UPPER BOUNDS FOR THE ESSENTIAL DIMENSION OF E_{7}

MARK L. MACDONALD

Abstract. This paper gives a new upper bound for the essential dimension and the essential 2-dimension of the split simply connected group of type E_{7} over a field of characteristic not 2 or 3. In particular, ed(E_{7}) ≤ 29, and ed(E_{7}; 2) ≤ 27.

The essential dimension of an algebraic group G is a numerical invariant which measures the complexity of its G-torsors. The essential dimensions of the exceptional algebraic groups remain largely open (see [Re10] for a recent survey). For example, even if the base field is the complex numbers we only know that 9 ≤ ed(E_{8}) ≤ 231 ([CS06], [Le04]).

We will say that a G-variety X is generically free if there is an open subset U ⊂ X(k̄) on which G(k̄) acts with trivial stabilizers, where k̄ is an algebraic closure of the base field k. We will define the essential dimension of G to be the minimum value of dim(X) − dim(G), where X is a generically free G-variety and there exists a Gequivariant dominant map V −−→ X (also called a compression), such that V is a generically free linear representation of G.

We assume throughout this paper that the characteristic of the base field k is not 2. Let E_{n} denote the split simply connected group of that type. We have the lower bound 7 ≤ ed(E_{7}) from [CS06] (for characteristic zero, see [RY00]), and over the complex numbers, we know that three copies of E_{7}’s faithful 56 dimensional representation V_{56} is generically free [Po86, 13], and so is V_{56} × ℙ(V_{56}) × ℙ(V_{56}). Therefore ed(E_{7}) ≤ (56 + 55 + 55) − 133 = 33, which is the best known upper bound. In this paper we will improve this bound, and only assume the base field has characteristic not 2 or 3.

For a prime p, we are also interested in the “localized” notion of essential p-dimension, which is often easier to compute. For primes p not equal to 2 or 3, we know ed(E_{7}; p) = 0. Also, ed(E_{7}; 3) = 3 ([GR09] or [Ga09]). For p = 2 we have 7 ≤ ed(E_{7}; 2) ≤ 33 ([RY00], [CS06], [Mac11]). In this paper we improve the upper bound.

We will take the subgroup E_{6} ⋊ μ_{4} ⊂ E_{7} as defined in [Ga01, 3.5]. For every field extension l/k, inclusion induces a surjection H^{1}(l, E_{6}⋊μ_{4}) → H^{1}(l, E_{7}) (see also [Ga09, 12.13]); in other words, we can reduce the structure of E_{7}-torsors. In particular, this implies

ed(E_{7}) ≤ ed(E_{6} ⋊ μ_{4}) and ed(E_{7}; 2) ≤ ed(E_{6} ⋊ μ_{4}; 2).

The new upper bounds will be achieved by using these inequalities. As always, we have ed(E_{7}; 2) ≤ ed(E_{7}), but we do not know if this inequality is strict.

Consider E_{7}’s fundamental faithful representation of dimension 56 (also known as its minuscule representation). As an E_{6} representation it decomposes as k⊕k⊕V ⊕V^{∗}, where V and V^{∗} are E_{6}’s two fundamental representations of dimension 27 (also known as its minuscule representations), and k denotes a one-dimensional trivial representation. Let w ∈ μ_{4} be defined as in [Ga01, 3.5], sending an element (α_{1}, α_{2}, v_{1}, v_{2}) to (iα_{2}, iα_{1}, iv_{2}, iv_{1}), where i is a square root of −1. Conjugation by w is an outer automorphism of E_{6}, it preserves a split maximal torus T ⊂ E_{6}, and it fixes a subgroup isomorphic to F_{4} [Ga01, 2.4 and 3.5]. So V ⊕V^{∗} is faithful and irreducible as an E_{6}⋊μ_{4} representation, and also as an N_{E_{6}}(T) ⋊ μ_{4} representation. Also notice that w^{2} is in the centre of E_{7}. We will denote the set of T -weights of V ⊕ V^{∗} as Λ, which consists of all weights in T̂ of length 4/3.

1. Essential dimension

Definition 1.1. [PV89, 7] Let k be an algebraically closed field, and let G be an algebraic group acting on an irreducible variety X over k. We say that H ⊂ G is a stabilizer in general position (SGP) if there is a dense open subset U ⊂ X such that Stab_{G}(x) is conjugate to H for every x ∈ U.

Lemma 1.2 (Popov). Let X and Y be irreducible G-varieties. Assume H_{1} is an SGP for the G actions on X, and H_{2} is an SGP for the H_{1} action on Y. Then H_{2} is an SGP for the G action on X × Y.

Proof. Assume Stab_{G}(x) = g^{−1}H_{1}g and Stab_{H_{1}}(y) = h^{−1}H_{2}h. Then Stab_{G}(x, y) = Stab_{Stab_{G}(x)}(y) = Stab_{g^{−1}H_{1}g}(y) = gh^{1}H_{2}hg^{−1}. □

There is a series of subgroups as follows.

{1} ⊂ SL_{2} ⊂ SL_{3} ⊂ G_{2} ⊂ Spin_{7} ⊂ Spin_{8} ⊂ F_{4} ⊂ E_{6} ⊂ E_{7}.

We will abuse notation slightly and use the above symbols to refer to specific subgroups of these types, as described in the proofs of Propositions 1.3 and 1.4.

Several of the results in the following two propositions may be found in [Bo11, 1.5] for the complex numbers, and the arguments there are analogous to the ones used here. For the rest of this section we will assume the characteristic of k is not 2 or 3.

Proposition 1.3. Assume that k = k̄.

(1) F_{4} is an SGP of the E_{6} action on V,

(2) Spin_{8} is an SGP of the F_{4} action on V (or V^{∗}),

(3) Spin_{8} is an SGP of the E_{6} ⋊ μ_{4} action on V ⊕ V^{∗},

(4) Spin_{8} ×μ_{6} is an SGP of the E_{6} ⋊ μ_{4} action on ℙ(V ⊕ V^{∗}).

Proof. Recall the E_{6} ⋊ μ_{4} action on the 54 dimensional faithful representation V ⊕ V^{∗} described above. As in [Ga01, 3.5], each of V and V^{∗} come equipped with isomorphic Jordan algebra structures, which include a cubic form called the norm. E_{6} preserves both of those cubic forms, but as E_{6} representations V and V^{∗} are not isomorphic. The subgroup of E_{6} which fixes the identity element e in V (equivalently, in V^{∗}), is the automorphism group of the Jordan algebra structure, and is of type F_{4} (see also [SV00, 5.9.4 and Section 7]). The subgroup of F_{4} which fixes a chosen triple (e_{1}, e_{2}, e_{3}) of pairwise orthogonal primitive idempotents in V, is isomorphic to Spin_{8} [Ja68, p.378].

(1) Two elements with non-zero norms in V are in the same E_{6}-orbit iff their norms are equal ([Ja68, Chap. IX.6, exer. 6], or [SV00, Proof of Thm. 7.3.2]).

(2) The subgroup F_{4} ⊂ E_{6} is the group of automorphisms of the Jordan algebra structure on V (which is the same as that on V^{∗}). By Jacobson [Ja68, p.381], any element can be “diagonalized”, in other words, the F_{4}-orbit of any element non-trivially intersects the subspace spanned by e_{1}, e_{2}, and e_{3}. If y is in that subspace and its e_{i} coefficients are all distinct (an open condition), then Stab_{F_{4}}(w) ≅ Spin_{8}.

(3) We have e ∈ V, and choose y ∈ V^{∗} as above, such that the norm of y is not a fourth root of unity. Then (e, y) can be considered to be in general position; by Lemma 1.2, we see that Spin_{8} is the SGP of E_{6} on V ⊕ V^{∗}. For any g ∈ E_{6}, and i = 1, 2, 3, the assumption on the norm of y, together with the fact that E_{6} is norm preserving, tells us that gw^{i}(e, y) ≠ (e, y). The result follows.

(4) For e and y as above, consider the element [e, y] ∈ ℙ(V ⊕ V^{∗}). Notice that Stab_{F_{4}}([e, y]) ⊂ N_{F_{4}}(Stab_{F_{4}}(e, y)) = Spin_{8} ⋊S_{3}. Here the symmetric group S_{3} permutes the diagonal coefficients of e_{i} in the Jordan algebra V (and V^{∗}). Since the e_{i} coefficients of y are distinct, we see Stab_{F_{4}}([e, y]) = Spin_{8}. Also Stab_{E_{6}}([e]) = F_{4} × μ_{3} [Ga01, 3.4] (see also [Ga09, 9.12]), so Stab_{E_{6}}([e, y]) = Spin_{8} ×μ_{3}. Finally, if g ∈ E_{6} such that gw fixes [e, y], then y would have to be a scalar multiple of e, which is not the case. Therefore Stab_{E_{6}⋊μ_{4}}([e, y]) = Spin_{8} ×μ_{6}, where μ_{6} is the product of the centres of E_{6} and E_{7}. □

Spin_{8} has three inequivalent 8 dimensional irreducible representations. We will choose one and call it the vector representation, and denote it by Σ. Then the other two, 𝕆^{+}, 𝕆^{−}, will be called the half-spin representations. Also, 𝕆 will denote the 8 dimensional spin representation of Spin_{7}.

Proposition 1.4. Assume k = k̄.

(1) Spin_{7} is an SGP of the Spin_{8} action on Σ,

(2) G_{2} is an SGP of the Spin_{7} action on 𝕆,

(3) SL_{3} is an SGP of the G_{2} action on 𝕆,

(4) SL_{2} is an SGP of the SL_{3} action on 𝕆,

(5) SL_{2} acts generically freely on 𝕆.

Proof. (1) The stabilizer of a point with non-zero norm in the vector representation is a subgroup of Spin_{8} isomorphic to Spin_{7}. This is the copy of Spin_{7} ⊂ Spin_{8} that we will use below.

(2) As Spin_{7} representations, 𝕆^{+} and 𝕆^{−} are isomorphic to 𝕆. The claim follows from [Ig70, Prop. 4]; here G_{2} is the automorphism group of an octonion algebra structure on 𝕆.

(3) As G_{2} representations, Σ and 𝕆 are isomorphic. In the 7-dimensional space of elements orthogonal to the identity (also called pure octonions), G_{2} acts transitively on norm one elements, and the stabilizer of such an element x_{1} is SL_{3} (see also [KMRT, §36 Exer. 6]).

(4) SL_{3} acts transitively on norm one elements orthogonal to both the identity and x_{1}. The stabilizer of such an element x_{2} is SL_{2} (see also [KMRT, §36 Exer. 7]).

(5) SL_{2} acts transitively on norm one elements orthogonal to all of 1, x_{1}, x_{2}, and x_{1}x_{2}. The stabilizer of such an element x_{3} is trivial, because an automorphism of the octonions is determined by its action on the “basic triple” (x_{1}, x_{2}, x_{3}), and therefore SL_{2} acts generically freely on 𝕆. □

Remark 1.5. As noted in [Ig70, Prop. 1], if H is an SGP of Spin_{8} acting on the half-spin representation 𝕆^{+}, then H is isomorphic (but not conjugate) to Spin_{7} by a triality automorphism. And the Spin_{8} representation Σ restricted to H is the spin representation (rather than the vector Spin_{7}-representation).

Theorem 1.6. ed(E_{7}) ≤ 29.

Proof. From the introduction, we know that ed(E_{7}) ≤ ed(E_{6} ⋊ μ_{4}). Consider the rational map

(V ⊕ V^{∗})^{2} −−→ W := V ⊕ V^{∗} × ℙ(V ⊕ V^{∗}).

It is dominant and E_{6} ⋊ μ_{4}-equivariant. To show the upper bound on essential dimension, we just need to show that W is generically free, and for this we may assume that k = k̄.

By Lemma 1.2 and Proposition 1.3, we see that an SGP of the E_{6} ⋊ μ_{4} action on W is equal to an SGP of the Spin_{8} action on V ⊕ V^{∗}. As a Spin_{8} representation this decomposes as the sum of six trivial representations, together with two copies each of Σ, 𝕆^{+}, and 𝕆^{−}. Therefore, by combining Lemma 1.2 with Proposition 1.4, we see the SGP of this Spin_{8} action is trivial, and hence the action of E_{6} ⋊ μ_{4} on W is generically free. Therefore ed(E_{7}) ≤ ed(E_{6} ⋊ μ_{4}) ≤ 54 + 53 − 78 = 29. □

2. Essential 2-dimension

Lemma 2.1. [MR09, Lemma 3.3] Assume we have algebraic groups 1 → T → N → F → 1, such that T is a split torus, and F is finite. Let V be a faithful N-representation which decomposes into a sum of one-dimensional weight spaces for T, and let Λ be the associated set of weights. Consider the N-invariant open subset U = (𝔸^{1} −0)^{Λ} ⊂ 𝔸^{Λ} = V.

(1) If f ∈ F acts non-trivially on ker(ℤ[Λ] → T̂), then f acts non-trivially U/T.

(2) N/T acts faithfully on U/T if and only if N acts generically freely on V.

Proof. For the first statement, assume f acts trivially on U/T and choose an arbitrary Σc_{λ}λ ∈ ker(ℤ[Λ] → T̂). Choose an isomorphism between the varieties U ≅ (𝔾_{m})^{Λ}, inducing a multiplicative structure on U (which is not necessarily compatible with the action of N). Consider the functions U → k^{∗} defined in terms of the λ-coordinates u_{λ} of an arbitrary u ∈ U, as

ϕ_{1}(u) = ∏_{λ∈Λ}(u_{λ})^{c_{λ}}, ϕ_{2}(u) = ∏_{λ∈Λ}(u_{f·λ})^{c_{λ}}.

These functions are invariant under the action of T on U. Now choose n_{f} ∈ N in the preimage of f. Since n_{f} acts linearly on V, there are constants b_{λ} and c such that ϕ_{1}(u) = ϕ_{1}(n_{f} · u) = ∏(b_{λ}u_{f·λ})^{c_{λ}} = cϕ_{2}(u). Since u ∈ (𝔾_{m})^{Λ} was arbitrary, the only way ϕ_{1} and ϕ_{2} can be constant multiples of each other is if c_{λ} = f · c_{λ} for every λ ∈ Λ; in other words, f fixes Σc_{λ}λ.

For the second statement, assume N/T acts faithfully on U/T. For each non-trivial f ∈ N/T, choose a representative n_{f} ∈ N. The action of f on U/T is non-trivial, and therefore there is an open dense set U_{f} ⊂ U on which n_{f}t has no fixed points for any t ∈ T. The finite intersection ∩U_{f} is open and dense in U, and N acts with trivial stabilizers on it. In other words, N acts generically freely on U, and hence on V. For the converse, we just need to notice that a faithful representation of a finite group is always generically free. □

Remark 2.2. The above Lemma was proved in [MR09] when the exact sequence is split, in which case the converse of the first statement is also true; the standard representation of N_{SL_{2}}(T) is a counter-example to the converse of the first statement.

Theorem 2.3. ed(E_{7}; 2) ≤ 27.

Proof. This upper bound will be proved using the following sequence of inequalities:

ed(E_{7}; 2) ≤ ed(E_{6} ⋊ μ_{4}; 2) ≤ ed(N_{E_{6}}(T) ⋊ μ_{4}; 2) ≤ 33 − 6 = 27.

Since conjugation by μ_{4} preserves T ([Ga01, 2.4]), we have N_{E_{6}⋊μ_{4}}(T) = N_{E_{6}}(T) ⋊ μ_{4}. Now the middle inequality follows from [Se02, III.4.3, Lemma 6].

Choose a Sylow 2-subgroup W^{(2)} ⊂ W (E_{6}), and let N_{2} ⊂ N_{E_{6}}(T) ⋊ μ_{4} be the preimage of F := W^{(2)} ⋊ μ_{4}. Let σ be the square of a generator of μ_{4}; it is the nontrivial element in the centre of E_{7}. One can verify the following facts using a computer algebra program such as Magma.

• The F orbits of Λ are of size 2, 4, 16 and 32.

• The weights in the largest orbit Λ_{32} additively generate the weight lattice

• The kernel of the F action on ker(ℤ[Λ_{32}] → T̂) is ⟨σ⟩. So the subspace V_{32} ⊂ V ⊕ V^{∗} generated by the weight spaces of Λ_{32} is an irre-

ducible N_{2}-representation of dimension 32. Let V_{1} be the one-dimensional non-trivial representation of μ_{4}, which we also consider as an N_{2} representation via projection to μ_{4}. We claim that N_{2} acts generically freely on V_{32} × V_{1}.

For U ⊂ V_{32} × V_{1} as in the above Lemma, the third point implies that the F action on U/T has kernel contained in ⟨σ⟩. But σ acts non-trivially on V_{1}, and hence on U/T, since T acts trivially on V_{1}. So F acts faithfully on U/T, and hence by Lemma 2.1, the N_{2} action on V_{32} × V_{1} is generically free. Now we have the upper bound ed(N_{E_{6}}(T) ⋊ μ_{4}; 2) ≤ 33 − 6 = 27, and the result follows. □

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