ∈ V. Let # denote the bilinear product defined implicitly by the formula

(7.4)  b(x#y, z) = N(x, y, z) for x, y, z ∈ V.

Using (7.3), we find

(7.5)  ϕ(g)(x#y) = (gx)#(gy) for g ∈ G and x, y ∈ V.

We write simply x^{#} for the “half square” (x#x)/2.

The same argument as at the beginning of this section shows that the α_{1}- spaces are 1-dimensional subspaces consisting of elements x ∈ V such that x^{#} = 0. We say a nonzero vector x ∈ V is singular if x^{#} = 0. (These 1dimensional subspaces are precisely the singular points for the hypersurface in ℙ(V) defined by N = 0.) We call a subspace of V singular if its nonzero elements are singular. By Prop. 3.4.1, the α_{i}-spaces are singular for i ≠ 6.

We will now investigate the restriction of the representation of G on V to the subgroup L_{α_{6}} of type D_{5}. This will give us finer information about the product # and lead us to a description of the α_{6}-spaces. To see how a weight of G restricts to L_{α_{6}}, one drops the last coordinate and moves the second coordinate to the end of the vector (to allow for the fact that weights of D_{5} and E_{6} are numbered somewhat incompatibly in [Bou 4–6]).

Let W′ be the subgroup of the Weyl group W of G generated by the reflections with respect to the roots α_{i} for i ≠ 6. It is the Weyl group of L_{α_{6}}, and it is the stabilizer of −ω_{6} in W [Hu 90, Th. 1.12c].

7.6. Lemma. The orbits of W′ in the weights of V are the weights ≥ λ_{6}, the weights between λ_{2} and (0, 0, 0, 0, −1, 1), and the weight −ω_{6}.

Proof. Since the highest weight ω_{1} of V is minuscule, we have ⟨μ, α⟩ = 1, 0 or −1 for every weight μ of V and every root α. If μ and μ − δ are both weights for some δ ∈ Δ, then ⟨μ, δ⟩ = 1, ⟨μ − δ, δ⟩ = −1, and the reflection s_{δ} with respect to the root δ interchanges μ and μ − δ. Consulting Figure 7.1, we see that W′ acts transitively on each of the three sets of weights named in the statement of the lemma.

Conversely, ω_{1} and λ_{2} restrict to the weights (1, 0, 0, 0, 0) and (0, 0, 0, 0, 1) on L_{α_{6}}, which are not congruent modulo the D_{5} root lattice. Therefore, they lie in different W′-orbits. □

By restricting the weights of V to L_{α_{6}}, we can decompose V as a direct sum of irreducible representations. The proof of Lemma 7.6 shows that the components of V are

• the standard representation V_{α_{6}} of V (with highest weight (1, 0, 0, 0, 0)),

• a half-spin representation (with highest weight (0, 0, 0, 0, 1)), and

• a 1-dimensional trivial representation (from the lowest weight vector −ω_{6}).

7.7. Corollary. The Weyl group of type E_{6} acts transitively on triples μ_{1}, μ_{2}, μ_{3} of weights of V such that μ_{1} + μ_{2} + μ_{3} = 0.

Proof. The Weyl group acts transitively on the weights of V, so we may assume that μ_{1} is −ω_{6}. Since μ_{3} = ω_{6} − μ_{2} is a weight, μ_{2} cannot have last coordinate equal to −1, otherwise μ_{3} would have last coordinate −2, which is impossible. In particular, μ_{2} cannot be −ω_{6} or λ_{2}. Since the set of triples −ω_{6}, μ_{2}, μ_{3} with sum 0 is stable under the action of W′, μ_{2} must lie in the W′-orbit with lowest weight λ_{6}. □

Our preliminary results about the action of the Weyl group can now give us concrete information about the product #.

7.8. Lemma. Let x_{1}, x_{2} be nonzero vectors in V of weight μ_{1}, μ_{2} respectively. The product x_{1}#x_{2} is nonzero if and only if ϕ(μ_{1} + μ_{2}) is a weight of V.

The equations

(7.9)  ϕ(λ_{2} + λ_{5}) = λ_{3} and ϕ(ω_{1} + λ_{6}) = ω_{1}

furnish specific examples where the product # is not zero.

Proof of Lemma 7.8. If ϕ(μ_{1} + μ_{2}) is not a weight of V, then the product is zero by (7.5). So suppose that ϕ(μ_{1} + μ_{2}) is a weight of V. Since −ϕ is in the Weyl group, μ_{3} := −μ_{1} − μ_{2} is a weight of V; let y be a nonzero vector with that weight.

We claim that N(x_{1}, x_{2}, y) is not zero, and hence that x_{1}#x_{2} is not zero. Indeed, fix a basis {b_{λ}} for V consisting of weight vectors and write N in terms of the dual basis {x_{λ}}. Since N is G-invariant, a monomial x_{ν_{1}}x_{ν_{2}}x_{ν_{3}} has zero coefficient if ν_{1} + ν_{2} + ν_{3} is not zero. On the other hand, since N is not identically zero, there exist weights ν_{1}, ν_{2}, ν_{3} such that the coefficient of x_{ν_{1}}x_{ν_{2}}x_{ν_{3}} is not zero. Since their sum ν_{1} + ν_{2} + ν_{3} is zero, there is an element w in the Weyl group such that wν_{i} = μ_{i} for each i by Cor. 7.7. A representative of w can be found in G, hence the coefficient of x_{μ_{1}}x_{μ_{2}}x_{μ_{3}} in N is not zero. In particular, N(x_{1}, x_{2}, y) is not zero, as claimed. □

Finally, we can give an explicit description of the α_{6}-space V_{α_{6}}.

7.10. Proposition. V_{α_{6}} = v#V, where v is the highest weight vector.

Proof. (⊇): Suppose that x ∈ V has weight μ, which is necessarily at least the minimum weight −ω_{6}. Since ϕ respects the partial ordering on the weights, v#x has weight at least ϕ(ω_{1} − ω_{6}) = λ_{6}. But this is the minimal weight of V_{α_{6}} and every weight space in V is 1-dimensional, so V_{α_{6}} contains every weight space for weights between λ_{6} and the maximal weight ω_{1}.

(⊆): Let v′ be a nonzero vector of the lowest weight −ω_{6}. Then v#v′ has weight λ_{6} ∈ V_{α_{6}}, and v#v′ is not zero by Lemma 7.8. Every weight μ of V_{α_{6}} is obtained as wλ_{6} for some w ∈ W′ by Lemma 7.6, and ϕ(w) fixes ω_{1}. Hence w(v#v′) = v#ϕ(w)v′. This shows that there is a nonzero vector of weight μ in v#V. □

For x singular in V, we call the subspace x#V a hyperline, following Tits’ss terminology from [T 57, p. 25]. Combining the proposition with (7.5), we find that every α_{6}-space is a hyperline.

7.11. Remark. As the standard representation of the group L_{α_{6}} of type D_{5}, the space V_{α_{6}} supports an L_{α_{6}}-invariant quadratic form, uniquely determined up to a scalar. We claim that it is the form q given implicitly by the equation

(7.12)  x^{#} = q(x)v for x ∈ V_{α_{6}}.

First, observe that for y, z weight vectors in V_{α_{6}}, y#z has weight at least

ϕ(2λ_{6}) = ω_{1} − (α_{2} + α_{3} + 2α_{4} + 2α_{5} + 2α_{6}),

and the only such weight is ω_{1}. Therefore, for every x ∈ V_{α_{6}}, the vector x^{#} is in the k-span of v and the recipe (7.12) defines a quadratic form on V_{α_{6}}.

For g ∈ L_{α_{6}}, we have q(gx)v = q(x)ϕ(g)v. Since ϕ(g) fixes v (by 3.2, if you like), the form q is L_{α_{6}}-invariant. Moreover, q is not the zero form since # is bilinear and v#x is not zero for x of weight λ_{6}, by Lemma 7.8. Combining the two previous sentences, q is the unique invariant quadratic form as claimed, and it is nondegenerate. In particular, the hyperline v#V cannot contain an isotropic 6-dimensional subspace. In terms of the geometry, an α_{6}-space cannot contain an α_{2}-space.

Enlarging the geometry. In a manner completely analogous to the D_{n} case, we define Γ̃_{P} to be the geometry Γ_{P} with the addition of parabolics of types {α_{2}, α_{5}} and {α_{2}, α_{6}}. Performing the same enlargement on Γ_{V} to obtain Γ̃_{V}, we find that {α_{2}, α_{5}}- and {α_{2}, α_{6}}-spaces are 4- and 5-dimensional respectively. Both types of spaces are singular because they are contained in V_{α_{2}}. The arguments in §6 show—for example—that for δ = α_{2} or α_{6}, an {α_{2}, α_{6}}-space and a δ-space are incident if and only if one is contained in the other, and furthermore each {α_{2}, α_{6}}-space is contained in precisely one α_{2}-space and one α_{6}-space.

Transitivity. As described in §5, we now allow ourselves to use methods from outside of representation theory. Specifically, we view V as the vector space underlying an Albert algebra. The cubic form N is—up to a scalar multiple—the generic norm (“determinant”) of the algebra, and b is—again up to a scalar multiple—the symmetric bilinear form induced by the trace.

The group G acts transitively on the i-dimensional singular subspaces for i = 1, 2, 3, and 6 by [SV 68, 3.12], [Fa, p. 33], or [A, 6.5(2)]. Thus the α_{1}-, α_{2}-, α_{3}-, and α_{4}-spaces are as described in Table 7.13 below. Every hyperline is by definition of the form x#V for a singular x ∈ V. Since G acts transitively on the 1-dimensional singular subspaces, it acts transitively on the hyperlines by (7.5). Therefore, the α_{6}-spaces are the hyperlines.

Before describing the α_{5}-spaces, we first treat the {α_{2}, α_{6}}-spaces. Let M denote the 5-dimensional singular subspaces L_{{α_{2},α_{6}}}v. (Recall that L_{{α_{2},α_{6}}} is the subgroup of G generated by U_{α_{i}}, U_{−α_{i}} for i = 1, 3, 4, 5.) Note that M is not a maximal singular subspace because it is contained in V_{α_{6}}. Suppose now that X is a 5-dimensional singular subspace of V contained in a 6dimensional singular subspace of V. Since G acts transitively on the 6dimensional singular subspaces, we may assume that X is also contained in V_{α_{2}}. But L_{α_{2}} is of type A_{5}, so X is G-equivalent to M. We have just proved that the {α_{2}, α_{6}}-spaces are the 5-dimensional, non-maximal singular subspaces of V.

Consider V_{α_{5}}; it is a 5-dimensional singular subspace and its stabilizer is P_{α_{5}}. In contrast, the stabilizer of M is P_{{α_{2},α_{6}}}, hence V_{α_{5}} is not in the same G-orbit as M, i.e., V_{α_{5}} is a 5-dimensional, maximal singular subspace. Since G acts transitively on such subspaces by [SV 68, 3.14], we have proved: the α_{5}-spaces are the 5-dimensional, maximal singular subspaces of V.

We summarize the descriptions of the α_{i}-spaces in Table 7.13.

Incidence. Let X′ be an α_{i}-space and let X be an α_{j}-space with i ≤ j. If (i, j) is not (2, 5) or (2, 6), Prop. 3.4.2 applies and incidence is the same

-space description  name in [SV 68]
α_{1}  1-dim’l singular  point
α_{2}  6-dim’l singular  max’l space of 2nd kind
α_{3}  2-dim’l singular  space of prdim 1
α_{4}  3-dim’l singular  space of prdim 2
α_{5}  5-dim’l, maximal singular  max’l space of 1st kind
α_{6}  hyperline  line

Table 7.13. α_{i}-spaces in the E_{6} geometry

as inclusion. As in the D_{n} case, we quickly find: An α_{2}- and an α_{5}-space are incident if and only if their intersection is 4-dimensional. An α_{2}- and an α_{6}-space are incident if and only if their intersection is a 5-dimensional (non-maximal) singular subspace.

Bibliographic remarks. We deduced the existence of a G-invariant cubic form on V by decomposing (V^{∗})^{⊗3} as a direct sum of irreducible representations. The program LiE [vLCL] does this purely by computations with characters. But the cubic form was written down explicitly long before these mathematical tools were available, see e.g. [D 01a] and [D 01b]. For a modern derivation of an explicit formula for the cubic form, see [Lu].

Diagrams like Figure 7.1 for different groups and representations can be found, for example, in [PSV].

8. Interlude: more on E_{6}

We will now take a short break from examples of the geometries to derive some properties of the cubic form N and the product #. These results will only be used in the final section, §14.

Fix a nonzero vector c ∈ C. We claim that c^{#} is not zero. For the sake of contradiction, suppose that c^{#} is zero, i.e., C is an α_{1}-space. Then the stabilizer of C is a parabolic of type α_{1}, which has semisimple part of type D_{5} and dimension 45. However, F_{4} stabilizes C and has dimension 52, a contradiction.

Equation (7.5) gives that c^{#} is fixed by F_{4}. Therefore c^{#} equals λc for some scalar λ ∈ k^{×}. Combined with Prop. 7.2, we find:

N(c) = \frac{1}{6}N(c, c, c) = \frac{1}{6}b(c, c#c) ≠ 0.

The forms N and b were only determined up to a factor in k^{×}, so we are free to choose them so that

(8.1)  N(c) = 1 and b(c, c) = 3.

Since # was implicitly defined by equation (7.4), these choices affect # as well.

8.2. Proposition. With N and b chosen to satisfy (8.1), we have c^{#} = c and x^{##} = N(x)x for all x ∈ V.

Proof. The first equation follows from the various equations relating N, b, and #:

1 = N(c) = \frac{1}{6}b(c#c, c) = \frac{1}{3}b(λc, c) = λ.

For the second, we observe that x ↦ x^{##} and x ↦ N(x)x are both quartic G-invariant maps V → V. Representation theory gives that there is a unique such map up to a factor in k^{×}. Since both maps fix c, they agree on all of V. □

9. Example: type E_{7}

Dimensions and properties. For G of type E_{7}, the Dynkin diagram looks like

[figure: β = α_{7}; α_{6}; α_{5}; α_{4}; α_{3}; α_{1}; α_{2}; 1; 2; 3; 4; 6; 12; 7]

Representation theory shows that there exists a G-invariant symmetric quadrilinear form q (or, if you prefer, a quartic form) and a skew-symmetric bilinear form b on V; both are unique up to a factor in k^{×}. We define a symmetric trilinear map t: V × V × V → V (i.e., a linear map S^{3}V → V) implicitly via

q(x, y, z, w) = b(x, t(y, z, w)) for x, y, z, w ∈ V.

Since the infinite group G preserves q, one knows by general principles that the hypersurface in ℙ(V) defined by the equation q = 0 is singular, see e.g. [OS, §6]. That is, there are nonzero vectors x ∈ V such that t(x, x, x) is zero. But in this case, there is more than one kind of singular vector. We will say that an element x ∈ V is rank one if it is nonzero and the image of the linear map y ↦ t(x, x, y) has dimension ≤ 1.

9.1. Example. The highest weight vector v ∈ V is rank one. Indeed, let y ∈ V be a weight vector and write its weight as ω − α where α is a sum of positive roots. If t(v, v, y) is not zero it is a weight vector with weight 3ω−α, necessarily equal to ω − α′ for some sum of positive roots α′. However, the lowest weight of V is w_{0}ω where w_{0} is the longest element of the Weyl group of G, so α is at most ω − w_{0}ω. Putting these observations together with the fact that w_{0}ω = −ω, we have:

2ω = α − α′ ≤ α ≤ 2ω.

Hence α = 2ω and α′ = 0. In particular, t(v, v, V) is contained in the span of v.

By Prop. 3.4.1, the α_{i}-spaces consist of rank one elements except possibly for the α_{1}-spaces.

We claim that the δ-spaces are inner ideals for all δ ∈ Δ. An inner ideal is a subspace X of V such that t(X, X, V) is contained in X. It suffices to check that V_{δ} is an inner ideal. The lowest weight for the action of G on V_{δ} is ω_{7} − α, where α is a sum of simple roots in the δ-component, and the lowest weight for V is −ω_{7}. Therefore, every weight of t(V_{δ}, V_{δ}, V) is at least ω_{7} −2α. But every weight of V that differs from ω_{7} by a sum of simple roots in the δ-component belongs to V_{δ}. We have proved that V_{δ} is an inner ideal.

Transitivity. The group G acts transitively on the 1-dimensional subspaces of V spanned by rank one elements by [Fe, 6.2, 7.7]. It acts transitively on the subspaces of a given dimension consisting of rank one elements, and such subspaces necessarily have dimension ≤ 7 [Ga 01b, 6.12]. Consequently, the α_{i}-spaces are the subspaces consisting of rank one elements for i ≠ 1. The group also acts transitively on the collection of 12-dimensional inner ideals by [Ga 01b, 6.15], hence the α_{1}-spaces are the 12-dimensional inner ideals.

For the sake of brevity, we omit the “incidence” portion of this example. The reader might naturally wonder if the properties of rank one elements

are special to our particular representation V of G, or if they generalize to singular vectors for other quartic forms. We close this section with a relevant example.

9.2. Example. The most familiar example of a quartic form is the determinant of 4-by-4 matrices. Consider SL_{4} acting by conjugation on the vector space of 4-by-4 trace zero matrices. This representation is irreducible with highest weight ω_{1} + ω_{3}. The determinant is clearly SL_{4}-invariant, as is the nondegenerate symmetric bilinear form defined by (x, y) ↦ tr(xy). Let us see how the discussion of rank one elements applies. If we take the usual choices for the maximal torus and the Borel B—the diagonal and the upper-triangular matrices—the highest weight vector v can be taken to be the matrix whose entries are zero except for a 1 in the upper-right corner. Obviously this matrix is rank one in the classical linear algebra sense. Since w_{0} acts as −1 on the highest weight, the discussion in Example 9.1 shows that v is rank one in our sense.

We claim that in this case t(v, v, y) is zero for all y. Indeed, for τ an indeterminate, consider the map y ↦ det(τv + y). Examining the Laplace expansion of the determinant along the top row, we see that the coefficient of τ^{2} in det(τv + y) is zero. Therefore, when we polarize det to obtain a symmetric quadrilinear form, we find that the bilinear map (y, z) ↦ det(v, v, y, z) is identically zero. Since the bilinear form is nondegenerate, we have t(v, v, y) = 0 for all y.

10. Loose ends

To complete our discussion of the concrete realization of the geometries, we address some loose ends. Above, we have skipped the groups of type B and C; the reader should have no trouble filling them in from the previous examples.

10.1. Example: type F_{4}. We now sketch the case where G is of type F_{4}. We find the following diagram:

[figure: <; 1; 2; 3; 6; β = α_{4}; α_{3}; α_{2}; α_{1}]

Representation theory provides a G-invariant bilinear product on V, which we denote by #. (We have seen the objects G, V, # in §7, where they were known as F_{4}, V_{0}, #.) The usual argument shows that the product is identically zero on the α_{4}-spaces, and it is zero on the α_{3}- and α_{2}-spaces by Prop. 3.4.1.

The lowest weight of V_{α_{1}} is

ω_{4} − (2α_{4} + 2α_{3} + α_{2}) = α_{1} + α_{2} + α_{3}

by the arguments in 3.6. For x, y weight vectors in V_{α_{1}}, the product x#y has weight at least 2α_{1} + 2α_{2} + 2α_{3}. In particular, that character is not a weight of V since the α_{1}-coordinate of ω_{4} is 1. Therefore, the product # is identically zero on the α_{1}-spaces also. Freudenthal calls α_{1}-spaces “symplecta”, since they are associated with the standard representation of a group of type C_{3}.

The space V may be interpreted as the trace zero elements in an Albert algebra, i.e., a 27-dimensional simple exceptional Jordan algebra; this identifies G with the group of automorphisms of the algebra. Using this viewpoint, one can prove that the group G acts transitively on the d-dimensional subspaces of V on which the product # is identically zero for d = 1 (the α_{4}-spaces) by [Fr, 28.27], [J, Th. 4], or [A, 8.6], d = 2, 3 by [A, 9.5, 9.8], and for d = 6 (the α_{1}-spaces) by [Fr, 28.22]. (We remark that [A] does not use the interpretation in terms of Albert algebras.)

For all of the objects in Γ_{V}, incidence is the same as inclusion. Aside from α_{1}-spaces, this is Prop. 3.4.2. For cases involving an α_{1}-space, one can adapt the proof of 3.4.2, using the fact that a group of type C_{3} acts transitively on d-dimensional isotropic subspaces of the standard representation for d = 1, 2, 3.

10.2. Example: type G_{2}. The geometry for a group G of type G_{2} is similar to that for type F_{4}, but everything is easier. The dimensions are summarized in the following diagram:

[figure: <; 1; 2; β = α_{1}; α_{2}]

As in the F_{4} case, there is a G-invariant bilinear product on V, which we denote by #. The usual argument shows that the multiplication is identically zero on the α_{1}-spaces, hence also on the α_{2}-spaces by Prop. 3.4.

The vector space V may be viewed as the trace zero elements in the split octonion algebra; this identifies G with the group of automorphisms of the algebra. Using this viewpoint, it is easy to prove that G acts transitively on the 1-dimensional subspaces of V on which the multiplication is zero using [SV 00, 1.7.3]. The Cayley-Dickson process gives an explicit description of the octonion algebra, which one can use to prove that G acts transitively on the 2-dimensional subspaces of V on which the multiplication # is zero. (This essentially solves Problem 23.54 in [FH], cf. 10.4.)

In summary, the α_{i}-spaces are the i-dimensional subspaces of V on which the multiplication # is zero. Incidence is the same as inclusion by Prop. 3.4.2.

10.3. Example: type E_{8}. The recipe in §3 for giving a concrete realization of the geometry associated with a group has been very effective with the examples considered so far. But what of the least familiar case, where G has type E_{8}? The recipe still works, of course, and for each fundamental representation V, it is easy to write down the dimension of the δ-spaces for each δ ∈ Δ. This is already interesting. But a problem occurs when we attempt to describe the algebraic properties that characterize the δ-spaces.

For example, the smallest fundamental representation V of G is the adjoint representation, with highest weight ω_{8} and dimension 248. Representation theory shows that V does not support any obvious additional structure, e.g., there is no G-invariant quintic form on V. Therefore, the only description of the δ-spaces that suggests itself would be in terms of the Lie algebra structure.

The next smallest fundamental representation V has highest weight ω_{1} and dimension 3875. This representation has G-invariant bilinear and cubic forms, each determined uniquely up to a nonzero scalar multiple. Thus there is also a G-invariant commutative product on V. Unfortunately, one cannot simply translate the analysis in §7 to this case. For example, the proof of Lemma 7.8 does not translate because the weights of V are not all one orbit under the Weyl group and some of the weights occur with multiplicity greater than 1.

10.4. Projective homogeneous varieties. The above examples can all be viewed from the perspective of projective homogeneous varieties, i.e., projective varieties Y such that G acts on Y and the action is transitive on K-points for every algebraically closed extension K of k. We maintain our assumptions that G is split simply connected and V is a fundamental irreducible representation as in §3.

There is a bijection between subsets of Δ and isomorphism classes of projective homogeneous varieties given by sending S ⊆ Δ to Y_{S} := G/P_{S}. For example, Y_{∅} is a point because P_{∅} is all of G.

For S a singleton, say {δ}, the k-points of Y_{S} are the δ-spaces in V. Indeed, the δ-spaces are defined to be the orbit of V_{δ} in the appropriate Grassmannian, and V_{δ} has stabilizer P_{δ}.

10.5. Example. For G of type B or D, the variety Y_{α_{1}} is a conic. The other Y_{δ}’s are families of linear subspaces of the conic.

For an arbitrary subset S ⊆ Δ, a flag of type S is a collection of pairwise incident subspaces {X_{s} | s ∈ S} where X_{s} is an s-space. The flags of the extreme type Δ are called chambers. We call {V_{s} | s ∈ S} the standard flag of type S. (What we call the standard chamber is traditionally called the “fundamental chamber”.) We need the following consequence of the fact that Γ_{V} is a building:

10.6. Proposition. [T 74, 3.16] Every flag of type S in Γ_{V} is contained in a chamber and is in the G-orbit of the standard flag of type S.

In particular, G acts transitively on the collection of flags of type S. The stabilizer of the standard flag is the intersection ∩_{s∈S}P_{s}, which is P_{S} [Bou 4–6, IV.2.5, Th. 3c]. Hence the k-points of Y_{S} are the flags of type S in Γ_{V}.

We view the examples in the preceding sections as giving explicit descriptions of the geometry Γ_{V} as well as the projective homogeneous varieties under split groups G. When G is not split, the situation is somewhat more complicated. The absolute Galois group of k acts on the Dynkin diagram Δ, and there is a bijection between Galois-invariant subsets S of Δ and projective homogeneous varieties defined over k. The description in the split case can then be altered to give a description in the general case. For groups of type ^{1}A_{n}, one finds the generalized Severi-Brauer varieties as in [KMRT, §1]. Examples for groups of type D_{4} and E_{7} can be found in [Ga 99] and [Ga 01a] respectively.

11. Outer automorphisms

Every automorphism ϕ of G permutes the parabolic subgroups, hence induces an automorphism of Tits’s geometry Γ_{P}. Further, ϕ induces an automorphism of the concrete geometry Γ_{V} via the isomorphism between Γ_{P} and Γ_{V} from §2.

Every g ∈ G(k) defines an automorphism of G by sending h ↦ ghg^{−1}. Such automorphisms are called inner. It is easy to see the effect of such an automorphism on the geometry Γ_{V}: it sends an object X to gX. In particular, the types of objects in Γ_{V} are preserved. In classical projective geometry, such an automorphism is called a collineation.

On the other hand, some groups have automorphisms that are not of this type; such automorphisms are called outer. They have a more interesting action on the geometry Γ_{V} in that they do not preserve the types of objects. In classical projective geometry, they are called correlations. As an example, the map g ↦ (g^{−1})^{t} is an automorphism of SL_{3}, and it is outer because it does not fix the center elementwise. We will see in Example 11.2 below that the induced map ψ is the polarity with respect to a certain conic.

Generally speaking, the existence of an outer automorphism of G implies a principle of duality (for D_{4}, triality) in the geometry Γ_{V}. For SL_{3}— equivalently, ℙ^{2}—it takes the following form [Cox, 2.3]: “every definition remains significant, and every theorem remains true, when we interchange point and line, join and intersection.” See [W, p. 155] for an analogous statement of the principle of triality.

Let ϕ be an automorphism of G, and let SubSp(V) denote the collection of subspaces of V. We want an efficient way to check if a given function ψ: Γ_{V} → SubSp(V) is the automorphism of the geometry Γ_{V} induced by ϕ.

11.1. Theorem. If

(1) ψ(gX) = ϕ(g)ψ(X) for every X ∈ Γ_{V} and g ∈ G and

(2) there is a chamber {V_{i} | 1 ≤ i ≤ n} such that {ψ(V_{i}) | 1 ≤ i ≤ n} is also a chamber,

then ψ is the automorphism of the geometry Γ_{V} induced by the automorphism ϕ of G.

[The term “chamber” was defined in 10.4.]

Proof. Let X be an object in Γ_{V}. We first claim that ψ(X) is also in Γ_{V}. We find a chamber {X_{i} | 1 ≤ i ≤ n} containing X such that X_{i} is of type α_{i}. This chamber is conjugate to the chamber {V_{i} | 1 ≤ i ≤ n} from (2), i.e., there is some g ∈ G such that gV_{i} = X_{i} for every i. Therefore ψ(X_{i}) = ϕ(g)ψ(V_{i}), and ψ(X_{i}) is an object in the geometry for all i.

Let P be the stabilizer of X in G. For g ∈ ϕ(P), we have

gψ(X) = ψ(ϕ^{−1}(g)X) = ψ(X) by (1),

hence ϕ(P) is contained in the stabilizer of ψ(X). But ψ(X) is an object in Γ_{V}, hence by definition it is a nonzero, proper subspace of V. In particular, its stabilizer is a proper subgroup of G. Since P is a maximal proper subgroup of G, so is ϕ(P), hence ϕ(P) is the stabilizer of ψ(X). This proves that the diagram

Γ_{P} ⟶^{ϕ} Γ_{P}
↑  ↑
Γ_{V} ⟶^{ψ} Γ_{V}

commutes, where the vertical arrows send a subspace of V to its stabilizer in G. Since the vertical arrows are bijections (see §2), ψ is also a bijection. Moreover, ψ respects the notion of incidence in Γ_{V}, because that relation is the one transported from Γ_{P} by the vertical isomorphisms. We have proved that ψ is an automorphism of Γ_{V}, and the commutativity of the diagram shows that it is the one induced by ϕ. □

11.2. Example (Type A: projective duality). Let G be SL_{n} acting on k^{n}, and let ϕ be the automorphism g ↦ (g^{−1})^{t}. For a subspace X of k^{n}, we define ψ(X) to be the orthogonal complement of X with respect to the dot product.

Viewed algebraically, the dot product identifies X with the dual vector space X^{∗}. This identification pairs ψ(X) with the collection of linear forms vanishing on X.

Viewed geometrically, the map ψ is precisely the correspondence between points and hyperplanes giving projective duality in ℙ^{n−1} described in [Pe] and [Cox, 11.8]. It interchanges a point [a_{1} : a_{2} : …: a_{n}] in homogeneous coordinates with the hyperplane consisting of solutions to the equation ∑ a_{i}x_{i} = 0. For n = 3, it is the polarity with fundamental conic x^{2} + y^{2} + z^{2} = 0, cf. [VY, §98].

We now check that ψ satisfies the conditions of Theorem 11.1. The dot product is compatible with the automorphism ϕ in the sense that

x · y = (ϕ(g)x) · (gy) for g ∈ SL_{n} and x, y ∈ k^{n}.

Thus, a vector x is in ψ(gX) if and only if (ϕ(g)^{−1}x) · X = 0, i.e., if and only if x is in ϕ(g)ψ(X). Thus ψ satisfies (1).

Consider the collection {V_{1}, V_{2}, …, V_{n−1}} of subspaces such that V_{i} consists of the vectors whose bottom n − i coordinates are zero. Clearly, this is a chamber. Applying ψ, we find ψ(V_{i}) is the set of vectors whose top n − i coordinates is zero. Since these also form a chamber, ψ satisfies (2), hence ψ is the automorphism of Γ_{V} corresponding to the automorphism ϕ of SL_{n}.

12. Example: type D (orthogonal duality)

Let G be a group as in §6, constructed from the root system of type D_{n} for some n ≥ 4; it is traditionally denoted by Spin_{2n}. Its Dynkin diagram has an automorphism of order 2 given by interchanging the roots α_{n−1} and α_{n}. Here we will construct the corresponding automorphism ψ of the geometry Γ_{V}. 12.1. We can draw a Hasse diagram for the weights of V as we did for E_{6} in Figure 7.1. The case n = 4 is shown in Figure 13.1 below. Figure 12.2 shows the diagram in the general case, rotated counterclockwise by 90^{∘} for space considerations.

[figure: 1; 2; 1; 2; n − 2; n − 2; n − 1; n; n; n − 1]

Figure 12.2. Hasse diagram of weights of the standard representation of D_{n} from [PSV, Fig. 4]. Larger weights are on the left.

Fix nonzero vectors e_{1}, e_{2}, …, e_{n} in V such that e_{i} has weight

ε_{i} := ω_{1} − ∑_{j=1}^{i−1} α_{j}.

The longest element of the Weyl group is −1; it is the unique automorphism of the diagram of order 2 that stabilizes none of the weights. The weights ε_{1} through ε_{n−1} are those in the string on the left of the diagram and ε_{n} is the bottom weight in the middle square. The other weights of V are of the form −ε_{i} for some i. Let f_{i} be a nonzero vector of weight −ε_{i}, so the vectors e_{1}, e_{2}, …, e_{n}, f_{1}, f_{2}, …, f_{n} are a basis of V.

Let b be the G-invariant symmetric bilinear form on V as in §6. Clearly, since ε_{i} is not −ε_{j} for any pair i, j, the subspace of V spanned by the e_{i}’s (respectively, by the f_{i}’s) is isotropic, i.e., b(e_{i}, e_{j}) = b(f_{i}, f_{j}) = 0 for all i, j. Also, b(e_{i}, f_{j}) is nonzero if and only if i = j. By scaling the f_{j}’s, we may assume that b(e_{i}, f_{j}) = δ_{ij} (Kronecker delta). (We have now obtained the description of G and SO(b) given in [Br, §V.7].) The construction in §6 gives:

V_{α_{i}} := k-span {e_{1}, e_{2}, …, e_{i}} for i ≤ n − 2,

V_{α_{n−1}} = k-span {e_{1}, e_{2}, …, e_{n}}, and (12.3)

V_{α_{n}} = k-span {e_{1}, e_{2}, …, e_{n−1}, f_{n}}.

12.4. Since b is G-invariant and G is connected, the representation of G on V is a homomorphism ρ: G → SO(b), where SO(b) denotes the subgroup of SL(V) preserving the bilinear form b. We claim that ρ is a central isogeny. Indeed, every proper, closed normal subgroup of G is central, hence ker ρ is finite and the image of ρ has the same dimension as G. Root system data gives that the dimension of the Lie algebra of G (equivalently, the dimension of G) is \binom{2n}{2}. On the other hand, the Lie algebra of SO(b) is isomorphic over an algebraic closure of k to the space of skew-symmetric 2n-by-2n matrices, which also has dimension \binom{2n}{2}. Since im ρ and SO(b) are connected and have the same dimension, they are the same. That is, ρ is surjective. The claim now follows because we are in characteristic 0, hence ρ is automatically separable.

Let s denote the matrix in GL(V) that fixes e_{i} and f_{i} for 1 ≤ i < n and interchanges e_{n} and f_{n}. It leaves b invariant, and the map ϕ: SO(b) → SO(b) defined by ϕ(g) = sgs^{−1} is an automorphism of order 2. There is a unique lift of ϕ to an automorphism of G [BT 72, 2.24(i)], which we also denote by ϕ. The description of the root subgroups in SO(b) in [Bor, 23.4] shows that ϕ is, in fact, the automorphism of G induced by the automorphism of the Dynkin diagram that interchanges α_{n−1} and α_{n}.

For each subspace X of V, put ψ(X) := sX. It is a triviality that ψ satisfies condition (1) of 11.1 and that fundamental chamber exhibited in (12.3) is permuted by ψ, hence that ψ satisfies condition (2). That is, ψ is the automorphism of Γ_{V} induced by the automorphism ϕ of G and SO(b).

13. Example: type D_{4} (triality)

Continue the notation of the preceding section, §12, except suppose now that n = 4. The Dynkin diagram of G looks like

[figure: α_{1}; α_{2}; α_{3}; α_{4}]

Let ϕ be the automorphism of order 3 that permutes the arms counterclockwise. We will now describe explicitly the corresponding automorphism ψ of the geometry Γ_{V}.

Let ρ_{0} be the representation of G on V with highest weight ω_{1}. For i = 1, 2, we set ρ_{i} := ρ_{0}ϕ^{−i}; it is a representation of G on V. The highest weight of ρ_{1} is ϕ(ω_{1}) = ω_{3}, and the highest weight of ρ_{2} is ϕ^{2}(ω_{1}) = ω_{4}.

The weights of V with respect to ρ_{0} are listed in Figure 13.1. The weights relative to ρ_{1} and ρ_{2} are the same except with ϕ or ϕ^{2} applied, respectively. Let e_{i}, f_{j} be a basis for V as in 12.1. The vector e_{i} has weight ϕ^{j}(ε_{i}) relative to ρ_{j}. Moreover, the image ρ_{i}(G) of G in GL(V) is the same for all i, so the symmetric bilinear form b from 12.1 is ρ_{i}(G)-invariant for all i.

[figure: (1, 0, 0, 0) = ω_{1}; 1; (−1, 1, 0, 0); 2; (0, −1, 1, 1); 3; 4; (0, 0, −1, 1); 4; (0, 0, 1, −1); 3; (0, 1, −1, −1); 2; (1, −1, 0, 0); 1; (−1, 0, 0, 0) = −ω_{1}]

Figure 13.1. Hasse diagram of the weights of V relative to ρ_{0}.

By representation theory, there is a unique linear map t: V ⊗ V ⊗ V → k that is G-invariant in the sense that

(13.2)  t(ρ_{0}(g)x_{0}, ρ_{1}(g)x_{1}, ρ_{2}(g)x_{2}) = t(x_{0}, x_{1}, x_{2})

for all g ∈ G and x_{0}, x_{1}, x_{2} ∈ V. We can prove that t is nonzero for certain arguments.

13.3. Lemma. Let x_{i} ∈ V be a nonzero vector of weight μ_{i} relative to ρ_{i} for i = 0, 1, 2. We have: t(x_{0}, x_{1}, x_{2}) is nonzero if and only if μ_{0} + μ_{1} + μ_{2} = 0.

Proof. “Only if” is clear, so we prove “if”. Suppose that ∑ μ_{i} = 0. Since ρ_{2}(G) acts transitively on the weights of V relative to ρ_{2}, we may assume that μ_{2} is ω_{4}. By the argument in the proof of Lemma 7.6, the subgroup of the Weyl group fixing ω_{4} has two orbits on the weights of V relative to ρ_{1}, with representatives ±ω_{3}. Since −ω_{4} − ω_{3} is not a weight of V relative to ρ_{0}, we must have μ_{1} = −ω_{4} + ω_{3}. We have just proved that the Weyl group acts transitively on the triples μ_{0}, μ_{1}, μ_{2} such that ∑ μ_{i} = 0. As in the proof of Lemma 7.8, it follows that t(x_{0}, x_{1}, x_{2}) is nonzero. □

Moreover, t is invariant under cyclic permutations.

13.4. Lemma. The value of t is unchanged if its arguments are permuted cyclically.

Proof. Let i = 0, 1, or 2. Consider the linear map d: V ⊗ V ⊗ V → k defined by

d(x_{0}, x_{1}, x_{2}) := t(x_{0}, x_{1}, x_{2}) − t(x_{i}, x_{i+1}, x_{i+2}),

with the subscripts taken modulo 3. This map is G-equivariant because the permutation x_{j} ↦ x_{j+i} is cyclic. Indeed, we have:

t(ρ_{i}(g)x_{i}, ρ_{i+1}(g)x_{i+1}, ρ_{i+2}(g)x_{i+2}) = t(ρ_{0}(g′)x_{i}, ρ_{1}(g′)x_{i+1}, ρ_{2}(g′)x_{i+2})
= t(x_{i}, x_{i+1}, x_{i+2}),

where g′ := ϕ^{−i}(g).

By the uniqueness of t, the map d must be a scalar multiple of t. The vector e_{4} ∈ V is nonzero of weight −ω_{3} + ω_{4} relative to ρ_{0}. Then t(e_{4}, e_{4}, e_{4}) is not zero by the previous lemma, yet d(e_{4}, e_{4}, e_{4}) is zero. Therefore, d is identically zero. □

We now define products ·_{i} on V for i = 0, 1, 2 implicitly via

t(x_{0}, x_{1}, x_{2}) = b(x_{i}, x_{i+1} ·_{i} x_{i+2}).

By Lemma 13.4, all three products agree, so we write simply ·. Because t and b are G-equivariant, so is the product, i.e.,

(13.5)  (ρ_{i}(g)x) · (ρ_{i+1}(g)y) = ρ_{i+2}(g)(x · y).

This allows us to compute the multiplication, at least up to a scalar factor. Let x_{i} ∈ V be nonzero with weight μ_{i} relative to ρ_{i} for i = 1, 2. It follows from Lemma 13.3 that x_{1} · x_{2} is nonzero if and only if μ_{1} + μ_{2} is a weight of V relative to ρ_{0}, in which case x_{1} · x_{2} has weight μ_{1} + μ_{2}. We summarize these computations in the table below, where the entry in the row x_{1} and column x_{2} is “·” if x_{1} · x_{2} is zero and, for example, e_{3} if x_{1} · x_{2} is a nonzero scalar multiple of e_{3}. The left column lists the weight of x_{1} for the reader’s convenience; we omit the weight of x_{2} due to space considerations. Since the product is G-equivariant and the weights of ρ_{i} are preserved under multiplication by −1, one needs only to compute the first four columns of entries; the remaining four columns can be filled in by symmetry.

(13.6)  \begin{aligned}& & & & & & & x_{2} \\ & & & e_{1} & e_{2} & e_{3} & e_{4} & f_{4} & f_{3} & f_{2} & f_{1} \\ & (0, 0, 1, 0) & e_{1} & · & · & · & e_{1} & · & e_{2} & e_{3} & f_{4} \\ & (0, 1, −1, 0) & e_{2} & · & · & e_{1} & · & e_{2} & · & e_{4} & f_{3} \\ & (1, −1, 0, 1) & e_{3} & · & e_{1} & · & · & e_{3} & e_{4} & · & f_{2} \\ x_{1} & (1, 0, 0, −1) & e_{4} & e_{1} & · & · & · & f_{4} & f_{3} & f_{2} & · \\ & (−1, 0, 0, 1) & f_{4} & · & e_{2} & e_{3} & e_{4} & · & · & · & f_{1} \\ & (−1, 1, 0, −1) & f_{3} & e_{2} & · & f_{4} & f_{3} & · & · & f_{1} & · \\ & (0, −1, 1, 0) & f_{2} & e_{3} & f_{4} & · & f_{2} & · & f_{1} & · & · \\ & (0, 0, −1, 0) & f_{1} & e_{4} & f_{3} & f_{2} & · & f_{1} & · & · & · \end{aligned}

(There is a strong resemblance between this multiplication table and the one found by Seligman using the Zorn vector matrices in [Se, p. 287].)

Although the table above is not fine enough to allow us to actually multiply two vectors in V, it is sufficient to describe how the objects in the geometry Γ_{V} interact with the multiplication. Specifically, we can recover some of the results of [vdBS] without discussing octonion algebras.

13.7. Proposition. (Cf. [vdBS, §2])

(1) If X is an α_{1}-space (a “point”), then V · X is an α_{3}-space and X · V is an α_{4}-space.

(2) If X is an α_{2}-space (a “line”), then (X · V) · X and X · (V · X) are α_{2}-spaces.

(3) If X is an α_{3}-space (resp., an α_{4}-space), then there is a unique α_{1}- space U such that X = V · U (resp., X = U · V).

Proof. By (13.5), it suffices to check (1) for the case where X is the k-span of e_{1}, i.e., V_{α_{1}}. In that case, (1) is clear from the multiplication table. A similar argument handles (2).

We now prove (3) for α_{3}-spaces. As in the previous paragraph, it suffices to check the case where X is the k-span of e_{1}, e_{2}, e_{3}, e_{4}, i.e., V_{α_{3}}. The multiplication table shows that X is V ·e_{1}, so suppose that u ∈ V is nonzero and satisfies V · u = X. Since V and X are T -invariant, we may assume that u is a weight vector. The multiplication table gives that u is a multiple of e_{1}. This completes the proof of (3). □

Define ψ via

(13.8)  ψ(ka) = a · V, ψ(a · V) = V · a, and ψ(V · a) = ka

for a isotropic in V. This is well-defined by the proposition. For X an α_{2}-space, we define

(13.9)  ψ(X) = X · (V · X).

Applying (13.5), it is easy to check that ψ satisfies condition (1) of Th. 11.1. On the other hand, we checked in the proof of Prop. 13.7 that ψ maps the fundamental chamber to the fundamental chamber, so ψ also satisfies condition (2). Thus ψ is the automorphism of Γ_{V} corresponding to ϕ.

Remarks. Equation (13.8) defines ψ^{2} on the α_{1}-, α_{3}-, and α_{4}-spaces, but (13.9) omits the α_{2}-spaces. Appealing again to Th. 11.1, we can check that ψ^{2}(X) is (X · V) · X for X an α_{2}-space.

We remark that we have recovered a multiplication of the octonions—at least approximately—entirely from first principles of representation theory.

14. Example: type E_{6} (duality)

Let G be the split simply connected group of type E_{6} with standard representation V as in §7. In this section, we will give an explicit description of the automorphism ψ of the geometry Γ_{V} corresponding to the automorphism ϕ of the group G.

We define the brace product on V following [McC 04, p. 190]:

{x, y, z} := b(x, y)z + b(z, y)x − (x#z)#y.

Note that x and z are interchangeable. From (7.5), we find that

(14.1)  g{x, y, z} = {gx, ϕ(g)y, gz} for g ∈ G and x, y, z ∈ V.

For each subspace W of V, we set

ψ(W) := {x ∈ V | {W, x, V } ⊆ W }

An argument nearly identical to the one in Example 11.2 shows that ψ satisfies hypothesis (1) of Th. 11.1. The rest of this section is spent proving that ψ(V_{δ}) = V_{ϕ(δ)} for all δ ∈ Δ, i.e., ψ permutes the objects in the fundamental chamber. This will show that ψ satisfies hypothesis (2) of the theorem.

14.2. Example. Let U denote the set of weights μ of V such that ϕ(ω_{1} + μ) is not a weight of V. Let z be a nonzero vector of weight μ ∈ U; we claim that z is in {v, v′, V }, where v′ is a lowest weight vector of V. First observe that since z does not have weight −ω_{6}, b(v, z) is zero. Since μ is in U, v#z is zero and we have: {v, v′, z} = b(v, v′)z. But b(v, v′) is not zero because b is nondegenerate. This proves the claim.

The subspace of V spanned by weight vectors with weights in U is 17dimensional, as follows from the proof of Prop. 7.10. Therefore,

dim{v, v′, V } ≥ 17.

Connection with Jordan theory. Let c, N, and b be as in §8, so in particular they satisfy (8.1). It is well-known that V is the vector space underlying an exceptional Jordan (a.k.a. Albert) algebra with identity c such that G is the group of isometries of the cubic norm on the algebra. In particular, there exists a G-invariant cubic form N′ on V such that N′(c) = 1 and a symmetric bilinear form b′ on V such that b′(c, c) = 3 and b′ is compatible with G in the sense of (7.5). By the uniqueness of N and b, the cubic forms N and N′ and the bilinear forms b and b′ are the same. Therefore we may apply results about Jordan algebras and cubic norm structures.

Specifically, we will use the 5-linear identity from [McC 04, p. 202]

{x, y, {z, w, u}} = {{x, y, z}, w, u} − {z, {y, x, w}, u} + {z, w, {x, y, u}}.

Also, we will use the classification of the inner ideals of V. A subspace X is an inner ideal if {X, V, X} is contained in X. By [McC 71, §7], the proper inner ideals are the singular subspaces and the hyperlines. The maximal proper inner ideals are the α_{2}- and α_{6}-spaces (this follows, for example, from §7).

A straightforward application of the 5-linear identity gives: If I is an inner ideal in V, then ψ(I) is also an inner ideal. 14.3. Computation of ψ(V_{α_{1}}). Linearizing the equation x^{##} = N(x)x from Prop. 8.2 as in [McC 69, p. 496], we find the identity (McCrimmon’s Equation (12)):

(x#y)#(x#z) = b(x^{#}, y)z + b(x^{#}, z)y + b(y#z, x)x − x^{#}#(y#z)

Since v^{#} is zero, we have:

(v#y)#(v#z) = b(y#z, v)v.

and

{v, v#z, y} = b(v, v#z)y + b(y, v#z)v − b(y#z, v)v.

Since N(−, −, −) is symmetric, equation (7.4) shows that the first summand is zero and the second and third summands cancel. Therefore, {v, v#z, y} is zero and ψ(kv) contains v#V.

Since ψ(kv) is an inner ideal containing the hyperline v#V and it is proper (Example 14.2), the ideal must be the hyperline. Since G acts transitively on the singular 1-dimensional subspaces and ψ satisfies 11.1.1, we obtain the following lemma:

14.4. Lemma. If X is a 1-dimensional singular subspace (= an α_{1}-space), then ψ(X) is the hyperline X#V.

14.5. Computation of ψ(V_{α_{2}}). We now show that ψ(V_{α_{2}}) is V_{α_{2}}. For x, y weight vectors in V_{α_{2}} and w a weight vector in V, we find that {x, y, w} has weight at least

λ_{2} + ϕ(λ_{2}) − ω_{6} = ω_{1} − 2(α_{1} + α_{3} + α_{4} + α_{5} + α_{6}).

But every weight of V of the form ω_{1} − (c_{1}α_{1} + c_{3}α_{3} + c_{4}α_{4} + c_{5}α_{5} + c_{6}α_{6}) with each c_{i} a nonnegative integer belongs to L_{α_{2}}v, i.e., V_{α_{2}}. Hence V_{α_{2}} is contained in ψ(V_{α_{2}}).

Since v′ is not in ψ(V_{α_{2}}) by Example 14.2 and ψ(V_{α_{2}}) is an inner ideal, the classification of inner ideals gives that ψ(V_{α_{2}}) is precisely V_{α_{2}}.

We are left with computing ψ(V_{α_{i}}) for i ≠ 1, 2. We will use crucially the fact from 3.6 that the weights of V_{α_{i}} are precisely those lying between the highest weight ω_{1} and the weight labeled λ_{i} in Figure 7.1.

14.6. Example. Let y be a nonzero vector of weight λ_{2}, i.e., a lowest weight vector for L_{α_{2}} acting on V_{α_{2}}. We claim that {v, y, V } is V_{α_{2}}. Since y is in ψ(V_{α_{2}}) by 14.5, we need only show that V_{α_{2}} is contained in {v, y, V }.

Consulting Figure 7.1, we see that the weights of V_{α_{2}} are symmetric in the following sense: If ω_{1} − α is a weight of V_{α_{2}}, then λ_{2} + ϕ(α) is also a weight of V_{α_{2}}. Consequently, for every weight λ of V_{α_{2}},

f(λ) := −ϕ(λ_{2} + ϕ(ω_{1} − λ)) = −ϕ(λ_{2}) + λ − ω_{1}

is a weight of V. For each weight λ of V_{α_{2}}, fix a nonzero vector z_{λ} of weight f(λ). The vector {v, y, z_{λ}} has weight λ, and it suffices to prove that it is not zero for each λ.

For λ = ω_{1}, we note that ω_{1}+f(λ) = ω_{1}−ϕ(λ_{2}) has 2 as one of its entries, hence ϕ(ω_{1} + f(λ)) is not a weight of V. Therefore, v#z_{λ} is zero. We have

{v, y, z_{λ}} = b(z_{λ}, y)v,

which is not zero because f(λ) = −ϕ(λ_{2}).

For the other five weights λ of V_{α_{2}}, we claim that v#z_{λ} is not zero. It has weight μ := ϕ(ω_{1} + f(λ)) = ϕ(λ) − λ_{2}. For λ = λ_{3}, Equation (7.9) gives that μ = λ_{5}, a weight of V. For λ = λ_{4} = λ_{3} − α_{3}, we find that μ is λ_{5} − α_{5}. Similarly, we find that for each of the the three remaining λ’s, the weight μ is a weight of V. That is, v#z_{λ} is nonzero. The function f was defined so that (v#z_{λ})#y would have weight λ, hence that product is also not zero, i.e.,

{v, y, z_{λ}} = (v#z_{λ})#y ≠ 0.

We have proved that {v, y, V } is V_{α_{2}}.

14.7. We can now give a reasonably good description of the space {v, w, V } for w a weight vector in V. We say that w and v#V are connected if there is a singular vector x ∈ v#V such that x and w are “collinear”, i.e., such that x and w span an α_{3}-space. (In this case, Tits says that w and v#V are “liés” in [T 57, 3.9].) For example, the vector y from (14.6) and v#V are connected because y and v are in the α_{2}-space V_{α_{2}}. In contrast, the lowest weight vector v′ and v#V are not connected, as it is easily checked that ϕ(−ω_{6} + μ) is a weight of V for every weight μ of v#V. We find

dim\{v, w, V \} \begin{cases}= 0 & if w and v#V are incident \\ = 6 & if w and v#V are not incident but are connected \\ ≥ 17 & if w and v#V are neither incident nor connected. \end{cases}

We remind the reader that w and v#V are incident if and only if w is contained in v#V, so the first equality follows from 14.3. The second equality and the inequality are consequences of Lemma 7.6 and Examples 14.2 and 14.6.

14.8. Remark. Since G acts “strongly transitively” on the geometry Γ_{V}, it is sufficient to only consider the case where w is a weight vector. That is, (14.7) holds for every pair of singular vectors v, w. We will not use this fact.

14.9. Lemma. Fix a δ in Δ\{α_{2}}. If X is a basis of V_{δ} consisting of weight vectors, then

ψ(V_{δ}) = ⋂_{x∈X} x#V.

Proof. Fix a nonzero weight vector y ∈ ψ(V_{δ}). We claim that {v, y, V } is the zero subspace. Otherwise, by 14.7 {v, y, V } is an α_{2}-space or has dimension at least 17. If δ is not α_{6}, then V_{δ} has dimension at most 5, and we have a contradiction. When δ is α_{6}, V_{δ} does not contain an α_{2}-space by Remark 7.11, and again we find a contradiction. This proves that {v, y, V } is zero. Since V_{δ} and V are T -invariant, so is ψ(V_{δ}); that is ψ(V_{δ}) is a direct sum of weight spaces. All together, we have that {v, ψ(V_{δ}), V } is zero, i.e., ψ(V_{δ}) is contained in v#V.

For each x ∈ X, there is some g ∈ G such that gv is in the span of x and g leaves V_{δ} invariant. We have

ψ(V_{δ}) = ϕ(g)ψ(V_{δ}) ⊆ ϕ(g)(v#V) = x#V.

Conversely, suppose that y is in the intersection of the x#V ’s. Then {x, y, V } is zero for every x. Since the x’s span V_{δ}, y is in ψ(V_{δ}). □

We can now compute ψ(V_{α_{i}}) for i ≠ 1, 2. 14.10. Computation of V_{α_{3}} and V_{α_{4}}. The space V_{α_{3}} is spanned by the highest weight vector v and a vector x of weight λ_{3}. We wish to compute ψ(V_{α_{3}}), which is V_{α_{6}} ∩ (x#V) by Lemma 14.9. Each weight τ of V_{α_{6}} is a weight of x#V if and only if ϕ(τ) − λ_{3} is a weight of V. The five weights τ of V_{α_{6}} with a 1 as their last coordinate cannot belong to x#V because ϕ(τ) − λ_{3} has a 2 as its first coordinate. That is, ψ(V_{α_{3}}) is contained in V_{α_{5}}.

Since −ϕ(λ_{2}) is a weight of V and λ_{5} = ϕ(λ_{3} −λ(ϕ_{2})) by (7.9), the weight λ_{5} belongs to x#V. Figure 7.1 shows that −ϕ(λ_{2}) + α_{2} is also a weight of V, hence λ_{5} + α_{2} belongs to x#V. Continuing in this manner, we find that V_{α_{5}} is contained in x#V, hence that ψ(V_{α_{3}}) is V_{α_{5}}.

The space V_{α_{4}} is spanned by V_{α_{3}} and a vector y of weight λ_{4}. The two weights τ of V_{α_{5}} that do not belong to V_{α_{4}} each have a 1 as their 5th coordinate, hence ϕ(τ) − λ_{4} has a 2 as its 3rd coordinate, and such weights τ do not belong to ψ(V_{α_{4}}). The three weights of V_{α_{4}} are easily checked to be weights of y#V, hence ψ(V_{α_{4}}) is V_{α_{4}}.

14.11. Computation of V_{α_{5}} and V_{α_{6}}. By Lemma 14.9 and 14.10, ψ(V_{α_{5}}) is contained in V_{α_{4}}. Moreover, the 5th coordinate of ϕ(λ_{4}) − λ_{5} is −2, hence ψ(V_{α_{5}}) is contained in V_{α_{3}}.

Equation (7.9) gives that the weight λ_{3} belongs to z#V for z a nonzero vector of weight λ_{5}. Also,

λ_{1} = λ_{3} + α_{1} = ϕ(λ_{5} + (λ_{2} + α_{6})),

so v belongs to z#V. Similar calculations show that V_{α_{3}} is contained in x#V for x of weight λ_{5} + α_{2}, hence V_{α_{3}} is equal to ψ(V_{α_{5}}).

Lemma 14.9 gives that ψ(V_{α_{6}}) is contained in the 2-dimensional space ψ(V_{α_{5}}) = V_{α_{3}}. The 6th coordinate of ϕ(λ_{3}) − λ_{6} is −2, hence λ_{3} does not belong to u#V for every vector u of weight λ_{6}, and ψ(V_{α_{6}}) is contained in the k-span of the highest weight vector v.

We now show that v is in ψ(V_{α_{6}}). Linearizing the identity x^{##} = N(x)x as in [McC 69] again (and going through his Equation (19)), we find the identity

(14.12) z#(y#(x#z)) = b(x, z^{#})y +b(x, y)z^{#} +b(y, z)(x#z)−x#(y#z^{#}),

which holds for every x, y, z ∈ V. Substituting z ↦ v, we find

v#(y#(x#v)) = b(y, v)(x#v).

Recalling that b(v#x, v) is zero, we find that {v#x, v, y} is zero for all x, y ∈ V. That is, v is in ψ(V_{α_{6}}) and ψ(V_{α_{6}}) is V_{α_{1}}.

We have proved that ψ(V_{α_{i}}) = V_{ϕ(α_{i})} for all i. In particular, the image of the fundamental chamber under ψ is just the fundamental chamber. This proves that ψ is the automorphism of Γ_{V} induced by the automorphism ϕ of G.

In the language of classical projective geometry, ψ is a hermitian polarity. Indeed, since ϕ^{2} is the identity on G, ψ^{2} is the identity on Γ_{V}, i.e., ψ is a polarity. One says that ψ is hermitian because the “point” V_{α_{1}} is contained in its “polar” V_{α_{6}}.

For the sake of a finer description of the map ψ, we record the following corollary to Lemma 14.9.

14.13. Corollary. If X ∈ Γ_{V} is not of dimension 6, then

ψ(X) = {y ∈ V | {X, y, V } = 0}.

Proof. Let δ ∈ Δ be such that X is a δ-space. Fix a g ∈ G such that gX is V_{δ}. For every y ∈ ψ(X), we have

g{x, y, V } ⊆ {V_{δ}, ψ(V_{δ}), V },

which is the zero subspace by the proof of Lemma 14.9. □

Faulkner discussed the geometry Γ_{V} in terms of the brace product in [Fa], although he focussed on the points (α_{1}-spaces) and hyperlines. He described the duality on points and hyperlines by the equation displayed in the corollary. However, that definition does not work for our purposes, as the following example shows.

14.14. Example. If X ∈ Γ_{V} has dimension 6, then the set X^{c} := {y ∈ V | {X, y, V } = 0} is the zero subspace. To prove this, it suffices to check the case X = V_{α_{2}}. As in the proof of Lemma 14.9 we find that X^{c} is the intersection of the sets z#V as z ranges over nonzero vectors of each of the six weights of V between ω_{1} and λ_{2}. Since V_{α_{2}} contains V_{α_{4}}, we have V_{α_{2}}^{c} is contained in ψ(V_{α_{4}}) = V_{α_{4}}. Arguing as in 14.10, one quickly sees that the three weights λ_{1}, λ_{3}, λ_{4} of V_{α_{4}} do not belong to X^{c}. Hence X^{c} is zero as claimed.

14.15. Proposition. For X ∈ Γ_{V}, we have

b(X, ψ(X)) = 0 and {ψ(X), X, ψ(X)} = 0.

Proof. By the transitivity of the G-action, we may assume that X is V_{α_{i}} for some i. Let j be such that α_{j} = ϕ(α_{i}). Further, let d_{i} be such that ω_{1} − d_{i} = λ_{i}; it is a sum of positive roots.

We first argue that b(X, ψ(X)) is zero. By (7.3), it can only be nonzero if there are vectors x ∈ X and x′ ∈ ψ(X) of weights λ and λ′ such that λ + ϕ(λ′) = 0. But every weight of X (resp. ψ(X)) is at least λ_{i} (resp. λ_{j}), and

λ_{i} + ϕ(λ_{j}) = (ω_{1} + ω_{6}) − (d_{i} + ϕ(d_{j}));

we will show that this is > 0 for all i. Consulting the tables in [Bou 4–6], we find:

ω_{1} + ω_{6} = 2α_{1} + 2α_{2} + 3α_{3} + 4α_{4} + 3α_{5} + 2α_{6}.

When i = 1, d_{1} is zero and d_{6} is a sum of positive roots with no root occurring more than twice, as can be seen from Figure 7.1. Therefore, λ_{1} + ϕ(λ_{6}) > 0. Applying ϕ to both sides of the equation, we find that λ_{6} + ϕ(λ_{1}) > 0. When i = 2, 3, 4, or 5, no root appears more than once in d_{i}. But 2 ≤ j ≤ 5, hence the same is also true of d_{j}. Therefore no root appears in d_{i} + ϕ(d_{j}) more than twice. We have proved that λ_{i} + ϕ(λ_{j}) > 0 for all i, hence b(X, ψ(X)) is necessarily zero.

We now prove that the second equation holds. The space {ψ(X), X, ψ(X)} is a direct sum of its weight spaces, and each weight μ is at least ϕ(λ_{i}) + 2λ_{j} by (14.1). That is, each weight μ is of the form ω_{1} − d for

(14.16)  0 ≤ d ≤ ω_{1} − (ϕ(λ_{i}) + 2λ_{j}) = (ϕ(d_{i}) + 2d_{j}) − (ω_{1} + ω_{6}).

For each j, we see from Figure 7.1 that d_{j} does not include the root α_{j}, hence neither does ϕ(d_{i}). Therefore, the coefficient of α_{j} on the right side of (14.16) is negative. In particular, the equation d ≥ 0 is impossible, and μ cannot be a weight of V. This proves that {ψ(X), X, ψ(X)} is zero. □

In [LN, p. 260], Loos and Neher defined

Inid(X) := {y ∈ V | {y, X, y} = 0 and {X, y, V } ⊆ X},

for X a subspace of V. Clearly, ψ(X) contains Inid(X), and the preceding proposition shows that the two concepts agree for X ∈ Γ_{V}.

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Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109

E-mail address: mpcarr@umich.edu

Department of Mathematics & Computer Science, Emory University, Atlanta, GA 30322

E-mail address: skip@member.ams.org
URL: http://www.mathcs.emory.edu/~skip/