ESSENTIAL p-DIMENSION OF THE NORMALIZER OF A MAXIMAL TORUS

MARK L. MACDONALD

Abstract. We compute the exact value for the essential p-dimension of the normalizer of a split maximal torus for most simple connected linear algebraic groups. These values give new upper bounds on the essential p-dimension of some simple groups, including some exceptional groups.

For each connected simple algebraic group, we also give an upper bound on the essential p-dimension of any torus contained in that group. These results are achieved by a detailed case-by-case analysis.

Introduction

Let p be a prime, k a field of characteristic not p (an assumption we make on all base fields), and let G be a split simple algebraic group over k. For us a simple algebraic group is connected, linear, and has no proper connected normal subgroups. Then one can ask what is the essential p-dimension of the normalizer of a split maximal torus in G, ed(N; p)? This question was answered in [MR09] for G = P GL_{n}. Their main motivation came from the upper bound ed(G; p) ≤ ed(N; p). In the present paper we give exact values for ed(N; p) for most split simple algebraic groups G. These results are consistent with Reichstein’s conjecture (see [Re10]). In some cases we obtain new bounds on ed(G; p). We also obtain upper bounds for the essential p-dimension of any torus inside a given (not necessarily split) simple algebraic group (see Theorem 1.7).

The four Tables in this paper summarize our main results, which follow from a detailed case-by-case analysis. In the remainder of this introduction, we explain how these tables give new information about essential dimension.

0.1. Essential dimension of normalizers. The essential dimension (resp. essential p-dimension) of an algebraic group G over a field k is a non-negative integer invariant, denoted by ed_{k}(G) (resp. ed_{k}(G; p)). Roughly speaking, it is the number of independent parameters needed to specify a G-torsor up to isomorphism (resp. and allowing for prime to p field extensions). In recent years, much work has gone into computing these numbers (see [Re10] for a recent survey).

Let T ⊂ G be a split maximal torus in a simple algebraic group over a field k. Let N = N_{G}(T) be its normalizer, then N/T ≅ W = W (R) the Weyl group of the root system R of G.

For p any prime, computing ed(N; p) is roughly equivalent to computing the symmetric p-rank, SymRank(ϕ; p), of the Weyl group action on the characters of a split maximal torus (see Definition 1.1). We will now explain how to find the exact value of, or at least the best bounds for ed(N; p).

Every split simple algebraic group G has an entry in Table I, II, or III (with the exception of Remark 2.3). For a prime p, the pair (G, p) occurs in the Tables iff the representation^{1}T̂ has symmetric p-rank strictly bigger than the rank of T. So if the pair (G, p) does not occur in the Tables, then by Theorem 1.9 we have ed(N; p) = ed(W; p). The essential p-dimensions of the Weyl groups are evaluated in Table IV.

If the pair (G, p) has an entry in one of these Tables, and the final column of this entry contains a “y”, then by Corollory 1.11 we have that ed(N; p) = SymRank(^{1}T̂; p)− dim(T). The only remaining possibility is that the final column of this entry contains an “n”, in which case the exact value for ed(N; p) is not always known; see Theorem 1.9 for upper and lower bounds, and the final section for a discussion of individual cases.

This work extends the results of [MR09], where ed(N; p) was computed for G = P GL_{n}, which appears in Table I as the cases ^{1}A_{p^{r}−1}^{+1} and ^{1}A_{p^{r}s−1}^{+1} (using notation of 1.2).

0.2. New upper bounds for ed(G; p). By [Se02, III.4.3 Lemma 6]^{1} together with [BF03, Lemma 1.3] we have

ed(G; p) ≤ ed(N; p).

In some cases this bound is sharp, such as G = SO_{2n+1}, p = 2, but in other cases the inequality is known to be strict. Nevertheless, it sometimes provides the best known upper bound for ed(G; p). In the following cases, the best bounds which had previously been written down were also bounds for the absolute essential dimension, given in [Le04]. For the classical groups:

ed(P SO_{2n}; 2), ed(P Sp_{2n}; 2) ≤ \{ \begin{matrix}4 · 2^{r}(s − 1) & for n = 2^{r}s, s > 1, \\ n^{2} & for n = 2^{r}. \end{matrix}

For n odd we already knew ed(P Sp_{2n}; 2) = n + 1, [Mac08]. Let E_{n} denote the split adjoint group of type E_{n}, then we have:

ed(E_{6}; 3) ≤ 21 ed(2E_{7}; 2) ≤ 33 ed(E_{7}; 2) ≤ 57
ed(E_{8}; 2) ≤ 120 ed(E_{8}; 3) ≤ 73.

1. Preliminaries

The rank of a free ℤ-module L is defined to be the minimal size of a generating subset of L. If F is a finite group which acts on L (in other words ϕ : F → GL(L) is an integral representation), then we can define the symmetric rank.

SymRank(ϕ) := min{ size of an F -invariant subset of L which is generating}.

For example, if F is the Weyl group of some irreducible root system R, and ϕ is the induced action on the root lattice L, then SymRank(ϕ) = |R_{0}|, the number of roots of minimal length. This follows from a case-by-case analysis as in [Le04]. Notice that the Weyl group acts transitively on the minimal length roots.

In the present paper we will be interested in the p-local version of this invariant, for a prime p. We will say a subset Λ ⊂ L is p-generating if it generates a ℤ-submodule X_{Λ} whose rank is rank L, and whose index (defined as |L/X_{Λ}|) is prime to p. Choose a Sylow p-subgroup Γ ⊂ F.

1This Lemma assumes the base field is perfect, but in our case this assumption can be dropped.

Definition 1.1. The symmetric p-rank of ϕ is defined as

SymRank(ϕ; p) := min{ size of a Γ-invariant subset of L which is p-generating}.

Notice that this number depends only on the integral representation ϕ, and the prime p, and not on the choice of Sylow p-subgroup (since all Sylow subgroups are conjugate). Clearly we have

rank(ϕ) ≤ SymRank(ϕ; p) ≤ SymRank(ϕ) ≤ |F | rank(ϕ).

1.2. Lattices. A lattice is often defined as a free ℤ-module with a positive-definite bilinear form. In the present paper, all lattices come from some irreducible root system, as the root lattice, the weight lattice, or as some intermediate lattice (see [Bou68], [Hu92]). Notice that different root systems may produce isomorphic lattices.

Given an irreducible root system R, its group of automorphisms obeys A(R) = W (R) ⋊ Aut(D), where W (R) is the Weyl group and Aut(D) is the group of graph automorphisms of the Dynkin diagram. If L is some lattice associated to R, then A(R) has an induced action on L.

Let M_{r} denote the (unique) subgroup of Aut(D) of order r. We will imitate the notation of [Ti66] for the following integral representation, given by the restriction of the induced A(R) action

(1.3)  ^{r}L := ϕ : W ⋊ M_{r} → GL(L).

1.3If L is a root lattice of type A, D, or E, then L^{+d} will denote an intermediate lattice in which L has index d (this is unique, except for type D, and d = 2, see Section 3). This follows the notation of [CS88].

We find it convenient to embed our lattices L inside a real vector space, with a fixed orthonormal basis {ϵ_{i}}.

1.4. Algebraic tori. An (algebraic) torus over a field k is an algebraic group T such that T_{sep} := T ×Spec(k_{sep}) is isomorphic to some copies of the multiplicative group 𝔾_{m} (see [Vo98] or [Bo92]). A torus which is isomorphic to 𝔾_{m}^{n} over k will be called split. Any torus becomes split after a finite Galois extension K. For any torus T defined over k, let A_{T} := Gal(K/k), where K is a minimal Galois splitting field of T. Then A_{T} is well-defined, and it is called the decomposition group of T. A_{T} naturally acts on the character module T̂ = Hom(T (K), K^{∗}), and we can associate the Galois representation ψ_{T} : A_{T} → GL(T̂). The main result from [LMMR10b, Cor. 5.1] is that

(1.5)  ed(T; p) = SymRank(ψ_{T}; p) − dim T.

A torus T whose character representation obeys SymRank(ψ_{T}) = dim(T) is called quasi-split. If we can choose a maximal torus T ⊂ G which is (quasi-)split, then G is said to be (quasi-)split. We also know that ed(T) ≤ SymRank(ψ_{T}) − dim T, but equality does not always hold (there are non-quasi-split tori with ed(T) = 0).

1.6. Tori in algebraic groups; generic tori. Let T ⊂ G be a maximal torus in a simple algebraic group. Then any element of the decomposition group A_{T} acts on T̂ by preserving the root system R of G_{sep}. In other words, A_{T} ⊂ A(R). Notice that the integral representation ^{r}T̂ from (1.3) is independent of whether T splits, and depends only on G_{sep} and r.

Theorem 1.7. Let T_{0} ⊂ G be any torus in a simple algebraic group over k, and let r = | Aut(D)| for the Dynkin diagram associated to the root system of G_{sep}. Choose a maximal torus T which contains T_{0}. Then

ed(T_{0}; p) = SymRank(ψ_{T_{0}}; p) − dim(T_{0}) ≤ SymRank(^{r}T̂; p) − dim(T_{0}).

Moreover, if char(k) = 0 then there exists a field extension K/k and a “generic” maximal torus T_{0} ⊂ G_{K} which achieves this upper bound.

Proof. We can choose a Sylow p-subgroup Γ_{0} ⊂ A_{T_{0}} contained in a Sylow p-subgroup Γ ⊂ A(R). Then a Γ-invariant p-generating subset of T̂ induces a Γ_{0}-invariant pgenerating subset of T̂_{0}, as required.

Now we will show that the upper bound is achieved for generic tori. Given a quasisplit maximal torus T ⊂ G, with N its normalizer, then the variety G/N may be thought of as the variety of maximal tori in G, and we can construct the tautological fibration S → G/N. The fibre over a point x ∈ G/N is a maximal torus S_{x} ⊂ G_{k(x)}. The fibre over the generic point is called the generic torus, and we will denote it T_{gen} ⊂ G_{k(G/N)}. Voskresenskii showed that, for char(k) = 0, we have A_{T_{gen}} ≅ W (R) ⋊ A_{T} [Vo88, Theorem 2].

We can choose a quasi-split maximal torus T ⊂ G_{k′} for some field extension k′/k, such that r := |A_{T} | = | Aut(D)|. By (1.5) we have ed_{K}(T_{gen}; p) = SymRank(^{r}T̂; p) − dim(T) for K = k′(G/N). Hence the upper bound is achieved for T_{0} = T_{gen}. □

Theorem 1.7 gives us motivation to compute the symmetric p-rank of the A(R)- action on the character lattices; the results can be found in Tables I, II, and III.

1.8. Normalizers of split tori. A representation ϕ : N → GL(V) is said to be pfaithful (resp. p-generically free), if ker(ϕ) is finite of order prime to p, and ϕ descends to a faithful (resp. generically free) representation of N/ ker(ϕ).

For the remainder of this section we will assume T ⊂ G is a split maximal torus in a simple algebraic group over k, and N = N_{G}(T), and W = W (R) ≅ N/T, where R is the root system of G. Also, H will denote a Sylow p-subgroup of W.

Theorem 1.9. Let T ⊂ G be a maximal and split torus over k, let N be its normalizer, and ^{1}T̂ the representation of the Weyl group from (1.3). Then we have the lower bounds:

max{SymRank(^{1}T̂; p) − dim(T), ed(W; p)} ≤ ed(N; p)

We also have the upper bound:

ed(N; p) ≤ SymRank(^{1}T̂; p) − dim(T) + ed(W; p).

In particular, if SymRank(^{1}T̂; p) = dim(T) then ed(N; p) = ed(W; p).

For the values of SymRank(^{1}T̂; p) see Tables I,II, and III, and for ed(W; p) see Table IV. Notice that ed(N; p) > 0 iff p divides the order of W.

Proof of lower bounds in Theorem 1.9. Firstly, notice that ed(W; p) ≤ ed(N; p) follows from the surjection H^{1}(k, N) → H^{1}(k, W) for quasi-split groups (see [Gi04, Thm. 5.1] or [Ra04]).

For a Sylow p-subgroup H ⊂ W, let N_{H} denote the preimage of H in N. Then by [MR09, Lemma 4.1] we have that ed(N_{H}; p) = ed(N; p). From [LMMR10a, Thm. 1.3(a)] we know there is a p-faithful (N_{H})_{alg}-representation V, defined of over an algebraic closure, such that dim(V) − dim(T) ≤ ed_{k}(N_{H}; p). Such a representation is also a p-faithful T_{alg}-representation. Decompose V = ⊕_{λ∈Λ}V_{λ} into weight spaces of T_{alg}, where Λ ⊂ T̂ is the set of non-trivial weights. Now Λ is p-generating (since V is p-faithful), and it is invariant under H. Therefore SymRank(^{1}T̂; p) ≤ |Λ| ≤ dim(V), as required. □

We will prove the upper bound of Theorem 1.9 by constructing a generically free representation from a subset of T̂. Let ϕ : F → GL(L) be a representation of a finite group, and let Λ ⊂ L be an F -invariant subset. The following condition on Λ will ensure generic freeness in Lemma 1.10(iii).

(K_{F}): The kernel of the F -map ℤ[Λ] → L is faithful as an F -module.

Fix a Sylow p-subgroup H ⊂ W, and let N_{H} ⊂ N be its preimage under π : N → W. The following construction slightly generalizes [MR09, Section 3], where only the case when N_{H} is a semi-direct product was considered.

Lemma 1.10. Let T be a split torus over k, and let Λ ⊂ T̂ be a finite H-invariant subset. Then there is an N_{H}-representation, V_{Λ}, obeying the following properties.

(i) V_{Λ} has a basis {v_{λ}}_{λ∈Λ}, on which T acts by tv_{λ} = λ(t)v_{λ}, (ii) Λ is p-generating iff V_{Λ} is p-faithful, (iii) Λ is p-generating and satisfies (K_{H}) iff V_{Λ} is p-generically free.

Proof. Choose a set of representatives {λ_{i}} ⊂ Λ, one for each H-orbit; let H_{i} := Stab_{H}(λ_{i}), and let N_{i} be the preimage of H_{i} in N_{H}. Now consider the irreducible Grepresentation L(λ_{i}) of highest weight λ_{i} [Ja87, II 2.4]. Since elements of N permute the weight spaces of any representation, we can define V_{i} := L(λ_{i})_{λ_{i}}, a one-dimensional N_{i}-representation (see [Ja87, II 2.4 Prop. (b)]). This is defined over k because λ_{i} is (since T is split). Then define

V_{Λ} := ⊕_{i} ind_{N_{i}}^{N_{H}} (V_{i}).

For each i, the corresponding summand has a basis {v_{λ}}_{λ∈H·λ_{i}}, where T acts as in (i).

For (ii), let M be the kernel of the representation N_{H} → GL(V_{Λ}) defined above; notice that M ⊂ T. Also, T → GL(V_{Λ}) factors through Diag(X_{Λ}), which acts faithfully on V_{Λ}. So by the anti-equivalence Diag, we see that M ≅ Diag(T̂/X_{Λ}). Therefore M is order prime to p iff Λ is p-generating.

To show (iii), first assume that Λ is generating. In this case, we know from [MR09, Lemma 3.3]^{2} that Λ satisfies (K_{H}) iff V_{Λ} is generically free. By (ii), the general case now follows. □

Proof of Thm. 1.9. All that remains is the upper bound; we want to construct a pgenerically free representation of N_{H} of the right dimension (by [LMMR10a, Section 1]). Given a Sylow p-subgroup H ⊂ W, we can choose an H-invariant p-generating subset of minimal size |Λ| = SymRank(^{1}T̂; p). By Lemma 1.10 this gives us a p-faithful N_{H}-representation, V_{Λ}. Let V be a faithful H-representation of dimension ed(W; p) (which is possible by [KM08], extending the base field if necessary). Then, as in [MR09, Lemma 3.2], V_{Λ} × V is a p-generically free N_{H}-representation, and thus we have the required upper bound on ed(N; p) = ed(N_{H}; p). □

Corollary 1.11. Let T ⊂ G be a split maximal torus, N its normalizer, and H ⊂ W a Sylow p-subgroup. If a minimal H-invariant p-generating subset of T̂ satisfies condition (K_{H}), then we have ed(N; p) = SymRank(^{1}T̂; p) − dim(T).

Proof. By Lemma 1.10 we have an N_{H}-representation of dimension SymRank(^{1}T̂; p) which is p-generically free. The result now follows from the lower bound of Theorem 1.9 and [LMMR10a, Section 1]. □

For F ⊂ A(R), we will now give a useful condition on an F -invariant subset Λ ⊂ T̂ which will imply the condition (K_{F}). We will say that λ_{0} ∈ Λ is Λ-independent if for any λ_{0} ≠ λ_{1} ∈ Λ we can choose coefficients a_{λ} ∈ ℤ such that ∑_{λ∈Λ} a_{λ}λ = 0 ∈ T̂ with a_{λ_{0}} ≠ 0 and a_{λ_{1}} = 0.

Lemma 1.12. Let Λ ⊂ T̂ be an F -invariant subset, such that rank(X_{Λ}) = rank T̂. If λ is Λ-independent, and h ∈ F, then h · λ is also Λ-independent. If every λ ∈ Λ is Λ-independent, then Λ satisfies (K_{F}).

Proof. The first statement is clear.

Assume every λ ∈ Λ is Λ-independent. Take an h ∈ F which acts trivially on the kernel of ℤ[Λ] → T̂. For every linear dependency ∑_{λ∈Λ} a_{λ}λ = 0 ∈ T̂, F permutes the elements {λ|a_{λ} ≠ 0}. By assumption, this implies h fixes every λ ∈ Λ. The span of Λ is full rank, and T̂ is a faithful F -module, so ℤ[Λ] is a faithful F -module, hence h = 1.

□

Lemma 1.13. Let ϕ : F → GL(L) be an integral representation of a finite group, and let L′ ⊂ L be an index prime to p sublattice (of equal rank). Then SymRank(ϕ; p) = SymRank(ϕ|_{L′}; p). Also, if Γ ⊂ F is a Sylow p-subgroup, then a minimal Γ-invariant p-generating subset of L satisfies (K_{Γ}) iff such a subset of L′ satisfies (K_{Γ}).

Proof. Firstly notice that if Λ is a minimal Γ-invariant p-generating subset of L′, then Λ is also p-generating for L; so SymRank(ϕ; p) ≤ SymRank(ϕ|_{L′}; p). Now let Λ be a minimal Γ-invariant p-generating subset of L. If we multiply every element in Λ by the index c = |L/L′|, then cΛ is Γ-invariant p-generating subset of L′. The final sentence of the Lemma now follows because ker(ℤ[Λ] → L) is a faithful Γ-module iff ker(ℤ[cΛ] → L′) is. □

2They assume N_{H} is a semi-direct product of T and H, but an identical proof works in general.

2. Type A_{n}

We will view the lattice A_{n} as the ℤ-submodule of Span_{ℤ}({ϵ_{i}}_{i=1}^{n+1}) of elements whose coordinates sum to zero. The Weyl group W (A_{n}) is the symmetric group on n+1 letters, which acts by permuting the coordinates ϵ_{i}. Negation is an outer automorphism of order 2.

This A_{n} is the character lattice of a maximal torus in P GL_{n+1}. To obtain the weight lattice, and the other intermediate lattices, let m|n + 1, and then adjoin the following vector:

(2.1)  v_{m} := \frac{1}{m}(1, ⋯, 1) − (0, ⋯, 0, 1, ⋯ 1).

Here the last \frac{n+1}{m} coordinates in the second summand are 1’s, so that the sum of the coordinates of v_{m} is zero. Define A_{n}^{+m} to be the span of A_{n} together with v_{m}. In particular A_{n}^{+1} is the root lattice, and A_{n}^{+(n+1)} is the weight lattice.

We may define an isomorphism A_{n}^{+m}/A_{n}^{+1} ≅ ℤ/mℤ by sending v_{m} to 1 ∈ ℤ/mℤ. Then, for any λ ∈ A_{n}^{+m}, we will denote by g(λ) ∈ ℤ/mℤ its projection to the quotient module, and we sometimes call this the glue part. Now one may check the following.

Lemma 2.2. For λ ∈ A_{n}^{+m}, we have

λ = \frac{g(λ)}{m}(1, ⋯, 1) + (a_{1}, ⋯, a_{n+1}),

such that a_{i} ∈ ℤ and ∑ a_{i} = −g(λ)\frac{n+1}{m}.

For the rest of this section we will write n = p^{r}s − 1, for s not divisible by p (and possibly equal to 1). Table I summarizes our computations of SymRank(ϕ; p) for A_{n}- type representations (using the notation as in 1.2). In this table we assume s ≠ 1, and that it is not divisible by p. The (K) column states whether the choice of minimal pgenerating Γ-invariant subset satisfies (K_{Γ}). Also, SU_{n+1} denotes the quasi-split (and not split) simply connected simple group of type A_{n} (see [Ti66, p. 55]).

Remark 2.3. For t|s and k ≤ r, we have that A_{p^{r}s−1}^{+p^{k}t} ⊂ A_{p^{r}s−1}^{+p^{k}} is a sublattice of index t, and hence prime to p. By Lemma 1.13, we have reduced to considering lattices with t = 1, because SymRank(^{1}A_{p^{r}s−1}^{+p^{k}}; p) = SymRank(^{1}A_{p^{r}s−1}^{+p^{k}t}; p).

The Sylow p-subgroup H_{r} := W (A_{p^{r}−1})^{(p)} is given by an iterated wreath product of cyclic p-groups, as in [MR09]. The H_{r}-action on ℝ^{p^{r}} naturally partitions the p^{r} coordinates into blocks of size p^{i}, for 0 ≤ i ≤ r. We will call these p^{i}-blocks, and we may consider them as elements of ℝ^{p^{i}}; for each i the number of p^{i}-blocks is p^{r−i}.

The H_{r}-action restricted to a single p^{i}-block is given by the H_{i} := W (A_{p^{i}−1})^{(p)}-action on ℝ^{p^{i}}. Two p^{i}-blocks are equivalent if there is an element of H_{i} that maps one into the other. We will say a p^{i}-block is j-stable if every p^{j}-block (0 ≤ j ≤ i) is equivalent to each other. We will call such a block scalar if it is 0-stable. If the coordinates are all integers, then we can consider the related terms mod p; for example, scalar mod p means all the coordinates are congruent mod p.

For s ≥ 1 not divisible by p, we may partition the coordinates of ℝ^{p^{r}s} into s different p^{r}-blocks. We choose a Sylow p-subgroup H := W (A_{p^{r}s−1})^{(p)}, such that H restricted

ϕ  G  p Conditions  SymRank(ϕ; p)  (K)
1A_{p^{r}−1}^{+p^{k}}  \frac{SL_{p^{r}}}{μ_{p^{r−k}}}  ≠ 2 1 ≤ k = r  p^{r}  n
1 ≤ k ≤ r − 1 p^{2r−1} + p^{k}  y
k = 0, r ≥ 2 p^{2r−1}  y
k = 0, r = 1 p^{2r−1} = p  n
1A_{2^{r}−1}^{+2^{k}} \frac{SL_{2^{r}}}{μ_{2^{r−k}}} 2 1 ≤ k = r 2^{r} n
2 ≤ k = r − 1 2^{2r−2} + 2^{k}  y
1 = k = r − 1 2^{2r−2} + 2^{k} = 6  n
1 ≤ k ≤ r − 2 2^{2r−1} + 2^{k}  y
k = 0, r ≥ 2 2^{2r−1}  y
k = 0, r = 1 2^{2r−1} = 2  n
2A_{2^{r}−1}^{+2^{k}} \frac{SU_{2^{r}}}{μ_{2^{r−k}}} 2 2 ≤ k = r 2^{r+1} y
2 ≤ k ≤ r − 1 2^{2r−1} + 2^{k+1}  y
1 = k ≤ r − 2 2^{2r−1} + 2  y
1 = k = r − 1 2^{2r−2} + 2^{k} = 6  n
k = 0, r ≥ 2 2^{2r−1}  y
1A_{p^{r}s−1}^{+p^{k}} \frac{SL_{p^{r}s}}{μ_{p^{r−k}s}} any 1 ≤ k = r p^{r}s n
1 ≤ k ≤ r − 1 p^{2r}(s − 1) + p^{k}  y
k = 0, r ≥ 1 p^{2r}(s − 1)  y
2A_{2^{r}s−1}^{+2^{k}} \frac{SU_{2^{r}s}}{μ_{2^{r−k}s}} 2 1 ≤ k = r 2^{r+1}s y
1 ≤ k ≤ r − 1 2^{2r+1}(s − 1) + 2^{k+1}  y
k = 0, r ≥ 1 2^{2r+1}(s − 1)  y
k = 0 = r  2^{2r+1}(s − 1) = 2(s − 1) n

Table I. A_{n} lattices

to any p^{r}-block is given by H_{r} (see [MR09]); so that (H_{r})^{s} ⊂ H. For s = 1, we have (H_{r})^{s} = H.

We will consider a map Σ : ℝ^{p^{r}s} → ℝ^{p^{r−1}s}, whose i^{th} coordinate is given by the sum of the coordinates of the i^{th} p-block. We will abuse notation slightly and write compositions of such maps as Σ^{i} : ℝ^{p^{r}s} → ℝ^{p^{r−i}s}. In particular, if s = 1, then Σ^{r} is simply the sum of all coordinates.

Lemma 2.4. Let λ ∈ ℝ^{p^{r}s}, where s is not divisible by p. For any 0 ≤ i ≤ r, we have

|(H_{r})^{s}λ| ≥ |Σ^{i}((H_{r})^{s}λ)| · |(H_{i})^{p^{r−i}s}λ|.

Proof. Notice that (H_{i})^{p^{r−i}s} stabilizes the fibers of Σ^{i}. The result follows. □

Lemma 2.5. Let λ ∈ ℝ^{p^{r}s}, where s is not divisible by p. If λ contains a non-scalar p^{k}-block (k ≤ r), then |(H_{r})^{s}λ| ≥ p^{r−k+1}.

Proof. First we prove the lemma for k = 1. Assume there is a non-scalar p-block, say B_{1}. For all 1 ≤ i ≤ r, let B_{i} be the unique p^{i}-block which contains B_{1}.

To get a lower bound on the number of non-scalar p-blocks, start with the block B_{1}, which is non-scalar. If B_{2} is 1-stable, then we know there are at least p non-scalar p-blocks, since the other p-blocks in B_{2} must also be non-scalar. Similarly, for each i such that B_{i} is not (i − 1)-stable, we get another factor of p. Let m be the number of indices 1 ≤ i ≤ r for which B_{i} is (i − 1)-stable. Then there are at least p^{m} non-scalar p-blocks in λ.

Now we also have r − m − 1 indices i ≥ 2 for which the p^{i−1}-blocks in B_{i} are not equivalent. For such an i, the p inequivalent blocks in B_{i} are cyclically permuted by ℤ/p ⊂ (H_{r})^{s}, and have different images under Σ^{1}. Now we can apply Lemma 2.4 to get

|(H_{r})^{s}λ| ≥ |Σ^{1}((H_{r})^{s}λ)| · |(H_{1})^{p^{r−1}s}λ| ≥ p^{r−m−1}p^{p^{m}} ≥ p^{r−m−1}p^{m+1} = p^{r}.

Here we have used the inequality p^{m} ≥ m + 1, which is valid for m ≥ 0 and p ≥ 2. This proves the k = 1 case.

Now assume λ contains a non-scalar p^{k}-block. If k = 1, then we have already shown the result, so assume every p-block is scalar. Then |(H_{r})^{s}λ| = |Σ^{1}((H_{r})^{s}λ)| = |(H_{r−1})^{s}Σ^{1}(λ)|, and Σ(λ) contains a non-scalar p^{k−1}-block. So, by induction, |(H_{r})^{s}λ| ≥ p^{(r−1)−(k−1)−1} = p^{r−k−1}. □

Lemma 2.6. Let p = 2 and k ≥ 2. For then any λ ∈ A_{2^{r}s−1}^{+2^{k}} such that s and g(λ) are both odd, we have −λ ∉ H · λ. In other words, the (H × ℤ/2)-orbit of λ is strictly bigger than the H-orbit.

Proof. This is simply because g(hλ) ≡ g(λ) mod 2^{k}, but g(−λ) ≡ −g(λ) ≢ g(λ) mod 2^{k}. □

Lemma 2.7. Let λ ∈ A_{p^{r}s−1}^{+p^{k}}, with s not divisible by p, and g(λ) ≢ 0 mod p.

(i) Then |Hλ| ≥ p^{k}.
(ii) If p = 2, and either k ≥ 2 or s > 1 then |(H × ℤ/2)λ| ≥ 2^{k+1}.

Proof. By Lemma 2.2, we have λ = \frac{g(λ)}{p^{k}}(1, ⋯, 1) + (a_{1}, ⋯, a_{p^{r}s}), such that ∑ a_{i} = −g(λ)p^{r−k}s, and a_{i} ∈ ℤ. Here ∑ a_{i} is not divisible by p^{r−k+1}, so there is a p^{r−k+1}-block which is non-scalar, and hence by Lemma 2.5 we have |Hλ| ≥ p^{r−(r−k+1)+1} = p^{k}.

The p = 2, k ≥ 2 case now follows, because the (H × ℤ/2)-orbit has order a power of 2, and is strictly bigger than the H-orbit, by Lemma 2.6.

For p = 2, s > 1, we have the H_{r}-orbit of some 2^{r}-block is at least 2^{k}. For any other 2^{r}-block, this copy of H_{r} acts trivially, but since g(λ) ≢ 0 mod 2, negation acts non-trivially, hence increasing the (H × ℤ/2)-orbit size by a factor of 2. □

Lemma 2.8. Assume λ ∈ A_{p^{m}s−1}.

(i) If s = 1 and Σ^{m−1}(λ) ≢ 0 mod p, then |Hλ| ≥ p^{2m−1}.

(ii) If s > 1 not divisible by p, and Σ^{m}(λ) ≢ 0 mod p, then |(H_{m})^{s}λ| ≥ p^{2m}. If

also p = 2, then |((H_{m})^{s} × ℤ/2)λ| ≥ 2^{2m+1}.

Proof. For (i) notice that Σ^{m−1}(λ) is non-scalar, and therefore |Σ^{m−1}(Hλ)| = p.

Since the sum of the coordinates is zero, there are two p^{m−1}-blocks whose coordinates sums are not divisible by p, and hence each contain a non-scalar p-block. Now combine Lemma 2.4 and Lemma 2.5 to see

|Hλ| ≥ |Σ^{m−1}(Hλ)| · |(H_{m−1})^{p}λ| ≥ p(p^{m−1})(p^{m−1}) = p^{2m−1}.

For (ii), similar to case (i), there must be two p^{m}-blocks each containing a non-scalar p-block. So by Lemma 2.5, we have |(H_{m})^{s}λ| ≥ (p^{m})(p^{m}) = p^{2m}.

For p = 2, notice (H_{m})^{s} acts trivially on Σ^{m}(λ), but negation acts non-trivially, hence increasing the orbit size by a factor of two. □

Remark 2.9. Part (i) is also proved in [MR09].

Given a set of elements S in ℤ^{N}, and a number x ∈ ℤ, the notation S = x (resp. S ≡ x mod p) will mean that every coordinate of every element in S is equal to (resp. congruent mod p to) the number x.

Lemma 2.10. Let λ ∈ A_{p^{r}s−1}^{+p^{k}}, with s not divisible by p, and 1 ≤ k ≤ r − 1.

(i) Assume s = 1. If Σ^{r−1}(λ) ≢ 0 mod p and Σ^{k}(λ) ≢ g(λ) mod p, then |Hλ| ≥

p^{2r−1}.

(ii) Assume s > 1. If Σ^{r}(λ) ≢ 0 mod p, then |(H_{r})^{s}λ| ≥ p^{2r}. If also p = 2, then

|((H_{r})^{s}) × ℤ/2)λ| ≥ 2^{2r+1}.

Proof. From Lemma 2.2 we can write any p^{k}-block as B = \frac{g(λ)}{p^{k}}(1, ⋯, 1)+(b_{1}, ⋯, b_{p^{k}}), where b_{i} ∈ ℤ. By our assumptions, there must be at least two p^{k}-blocks B_{1}, B_{2} in λ that are not sent to g(λ) mod p, under Σ^{k}. In other words, g(λ) + ∑ b_{i} ≢ g(λ) mod p. In particular, the p-blocks inside B_{i} are not all scalar. Therefore we can apply Lemma 2.5 to see |H_{k}B_{i}| ≥ p^{k}.

For (i), since Σ^{k}(λ) ∈ A_{p^{r−k}−1}, from Lemma 2.8(i), we see |Σ^{k}(Hλ)| ≥ p^{2(r−k)−1}. Now apply Lemma 2.4:

|Hλ| ≥ |Σ^{k}(Hλ)| · |(H_{k})^{p^{r−k}}λ| ≥ p^{2(r−k)−1}(p^{k})(p^{k}) = p^{2r−1}.

For (ii), similarly by Lemma 2.8(ii), we see |Σ^{k}((H_{r})^{s}λ)| ≥ p^{2(r−k)}. Therefore, by Lemma 2.4:

|(H_{r})^{s}λ| ≥ |Σ^{k}((H_{r})^{s}λ)| · |(H_{k})^{p^{r−k}s}λ| ≥ p^{2(r−k)}(p^{k})(p^{k}) = p^{2r}.

Finally, for p = 2, notice (H_{r})^{s} acts trivially on Σ^{r}(λ), but negation acts non-trivially, hence increasing the orbit size by a factor of two. □

2.11. Case ^{1}A_{p^{r}−1}^{+p^{k}}, p ≠ 2 or p = 2. The arguments for p = 2 and p ≠ 2 are mostly the same, so we consider them simultaneously. We use the Sylow subgroup H = H_{r}, as defined above.

Lemma 2.12. Let Λ ⊂ A_{p^{r}−1}^{+p^{k}} be H-invariant and p-generating, with 1 ≤ k ≤ r − 1. Then there are distinct H-orbits, Δ_{0}, Δ_{1} ⊂ Λ such that Σ^{r−1}(Δ_{0}) ≢ 0 mod p and g(Δ_{1}) ≢ 0 mod p, such that one of the two orbits obeys Σ^{r−1}(Δ_{i}) ≢ g(Δ_{i}) mod p.

Proof. Consider the map

π : (\frac{1}{p^{r}} ℤ)^{p^{r}} → (ℤ/p)^{p} × ℤ/p
λ ↦ (Σ^{r−1}(λ), g(λ)) mod p.

Then π(A_{p^{r}−1}^{+p^{k}}) is a rank p abelian group. Therefore, if Λ is p-generating, π(Λ) is also a rank p abelian group. But any H-orbit has image under π either rank p − 1 or rank 1. H-orbits whose image have rank 1 come from elements whose image under Σ^{r−1} have equal coordinates mod p.

So we can choose two orbits in Λ such that the two together generate a rank p image under the map π. Therefore one of the two orbits obeys Σ^{r−1}(Δ_{i}) ≢ g(Δ_{i}) mod p.

At least one of these two orbits has non-trivial glue part, g(Δ_{i}); choose one and call it Δ_{1}, and the other one Δ_{0}. At least one of the two orbits must obey Σ^{r−1}(Δ_{i}) ≢ 0 mod p. If Δ_{0} obeys this condition, then we are done.

Otherwise, Δ_{0} has image rank 1, and therefore it has non-trivial glue part. Now by swapping the names of Δ_{0} and Δ_{1}, we have found the required pair of orbits. □

We will say that a pair of orbits Δ_{0}, Δ_{1} from Lemma 2.12 can be swapped if Σ^{r−1}(Δ_{1}) ≢ 0 mod p, and g(Δ_{0}) ≢ 0 mod p. In this case we may relabel Δ_{0} as Δ_{1}, and vice versa.

Lemma 2.13. Given a pair of H-orbits, Δ_{0} and Δ_{1} from Lemma 2.12, one of the following two conditions is satisfied (possibly after swapping Δ_{0} and Δ_{1}):

(a) Σ^{k}(Δ_{0}) ≢ g(Δ_{0}) mod p.

(b) Σ^{k}(Δ_{0}) ≡ g(Δ_{0}) mod p, k = r − 1, and Σ^{r−1}(Δ_{1}) = 0.

Proof. Assume that such a pair doesn’t obey (a), even after swapping the roles of Δ_{0} and Δ_{1}, if possible. We must show the pair obeys (b).

So we have that Σ^{k}(Δ_{0}) ≡ g(Δ_{0}) mod p. Notice that if k < r − 1, then this would imply Σ^{r−1}(Δ_{0}) ≡ 0 mod p, a contradiction. So we have k = r − 1, and therefore g(Δ_{0}) ≢ 0 mod p.

Now we know by Lemma 2.12 that Σ^{r−1}(Δ_{1}) ≢ g(Δ_{1}), and so (by assumption), Δ_{0} and Δ_{1} can’t be swapped. Therefore Σ^{r−1}(Δ_{1}) ≡ 0 mod p. Now this implies Σ^{r−1}(Δ_{1}) = 0, and so we are done. □

Lemma 2.14. Let λ ∈ A_{p^{r}−1}^{+p^{k}} with k = r − 1 ≠ 0, such that Σ^{r−1}(λ) = 0 and g(λ) ≢ 0 mod p. Then |Hλ| ≥ p^{p(r−1)}.

Proof. Write λ = \frac{g(λ)}{p^{r−1}} (1, ⋯, 1) + (a_{1}, ⋯, a_{p^{r}}), as in Lemma 2.2. Consider the p^{r−1}- blocks of λ. Since Σ^{r−1}(Δ_{1}) = 0, we sum the coordinates of the first p^{r−1}-block to get ∑ a_{i} = −g ≢ 0 mod p. In particular, each p^{r−1}-block contains a p-block which is not scalar. So by Lemma 2.5, the H_{r−1}-orbit of each p^{r−1}-block is at least p^{r−1}, and therefore by Lemma 2.4, we have |Hλ| ≥ |(H_{r−1})^{p}λ| ≥ (p^{r−1})^{p}, as required. □

Corollary 2.15. Let Λ ⊂ A_{p^{r}−1}^{+p^{k}} be H-invariant and p-generating, with 1 ≤ k ≤ r − 1. Then |Λ| ≥ min{p^{2r−1} + p^{k}, p^{p(r−1)} + p^{k}}.

Proof. By Lemma 2.12 we can find two orbits Δ_{0} and Δ_{1}, which obey either case (a) or (b) from Lemma 2.13. In case (a), we have |Δ_{0}| ≥ p^{2r−1} by Lemma 2.10, and |Δ_{1}| ≥ p^{k} by Lemma 2.7. In case (b) we have |Δ_{0}| ≥ p^{k} by Lemma 2.7, and |Δ_{1}| ≥ p^{p(r−1)} by Lemma 2.14. □

Theorem 2.16. Let Λ ⊂ A_{p^{r}−1}^{+p^{k}} be H-invariant and p-generating. Then

|Λ| ≥ \{ \begin{matrix}p^{r} & 1 ≤ k = r \\ p^{2r−2} + p^{k} & 1 ≤ k = r − 1, p = 2 \\ p^{2r−1} & k = 0 \\ p^{2r−1} + p^{k} & otherwise \end{matrix}

Moreover, there exist Λ such that these bounds are achieved.

Proof. For 1 ≤ k = r, we must have an element λ such that g(λ) ≢ 0 mod p, so the lower bound follows from Lemma 2.7. Let Λ = H · v_{p^{r}}, where v_{p^{r}} is defined in (2.1). This set is p-generating and H-invariant, and has size p^{r}; so the bound is achieved.

For p = 2 and k = r − 1, notice this is the only case where the quantity p^{p(r−1)} + p^{k} is less than the quantity p^{2r−1} +p^{k}, from Corollory 2.15. To see this bound is achieved, take Δ_{0} = H · v_{2^{k}}, where v_{p^{k}} is defined in (2.1); then Δ_{0} has size 2^{r−1}. Also let Δ_{1} = H · (∑_{i=1}^{2^{r}} \frac{1}{2^{r−1}} ϵ_{i} − ϵ_{1} − ϵ_{1+2^{r−1}}), where the two 2^{k}-blocks are equivalent; this orbit has size 2^{r−1}2^{r−1}. One checks that Λ = Δ_{0} ∐ Δ_{1} generates the lattice, so the lower bound is achieved.

The k = 0 bound follows from Lemma 2.8. The bound is achieved by choosing Λ = H · λ for

(2.17)  λ := (\underbrace{1, 0, ⋯, 0,}_{p^{r−1}-block} \underbrace{−1, 0, ⋯, 0,}_{p^{r−1}-block} 0, ⋯, 0) = ϵ_{1} − ϵ_{1+p^{r−1}} .

This is the same choice as in [MR09, 6 and 7].

For all other cases, by Corollory 2.15 we have |Λ| ≥ p^{2r−1} + p^{k}. To see this bound is achieved, take λ as in (2.17). Then Δ_{0} = H · λ has size p^{2r−1}, as in the k = 0 case. Now take Δ_{1} = H · v_{p^{k}}, which has size p^{k}. One checks that Λ = Δ_{0} ∐ Δ_{1} generates the lattice. □

Now we show when the choices from the proof of Theorem 2.16 satisfy (K_{H}). Firstly, the choice for 1 ≤ k = r does not satisfy (K_{H}), since the only linear relation among the elements of Λ is that they all sum to zero.

For the p = 2, 1 ≤ k = r − 1 case, let σ := \frac{1}{2^{r−1}} ∑_{i=1}^{2^{r}} ϵ_{i}, and assume r ≥ 3. Then we have the following equations among elements of Λ:

(σ − ϵ_{1} − ϵ_{1+2^{r−1}}) + (σ − ϵ_{i} − ϵ_{i+2^{r−1}}) − (σ − ϵ_{i} − ϵ_{1+2^{r−1}}) − (σ − ϵ_{1} − ϵ_{i+2^{r−1}}) = 0 (σ − ϵ_{1} − ϵ_{2}) + (σ − ϵ_{j+2^{r−1}} − ϵ_{j+1+2^{r−1}}) − (σ − ϵ_{1} − ϵ_{j+2^{r−1}}) − (σ − ϵ_{2} − ϵ_{j+1+2^{r−1}}) = 0.

These equations are valid for i = 2, 3 and j = 1, 3 (here we have used that r ≥ 3). They show that (σ − ϵ_{1} − ϵ_{1+2^{r−1}}) is Λ-independent, and that (σ − ϵ_{1} − ϵ_{2}) is Λ-independent. So by Lemma 1.12 we see that (K_{H}) is satisfied. For r = 2, one checks that (K_{H}) is not satisfied.

For k = 0, let γ ∈ H be the order p permutation given by sending ϵ_{i} ↦ ϵ_{i+p^{r−1}}, where ϵ_{i+p^{r}} = ϵ_{i}. Assume r ≥ 2, and consider the following relations among elements of Λ:

(ϵ_{1} − ϵ_{j}) + (ϵ_{2} − ϵ_{j+1}) − (ϵ_{2} − ϵ_{j}) − (ϵ_{1} − ϵ_{j+1}) = 0 (2.18)

(ϵ_{1} − ϵ_{j}) + γ(ϵ_{1} − ϵ_{j}) + ⋯ + γ^{p−1}(ϵ_{1} − ϵ_{j}) = 0. (2.19)

Here j = 1 + p^{r−1}. This shows (ϵ_{1} − ϵ_{j}) is Λ-independent, and hence by Lemma 1.12 it satisfies (K_{H}). For r = 1, (K_{H}) is not satisfied.

In all other cases, we can argue as follows. Let σ = \frac{1}{p^{k}} ∑_{i=1}^{p^{r}} ϵ_{i}, and let γ be as in the k = 0 case. Then consider the following equations:

(σ − ∑_{i=1}^{p^{r−k}} ϵ_{i}) − (σ − ∑_{i=1}^{p^{r−k}} γ^{j} ϵ_{i}) + ∑_{i=1}^{p^{r−k}}(ϵ_{i} −γ^{j} ϵ_{i}) = 0 (2.20)

(σ − ∑_{i=1}^{p^{r−k}} ϵ_{i}) − (σ − ∑_{i=1}^{p^{r−k}} ϵ_{i+p^{r−k}}) + ∑_{i=1}^{p^{r−k}}(ϵ_{i} −γ ϵ_{1}) − ∑_{i=1}^{p^{r−k}}(ϵ_{i+p^{r−k}} −γ ϵ_{1}) = 0. (2.21)

If p ≥ 3 then we can use j = 1, 2 in the first equation. The second equation is only valid if k ≥ 2, in which case it is different from the first. So for p ≥ 3 or k ≥ 2 we have show that (σ − ∑_{i=1}^{p^{r−k}} ϵ_{i}) is Λ-independent. In the only remaining case, p = 2 and k = 1 we also have the following equation, which shows Λ-independence:

(σ − ∑_{i=1}^{2^{r−1}} ϵ_{i}) + (σ − ∑_{i=1}^{2^{r−1}} ϵ_{i+2^{r−1}}) = 0. (2.22)

We already know (ϵ_{1} − ϵ_{1+p^{r−1}}) is Λ-independent from the k = 0 case (notice that r = 1 doesn’t occur in the present case). So our choice of Λ satisfies (K_{H}), by Lemma 1.12.

2.23. Case ^{2}A_{2^{r}−1}^{+2^{k}}, p = 2. As above we will use the notation H := H_{r} = W (A_{2^{r}−1})^{(2)}, and let Γ = H × ℤ/2, where the generator for ℤ/2 is the outer automorphism on the lattice given by negation.

For k = 0 or 1, one checks that (for each value of r) the Λ from Theorem 2.16 is invariant under negation, so the value SymRank(ϕ; 2) is unchanged from the ^{1}A_{2^{r}−1}^{+2^{k}} case. Furthermore, the condition (K_{Γ}) is equivalent the condition (K_{H}), by Lemma 1.12.

For k ≥ 2, by Lemma 2.7 any λ in the lattice with g(λ) odd has Γ-orbit size at least 2^{k+1}. Now we can modify the proof of Theorem 2.16 to obtain the desired lower bounds. Furthermore, these bounds are achieved by taking the Γ-orbits of the same elements as in Theorem 2.16 (instead of just the H-orbits). Also (K_{H}) implies (K_{Γ}) by Lemma 1.12. Now Λ contains negatives of elements, so we get relations of the form λ + (−λ) = 0, which ensures that in the case k = r condition (K_{Γ}) is satisfied.

2.24. Case ^{1}A_{p^{r}s−1}^{+p^{k}}, with s > 1 not divisible by p. We will use H := W (A_{p^{r}s−1})^{(p)} as described near the start of this section. We will also denote σ := \frac{1}{p^{k}} ∑_{i=1}^{p^{r}s} ϵ_{i}.

Lemma 2.25. Let Λ ⊂ A_{p^{r}s−1}^{+p^{k}} be p-generating, with 1 ≤ k ≤ r, and s > 1 not divisible by p. Then either Σ^{r}(Λ) contains at least s elements whose reduction mod p is not the scalar zero, or Σ^{r}(Λ) contains s − 1 such elements, as well as one whose glue part is not zero mod p.

Proof. Consider the the map

π : (\frac{1}{p^{r}} ℤ)^{p^{r}s} → (ℤ/p)^{s} × ℤ/p
λ → (Σ^{r}(λ), g(λ)) mod p.

The image π(A_{p^{r}s−1}^{+p^{k}}) is a rank s abelian group, so for Λ to be p-generating, π(Λ) must generate a rank s group, and in particular the image contains s generators. At most one of these generators is zero mod p on the (ℤ/p)^{s} component. This proves the lemma. □

Theorem 2.26. Let Λ ∈ A_{p^{r}s−1}^{+p^{k}} be H-invariant and p-generating, where s > 1 is not divisible by p. Then

|Λ| ≥ \{ \begin{matrix}p^{r}s & 1 ≤ k = r \\ p^{2r}(s − 1) + p^{k} & 1 ≤ k ≤ r − 1 \\ p^{2r}(s − 1) & k = 0 \end{matrix}

Moreover, in each case there exists a Λ such that these bounds are achieved.

Proof. For k = r, assume Λ is p-generating. We can apply Lemma 2.25 and get s elements of Λ which obey either Σ^{r}(λ) ≢ 0 mod 2 or g(λ) ≢ 0 mod 2. If the latter, then by Lemma 2.7 we have |(H_{r})^{s}λ| ≥ p^{r}. If g(λ) = 0 mod 2, then λ ∈ A_{2^{r}s−1}^{+2^{k−1}}, and we can apply Lemma 2.10 we see |(H_{r})^{s}λ| ≥ p^{2r} > p^{r}. Since (H_{r})^{s} acts trivially on Σ^{r}(Λ), we get that |Λ| ≥ p^{r}s. To see this value is achieved, choose Λ = {σ − s ϵ_{i}}_{i=1}^{p^{r}s}.

For 1 ≤ k ≤ r − 1, we combine Lemma 2.25 and Lemma 2.7 and Lemma 2.10 to get |Λ| ≥ min{p^{2r}s, p^{2r}(s − 1) + p^{k}} = p^{2r}(s − 1) + p^{k}.

To show this is achieved, consider the set:

(2.27)  Λ_{0} = ⋃_{1≤i≤p^{r}<j≤p^{r}s}{ϵ_{i} − ϵ_{j}}.

This set is H-invariant, and has p^{2r}(s − 1) elements. All that remains is to find an element with g(λ_{1}) ≢ 0 mod p whose orbit is p^{k}. Since we have assumed the first p^{r}-block is a big block, then λ_{1} = σ − s ∑_{i=1}^{p^{r−k}} ϵ_{i} is such an element. So Λ = Λ_{0} ∪ Hλ_{1} will do.

For k = 0, as in Lemma 2.25, we can find s − 1 distinct non-zero elements in Σ^{r}(Λ), and by Lemma 2.10 each has (H_{r})^{s}-orbit size at least p^{2r}. Since (H_{r})^{s} acts trivially on Σ^{r}(Λ), we have shown the lower bound. To see is achieved, use Λ = Λ_{0} from (2.27) above. □

Remark 2.28. The k = 0 case was also covered in [MR09, 8].

Now we will determine when these choices satisfy (K_{H}). For 1 ≤ k = r, the condition is not satisfied.

For k = 0, similar to the k = 0 case when s = 1, we define γ ∈ H by sending ϵ_{i} ↦ ϵ_{i+p^{r−1}}, unless i is divisible by p^{r}, in which case ϵ_{i} ↦ ϵ_{i+p^{r−1}−p^{r}}. This γ has order p. Now for r ≥ 1, the equation (2.18) is valid if we instead use j = 1 + ip^{r}, for any 1 ≤ i ≤ s − 1. If s ≥ 3 then we get at least two equations, which show that (ϵ_{1} − ϵ_{j}) is Λ-independent for each of these j; furthermore Λ is the union of the H-orbits of these elements. If s = 2, then p ≥ 3, so we can modify equation (2.18) to get our second equation. In either case, by Lemma 1.12, we have shown (K_{H}) is satisfied. Notice that for r = 0 we have SymRank(ϕ; p) = s − 1 = rank(ϕ).

Finally, for 1 ≤ k ≤ r − 1, by using the argument from the k = 0 case we see that every element of Λ_{0} is Λ-independent. Similar to the s = 1 case, if we insert a factor of s to all of the summation signs in equations (2.20),(2.21),(2.22), then an identical argument goes through. So (K_{H}) is satisfied.

2.29. Case ^{2}A_{2^{r}s−1}^{+2^{k}}, with s > 1 odd, and p = 2. Let Γ = (H × ℤ/2). The lower bounds for the size of Γ-invariant 2-generating subsets of the lattice are the same as Theorem 2.26, except that negation adds another factor of two by Lemma 2.7 and Lemma 2.10.

To see they are achieved, simply take the Λ used in Theorem 2.26 together with their negatives. Notice that (K_{H}) implies (K_{Γ}). Furthermore, when 1 ≤ k = r, by considering the equations (λ)+(−λ) = 0, we see that every element of Λ is Λ-independent, and hence that (K_{Γ}) is satisfied by Lemma 1.12. For 0 = k = r, we have that Λ consists of s − 1 linearly independent elements together with their negatives, and hence (K_{Γ}) is not satisfied.

3. Type D_{n}

Let us describe the lattices of type D_{n}, following the notation of [CS93]. Denote by I_{n} the ℤ-module with (orthonormal) basis ϵ_{1}, ⋯, ϵ_{n}. It has an index 2 submodule, D_{n} = {∑ c_{i} ϵ_{i} | ∑ c_{i} is even. }. We construct the module D_{n}^{+4} by taking the span of I_{n} together with \frac{1}{2} ∑ ϵ_{i}. Finally, for n even, D_{n}^{+2} ⊂ D_{n}^{+4} is the index 2 submodule of elements whose coefficients sum to an even integer.

The action of the Weyl group W (D_{n}) = (ℤ/2)^{n−1} ⋊ S_{n} on D_{n} is given by S_{n} permuting the vectors ϵ_{i}, and (ℤ/2)^{n−1} making an even number of sign changes on {ϵ_{i}}. We have an outer automorphism given by negating only ϵ_{1}. When n = 4 there is also an outer automorphism of order 3, which is a phenomenon known as triality. The following table summarizes the possible situations. The column ϕ uses notation as from 1.2, and G is a corresponding split group. For the rest of this section, n = 2^{r}s where s is odd.

Unless the lattice is D_{4}, the primes p ≠ 2 do not appear in the table. This is because in these cases the size n set Λ = {ϵ_{i}} is Sylow invariant, and generates a sublattice of index a power of 2, so SymRank(ϕ; p) is the rank of the lattice.

We have n = 2^{r}s where s is odd, and we use the notation (H_{r})^{s} ⊂ H ⊂ S_{2^{r}s} as defined near the start of Section 2. Then for p = 2, our Sylow subgroup Γ is either (ℤ/2)^{n−1} ⋊ H or (ℤ/2)^{n} ⋊ H.

ϕ  G  p Conditions  SymRank(ϕ; p) (K)
1D_{n}  P SO_{2n}  2 n = 2^{r}s, s > 1 2^{2r+2}(s − 1)  y
n = 2^{r} ≥ 4  2^{2r}  y
2D_{n}  P Sp_{2n}  2 n = 2^{r}s, s > 1 2^{2r+2}(s − 1)  y
n = 2^{r} ≥ 4  2^{2r}  y
1D_{n}^{+2} HSpin_{2n}  2 n ≥ 6  2^{n−1}  y
1I_{n} SO_{2n}  2 n ≥ 4  2n  n
2I_{n} Sp_{2n}, SO_{2n+1} 2 n ≥ 2  2n  n
1D_{n}^{+4} Spin_{2n}  2 n ≥ 5 odd  2^{n−1}  y
n ≥ 4 even  2^{n−1} + 2^{r+1}  y
2D_{n}^{+4} Spin_{2n+1}  2 n ≥ 2  2^{n}  y
3D_{4} F_{4}  3  9  y

Table II. D_{n} lattices (which includes groups of types B_{n} and C_{n})

3.1. Case ^{1}D_{2^{r}s} or ^{2}D_{2^{r}s}, with s > 1 odd. We consider both the cases simultaneously. Consider the map π : D_{2^{r}s} → (ℤ/2)^{s} given by π(λ) = Σ^{r}(λ) mod 2. For Λ ⊂ D_{2^{r}s} to be 2-generating, we need π(Λ) to be of rank s − 1. Let λ ∈ Λ be such that π(λ) is one of the s − 1 generators. Since π(λ) is non-trivial, Σ^{r}(λ) must contain at least two non-trivial coordinates, and hence λ contains at least two 2^{r}-blocks both of which contain a non-scalar 2-block. Now by Lemma 2.5, the H_{r}-orbits of each of these 2^{r}-blocks is at least 2^{r}. Also, Σ^{r}((ℤ/2)^{n−1}λ) is at least size 4, since we can change the signs of either of these blocks (even in the case r = 0). Finally, (ℤ/2)^{n} ⋊ (H_{r})^{s} acts trivially on π(D_{2^{r}s}), and so we have |Λ| ≥ 4(2^{r})(2^{r})(s − 1) = 2^{r+2}(s − 1).

To see this bound is achieved, notice the following set is ((ℤ/2)^{n} ⋊ H)-invariant and generating:

Λ = ⋃_{1≤i≤2^{r}<j≤n}{± ϵ_{i} ± ϵ_{j}}.

To see this satisfies condition (K_{Γ}), we use Lemma 1.12. Let us consider ϵ_{1} + ϵ_{2^{r}+1} ∈ Λ. Then we have

(ϵ_{1} + ϵ_{2^{r}+1})+(− ϵ_{1} + ϵ_{2^{r}+1}) + (ϵ_{1} − ϵ_{2^{r}+2}) + (− ϵ_{1} − ϵ_{2^{r}+2}) = 0
(ϵ_{1} + ϵ_{2^{r}+1}) + (− ϵ_{1} − ϵ_{2^{r}+1}) = 0

So ϵ_{1} + ϵ_{2^{r}+1} is Λ-independent. A similar argument can be used on the other elements of Λ.

3.2. Case ^{1}D_{2^{r}} or ^{2}D_{2^{r}}, with r ≥ 2. We consider both the cases simultaneously. Consider the map π : D_{2^{r}} → (ℤ/2)^{2} given by π(λ) = Σ^{r−1}(λ) mod 2. For Λ ⊂ D_{2^{r}} to be 2-generating, we need π(Λ) to contain a non-trivial element. Let λ ∈ Λ be such that π(λ) is non-trivial. Therefore both 2^{r−1}-blocks in λ contain a non-scalar 2-block. Now by Lemma 2.5, the H_{r−1}-orbits of each of these 2^{r−1}-blocks is at least 2^{r−1}. Also, Σ^{r−1}((ℤ/2)^{2^{r}−1}λ) is at least size 4, since we can change the signs of either of these blocks (even in the case r = 0). Finally, (ℤ/2)^{2^{r}−1} ⋊ H_{r−1} acts trivially on π(D_{2^{r}}), and so we have |Λ| ≥ 4(2^{r−1})(2^{r−1}) = 2^{2r}.

This lower bound is achieved by choosing:

Λ = ⋃_{1≤i≤2^{r−1}<j≤2^{r}}{± ϵ_{i} ± ϵ_{j}}.

This Λ satisfies (K_{Γ}), which can be seen by using a similar argument to the s > 1 case to show ϵ_{1} + ϵ_{2^{r−1}+1} is Λ-independent.

3.3. Case ^{1}D_{n}^{+2}, and n ≥ 6 even. For Λ ⊂ D_{n}^{+2} to be 2-generating, there must be a λ ∈ Λ, all of whose coefficients of ϵ_{i} are half-integers. In other words, c_{i} ∈ \frac{1}{2}ℤ\ℤ; and in particular are non-zero. By considering how the coefficients change sign, the Γ-orbit of such an element is at least of size 2^{n−1}.

To ease notation, define σ := \frac{1}{2} ∑_{i=1}^{n} ϵ_{i}. For n ≡ 0 mod 4, this bound is achieved by choosing Λ = Γ·σ; for n ≡ 2 mod 4, this bound is achieved by choosing Λ = Γ·(σ −ϵ_{1}).

These both satisfy (K_{Γ}); we will only consider the n ≡ 0 mod 4 case, because the other case is similar. We have the following two equations.

σ + (σ − ϵ_{1} − ϵ_{2} − ϵ_{3} − ϵ_{4}) − (σ − ϵ_{1} − ϵ_{2}) − (σ − ϵ_{3} − ϵ_{4}) = 0
σ + (−σ) = 0.

Therefore, for n ≥ 5, we have that σ is Λ-independent, and thus Λ satisfies (K_{Γ}), by Lemma 1.12.

3.4. Case ^{1}I_{n}, and n ≥ 4. Λ must contain n linearly independent elements of I_{n}, and for even n, Γ sends these to their negatives, so |Λ| ≥ 2n. For odd n, if every element of Λ has at least one coordinate zero, then they are also all sent to their negatives. Otherwise, there is an element of Λ whose coordinates are all non-zero. The Γ-orbit of such an element is at least 2^{n−1}. For n ≥ 4, we have |Λ| ≥ 2^{n−1} ≥ 2n. This lower bound is achieved by choosing Λ = {± ϵ_{i}}. The condition (K_{Γ}) is not satisfied; the sign changes act trivially on the kernel of ℤ[Λ] → T̂.

3.5. Case^{2}I_{n}, and n ≥ 2. Λ must have n linearly independent elements of I_{n}, together with their negatives, and therefore |Λ| ≥ 2n. This lower bound is achieved by choosing Λ = {± ϵ_{i}}. As in the case ^{1}I_{n}, the condition (K_{Γ}) is not satisfied;

3.6. Case ^{1}D_{n}^{+4}, and n ≥ 5 odd. To be 2-generating, Λ must contain an element all of whose coefficients are half-integers, and in particular are non-zero. The Γ-orbit of such an element is at least of size 2^{n−1}.

Let σ := \frac{1}{2} ∑_{i=1}^{n} ϵ_{i}. Since n is odd, this bound is achieved by Λ = Γ·σ = \{\frac{1}{2}(± ϵ_{1} ⋯± ϵ_{n})}, where each element has an even number of minus signs. Notice that the generated lattice, X_{Λ}, contains ϵ_{1}, something which is not true for n even. To see that (K_{Γ}) is satisfied, we can use the first equation in the ^{1}D_{n}^{+2} case, but not the second, since the negatives are not in Λ. Since n ≥ 5, we can also use the following equation:

σ + (σ − ϵ_{1} − ϵ_{2} − ϵ_{3} − ϵ_{5}) − (σ − ϵ_{2} − ϵ_{5}) − (σ − ϵ_{1} − ϵ_{3}) = 0.

Then we see σ is Λ-independent, and by Lemma 1.12 we are done.

3.7. Case ^{1}D_{n}^{+4}, and n ≥ 4 even. Consider the map π : D_{n}^{+4} → ℤ/2 × ℤ/2 given by sending λ to its coordinate sum mod 2, and its projection D_{n}^{+4}/I_{n} ≅ ℤ/2. Then π is surjective, and Γ acts trivially on the image, so for any 2-generating subset Λ we have that π(Λ) generates (ℤ/2)^{2}.

Notice that for any λ whose projection to D_{n}^{+4}/I_{n} is non-trivial, in other words its coordinates consist of half-integers, then all of its coefficients are non-zero. In particular, |Γλ| ≥ 2^{n−1}.

Now assume λ is such that its coordinates sum to an odd number. If its coordinates are half-integers, then we know |Γλ| ≥ 2^{n−1} ≥ 2^{r+1}, so assume they are integers. We know one of its 2^{r}-blocks sum to an odd number (where n = 2^{r}s, for s odd), and hence this 2^{r}-block contains a non-scalar 2-block. By Lemma 2.5, we know |Hλ| ≥ 2^{r}. Finally, notice that the number of positive odd coefficients does not equal the number of negative odd coefficients (since there are an odd number of them), so −λ ∉ Hλ, and therefore |Γλ| ≥ 2^{r+1}.

This shows that |Λ| ≥ 2^{n−1} + 2^{r+1}.

Finally, this bound is achieved by Λ = (Γ·σ)∪{± ϵ_{i} |1 ≤ i ≤ 2^{r}}. To see this satisfies (K_{Γ}), consider the equations

(ϵ_{1}) + (ϵ_{2}) + ⋯ + (ϵ_{n}) + 2(σ) = 0
(ϵ_{1}) + (− ϵ_{1}) = 0, σ + (−σ) = 0.

This shows σ and ϵ_{i} are Λ-independent. By Lemma 1.12 we are done.

3.8. Case ^{2}D_{n}^{+4}, and n ≥ 2. For Λ ⊂ D_{n}^{+4} to be 2-generating, it must contain an element all of whose coefficients of ϵ_{i} are half-integers, and hence non-zero. The (ℤ/2)^{n}- orbit of such an element is of size at least 2^{n}, and therefore |Λ| ≥ 2^{n}.

This lower bound is achieved by Λ = Γ · σ = \{\frac{1}{2}(± ϵ_{1} ⋯ ± ϵ_{n})\}. Let us check that it satisfies (K_{Γ}). Let σ = \frac{1}{2} ∑_{i=1}^{n} ϵ_{i}; then we have the two equations

σ + (σ − ϵ_{1} − ϵ_{2}) + (−σ + ϵ_{1}) + (−σ + ϵ_{2}) = 0
σ + (−σ) = 0.

Therefore σ is Λ-independent, and thus Λ satisfies (K_{Γ}), by Lemma 1.12.

3.9. Case ^{3}D_{4}, and p = 3. Since |W (D_{4})^{(3)}| · 3 = |W (F_{4})^{(3)}|, and the root system of D_{4} is contained in that of F_{4}, we may work with the root lattice of F_{4}, together with its Weyl group action. Let α̃ = 2α_{1} + 4α_{2} + 3α_{3} + 2α_{2} be the highest root in F_{4}. Then we have a subroot system A_{2} × A_{2} ⊂ F_{4} generated by {α_{1}, α_{2}} and {α_{4}, −α̃}. These copies of A_{2} each have a copy of ℤ/3 is their Weyl groups, so we will use this choice for our Sylow 3-subgroup Γ := (ℤ/3)^{2} ⊂ W (F_{4}).

Notice that any 3-generating subset must contain an element λ whose coefficient of α_{3} (written in root coordinates) is not a multiple of 3; in particular such an element is in neither copy of A_{2}, and hence |Γ · λ| = 9.

Indeed, take Λ = Γ · α_{3}. We can choose generators {a, b} for each copy of ℤ/3 such that aα_{3} = α_{3} + 2α_{2} and bα_{3} = α_{3} + α_{4}. Then notice the equations

α_{3} + a^{i}b^{i}α_{3} − a^{i}α_{3} − b^{i}α_{3} = 0, for i = 1, 2.

This shows that α_{3} is Λ-independent, and hence by Lemma 1.12 we see Λ satisfies (K_{Γ}).

4. Exceptional Type

We will use E_{n} to denote both the root lattice of E_{n} as well as the split adjoint group of type E_{n}. We consider the exceptional groups at every prime which divides the order of the Weyl group, but we only include in the table the cases where SymRank(ϕ; p) > dim(ϕ).

ϕ  G p SymRank(ϕ; p) (K)
2A_{2} G_{2} 2  4  n
1A_{2} G_{2} 3  3  n
2D_{4} F_{4} 2  16  y
3D_{4} F_{4} 3  9  y
1E_{6} E_{6} 2  16  y
2E_{6}  ^{2}E_{6} 2  32  y
1E_{7} E_{7} 2  64  y
1E_{7}^{+2} 2E_{7} 2  40  y
1E_{8} E_{8} 2  128  y
1E_{6} E_{6} 3  27  y
1E_{6}^{+3} 3E_{6} 3  27  y
1E_{7} E_{7} 3  27  y
1E_{8} E_{8} 3  81  y
1E_{8} E_{8} 5  25  y

Table III. Exceptional lattices

We can find the G_{2} and F_{4} cases in the A_{n} and D_{n} tables.

For the p = 2 cases, we will use the description of the E_{6}, E_{7} and E_{8} root lattices given in [Bou68, p.213-220] or [Hu92], and often write them in these coordinates.

4.1. Case ^{1}E_{8} and p = 2. We have E_{8} = D_{8}^{+2}, and W (D_{8}) ⊂ W (E_{8}). Let H′ be the usual Sylow 2-subgroup of W (D_{8}), and choose a Sylow 2-subgroup H′ ⊂ H′′ ⊂ W (E_{8}), so that |H′′| = 2^{14}. Notice that the 248 roots of E_{8} decompose into H′-orbits of size 128, 64, 32, and 16. Since the H′′-orbits must have sizes which are powers of 2, these are also H′′-orbits.

Explicitly, using the usual coordinates for D_{8}^{+2}, we can choose Λ = \{\frac{1}{2}(±1, ⋯, ±1)\} with an even number of minus signs. Here Γ = H′′, so Λ is a Γ-invariant subset which generates the lattice. Therefore 128 ≥ SymRank(E_{8}; 2) ≥ SymRank(D_{8}^{+2}; 2) = 128. This Λ satisfies (K_{Γ}) for the same reason as the ^{1}D_{8}^{+2} case, above.

4.2. Case ^{1}E_{7} and p = 2. Defining H′′ ⊂ W (E_{8}) as above, consider the H′′-orbit of α = (0, 0, 0, 0, 0, 0, 1, 1) of size 16. By defining the E_{7} lattice as elements of E_{8} which are orthogonal to α, we have W (E_{7}) = W (E_{8})_{α}, the elements fixing α ([Hu92, Thm. 1.12(d)]). Furthermore, we can choose Γ = H′′_{α} as a Sylow 2-subgroup of W (E_{7}), because it has order 2^{14}/16 = |W (E_{7})^{(2)}|.

Consider the H′_{α} ⊂ Γ action on the 126 roots of E_{7}. They decompose into H′_{α}- orbits of size 64, 32, 16, 8, 4 and 2. Since the 126 roots are preserved by the Γ, these must also be Γ-orbits. In particular, the 64 element set Λ = Γ · \frac{1}{2}(1, −1, 1, 1, 1, 1, 1, −1) generates the E_{7} lattice, so SymRank(E_{7}; 2) ≥ 64. Indeed this is minimal, because any 2-generating subset must contain an element whose coefficients are all in \frac{1}{2}ℤ\ℤ, and (ℤ/2)^{6} ⊂ Γ has trivial stabilizer of such an element.

This Λ satisfies (K_{Γ}) by using a similar argument to the ^{1}D_{8}^{+2} case, above.

4.3. Case ^{1}E_{7}^{+2} at p = 2. E_{7}^{+2} is the span of the root lattice E_{7} together with \frac{1}{2}(1, 1, 1, 1, 1, 1, 0, 0), and the Γ action is the same as the E_{7} case. The 56 elements in E_{7}^{+2} of length 3/2 decompose into H′_{α}-orbits of size 36, 16, and 8. Therefore these are also Γ = H′′_{α}-orbits. We take the union of two of these orbits, Λ = Γ·\frac{1}{2}(1, 1, 1, 1, 1, 1, 0, 0)∪ Γ· \frac{1}{2}(0, 0, 0, 0, 0, 2, 1, −1) which is of size 40 = 32+8. One checks that this generates the lattice. To see it is minimal, notice Λ must contain an element whose first 6 coordinates are half-integers (which has Γ-orbit at least size 32) and an element whose last two coordinates are half integers (which has Γ-orbit at least size 8). Alternatively, notice that as lattices E_{7}^{+2} = A_{7}^{+4}, and we can embed W (A_{7}) ⋊ ℤ/2 ⊂ W (E_{7}). This implies 40 = SymRank(^{2}A_{7}^{+4}; 2) ≤ SymRank(^{1}E_{7}^{+2}; 2) ≤ 40.

To see our choice of Λ satisfies (K_{Γ}), notice that \frac{1}{2}(1, 1, 1, 1, 1, 1, 0, 0) is Λ-independent by a similar argument to the ^{1}D_{n}^{+2} case. We have the following relations of elements in Λ:

\frac{1}{2}(0, 0, 0, 0, 0, 2, 1, −1) + \frac{1}{2}(0, 0, 0, 0, 0, −2, 1, −1)

+ \frac{1}{2}(0, 0, 0, 0, 2, 0, −1, 1) + \frac{1}{2}(0, 0, 0, 0, −2, 0, −1, 1) = 0

\frac{1}{2}(0, 0, 0, 0, 0, 2, 1, −1) + \frac{1}{2}(0, 0, 0, 0, 0, −2, −1, 1) = 0.

Therefore \frac{1}{2}(0, 0, 0, 0, 0, 2, 1, −1) is Λ independent, and by Lemma 1.12, we are done.

4.4. Case ^{1}E_{6} at p = 2. Using the labellings as in [Bou68, p.218], the E_{6} simple roots α_{2}, ⋯, α_{6} form a root system of type D_{5}, so W (D_{5}) ⊂ W (E_{6}). In fact, |W (E_{6})^{(2)}| = |W (D_{5})^{(2)}| = 2^{7}, so we get a description of the Sylow action of E_{6}. In coordinates, we have α_{1} = \frac{1}{2}(1, −1, −1, −1, −1, −1, −1, 1), and Γ only changes the first 5 coordinates, so the Γ-orbit of α_{1} is of size 16. Furthermore, this orbit generates the E_{6} lattice. It is minimal, because any 2-generating subset of E_{6} must contain an element whose coefficients are all in \frac{1}{2}ℤ\ℤ, and such an element has orbit size at least 16.

Λ = Γ · α_{1} is the set \{\frac{1}{2}(±1, ±1, ±1, ±1, ±1, −1, −1, 1)\}, where each element has an even number of minus signs. Therefore (K_{Γ}) is satisfied, by using a similar argument to the ^{1}D_{5}^{+4} case.

4.5. Case ^{2}E_{6}, and p = 2. The Γ-action here is that same as the ^{1}E_{6} case, together with the outer automorphism given by negation. As in that case, there must be an element in Λ with half-integer coordinates, and the orbit of such an element is at least size 32. This is achieved by considering the Γ-orbit of α_{1}, which is size 32, and this orbit spans E_{6}.

One sees Λ = Γ · α_{1} satisfies (K_{Γ}) by a similar argument to the ^{2}D_{5}^{+4} case.

4.6. Cases ^{1}E_{6},^{1} E_{7}, and ^{1}E_{8}, at p = 3. We recall the description of the Sylow 3-subgroups W (E_{6})^{(3)} = W (E_{7})^{(3)} and W (E_{8})^{(3)} from [Ch06, 4]. We have a subroot system A_{2} × (A_{2})^{3} ⊂ A_{2} × E_{6} ⊂ E_{8}, and there are 5 order three generators {a, b, c, d, e} ⊂ W (E_{8})^{(3)}. We will now describe the action of these generators on the roots α_{1}, ⋯, α_{8}.

The extended Dynkin diagram of E_{6}, with highest root α_{E_{6}}, has a symmetry of order 3; d acts on these roots via this rotation, by sending α_{6} to −α_{E_{6}} to α_{1} to α_{6}, etc. a, b, c each cyclically permute the roots in their own copy of A_{2}; so a cyclically permutes {α_{3}, α_{1}, −α_{1} − α_{3}}; and then b = dad^{−1}, c = dbd^{−1}.

Now a, b, c, d act trivially on α_{8}, and e acts trivially on α_{1}, ⋯, α_{6}. e cyclically permutes {α_{8}, −α_{0}, α_{0} − α_{8}}, where α_{0} is the highest root of E_{8}. So all that remains is to describe the action of a, b, c, d, e on α_{7}. One checks that: aα_{7} = α_{7}, bα_{7} = α_{7}+α_{6}+α_{5}, cα_{7} = α_{7} + α_{E_{6}} − α_{2}, dα_{7} = α_{E_{7}} − α_{1}, eα_{7} = α_{7} + α_{8}. Here α_{E_{7}} is the highest root of E_{7} with respect to the simple roots α_{1}, ⋯, α_{7}.

With this complete description, one may check the following

Lemma 4.7. Let λ = ∑_{i} λ_{i}α_{i} ∈ ℝ^{8} be a vector in the vector space spanned by the roots of E_{8}, so λ_{i} ∈ ℝ. Then

(i) aλ = λ iff λ_{4} = 3λ_{1} and λ_{3} = 2λ_{1}.

(ii) bλ = λ iff 3λ_{5} = 2λ_{4} + 3λ_{7} and 3λ_{6} = λ_{4} + 2λ_{7}.

(iii) cλ = λ iff λ_{7} = λ_{2} and λ_{4} = 2λ_{2}.

(iv) dλ = λ iff λ_{1} = λ_{6} = 0 and λ_{2} = λ_{3} = λ_{5}.

(v) eλ = λ iff λ_{7} = λ_{8} = 0.

Corollary 4.8. Let λ = ∑_{i} λ_{i}α_{i} be an element of the E_{8} lattice, so λ_{i} ∈ ℤ. Then

(i) If λ_{4} ≢ 0 mod 3, then |⟨a, b⟩λ| = 9.

(ii) If λ_{7} ≢ 0 mod 3, then |W (E_{7})^{(3)}λ| ≥ 27 and |W (E_{8})^{(3)}λ| ≥ 81.

(iii) If λ_{4} ≢ 0 mod 3 and λ_{7} = λ_{8} = 0, then |W (E_{6})^{(3)}λ| ≥ 27.

Proof. Part (i) follows from 4.7. For Part (ii), assume λ_{7} ≢ 0 mod 3. Notice that the coefficient of α_{4} in dλ is λ_{4} +λ_{7} mod 3. So we can act on λ with d until its α_{4} coefficient is not divisible by 3. Then we can apply Part (i) to see |⟨a, b, d⟩λ| ≥ 27. Also e must increase the orbit size further, so |⟨a, b, d, e⟩λ| ≥ 81.

Finally, for Part (iii), we can rewrite such a λ in the basis {α_{1}, α_{3}, α_{5}, α_{6}, α_{2}, −β}, we see all 6 coefficients must be in \frac{1}{3}ℤ\ℤ, and in particular are non-zero. So 4.7 implies |W (E_{6})^{(3)}λ| ≥ 27 as required. □

This corollory immediately gives the desired lower bounds for SymRank(ϕ; 3). To see these bounds are achieved, take Λ_{E_{6}} := W (E_{6})^{(3)}α_{4}, Λ_{E_{7}} := W (E_{7})^{(3)}α_{7}, and Λ_{E_{8}} := W (E_{8})^{(3)}α_{7}. Then |Λ_{E_{6}}| = 27, |Λ_{E_{7}}| = 27 and |Λ_{E_{8}}| = 81, where each of these sets generate the E_{6}, E_{7}, and E_{8} lattices respectively.

Finally we will check these sets satisfy (K_{Γ}). Consider the following relations among elements of Λ_{E_{6}}:

α_{4} + (α_{4} + α_{3} + α_{5}) − (α_{4} + α_{3}) − (α_{4} + α_{5}) = 0
α_{4} + (α_{4} + α_{3} + α_{1} + α_{2}) − (α_{4} + α_{3} + α_{1}) − (α_{4} + α_{2}) = 0.

This shows α_{4} is Λ_{E_{6}}-independent, and hence by Lemma 1.12, we are done the E_{6} case. Consider the following relations among elements of Λ_{E_{7}}:

α_{7} + (α_{7} + α_{6} + β − α_{2}) − (α_{7} + α_{6}) − (α_{7} + β − α_{2}) = 0
α_{7} + (α_{7} + α_{6} + α_{5} + α_{E_{6}}) − (α_{7} + α_{6} + α_{5}) − (α_{7} + α_{E_{6}}) = 0.

This shows α_{7} is Λ_{E_{7}}-independent, and also Λ_{E_{8}}-independent. So by Lemma 1.12, we are done.

4.9. Case ^{1}E_{6}^{+3}, at p = 3. We will write elements of the lattice E_{6}^{+3} in coordinates of {α_{1}, α_{3}; α_{6}, α_{5}; −α_{E_{6}}, α_{2}}, corresponding to the three “arms” of the extended Dynkin diagram of E_{6}. Then the lattice E_{6} is additively generated by these 6 linearly independent vectors together with α_{4} = \frac{1}{3}(−1, −2; −1, −2; −1, −2), and E_{6}^{+3} is generated by E_{6} together with v := \frac{1}{3}(1, 2; 0, 0; −1, −2).

Observe that for any element, written in coordinates as above, λ = (λ_{1}, λ_{2}; λ_{3}, λ_{4}; λ_{5}, λ_{6}) ∈ E_{6}^{+3} we must have: λ_{1} + λ_{2}, λ_{3} + λ_{4}, λ_{5} + λ_{6} are all integers, and also λ_{1} + λ_{3} + λ_{5}, λ_{2} + λ_{4} + λ_{6} are both integers. These observations imply that if λ has a non-integral first coordinate, then the second coordinate is also non-integral and different from the first. Furthermore, either both λ_{3}, λ_{4} are non-intergral or both λ_{5}, λ_{6} are non-integral. Such a λ must exist in a 3-generating subset, and one checks that its W (E_{6})^{(3)}-orbit is at least size 27. This shows the lower bound. To see this bound is achieved, notice that Λ = W (E_{6})^{(3)} · v is size 27 and generates the lattice.

Let us check Λ satisfies (K_{Γ}). Notice the following relations in Λ:

v + acv + a^{2}c^{2}v = 0,
v + dv + d^{2}v = 0.

This shows v is Λ-independent, so by Lemma 1.12, we are done.

4.10. Cases^{1}E_{6},^{1} E_{7}, and^{1}E_{8}, at p = 5. Choose A_{4} ⊂ E_{6} generated by {α_{3}, α_{4}, α_{5}, α_{6}}. We can choose a Sylow 5-subgroup Γ := W (A_{4})^{(5)} = W (E_{6})^{(5)}. Then Γ fixes the highest root α_{E_{6}}, and sends α_{1} to α_{1} + α_{i} for i ∈ {3, 4, 5, 6}. So Λ := Γ · α_{1} ∪ {−α_{E_{6}}} is Sylow invariant, and generates an index 2 sublattice in E_{6} (since the α_{2} coefficient of the highest root is 2). Thus SymRank(ϕ; 5) = dim(E_{6}).

For E_{7}, similarly we can choose A_{4} ⊂ E_{7} generated by the simple roots {α_{4}, α_{5}, α_{6}, α_{7}}, and then we have the Sylow 5-subgroup Γ := W (A_{4})^{(5)} ≅ W (E_{7})^{(5)}, which fixes α_{1} and the highest root α_{E_{7}}. Furthermore, Γ sends α_{3} to α_{3} + α_{i} for i ∈ {4, 5, 6, 7}. Then Λ := Γ · α_{3} ∪ {α_{1}, α_{E_{7}}} is Sylow invariant, and it generates an index 2 sublattice in E_{7}. Thus SymRank(ϕ; 5) = dim(E_{7}).

For E_{8} we choose a sublattice A_{4} × A_{4} ⊂ E_{8} generated by {α_{1}, α_{3}, α_{4}, α_{2}} together with {α_{6}, α_{7}, α_{8}, α_{0}}, where α_{0} is the highest root of E_{8}. We have Γ := W (A_{4})^{(5)} × W (A_{4})^{(5)} ≅ W (E_{8})^{(5)}, but the sublattice is index 5. Indeed, any 5-generating subset Λ ⊂ E_{8} must contain an element whose coefficient of α_{5} (when written in the basis of simple roots) is not divisible by 5. One checks that such an element has Γ-orbit size

25. This bound is achieved with Λ := Γ · α_{5}.

Let us check that Λ satisfies (K_{Γ}). We can choose generates σ_{1}, σ_{2} of the two copies of W (A_{4})^{(5)} ≅ ℤ/5 such that σ_{1}α_{5} = α_{5} + α_{2} + α_{4}, and σ_{2}α_{5} = α_{5} + α_{6}. With the choices we have:

α_{5} + σ_{1}σ_{2}α_{5} − σ_{1}α_{5} − σ_{2}α_{5} = 0,
α_{5} + σ_{1}^{2}σ_{2}^{2}α_{5} − σ_{1}^{2}α_{5} − σ_{2}^{2}α_{5} = 0.

So α_{5} is Λ-independent, and hence by Lemma 1.12 we are done.

4.11. Cases^{1}E_{7} and^{1}E_{8}, at p = 7. Take A_{6} ⊂ E_{7} ⊂ E_{8} generated by {α_{1}, α_{3}, α_{4}, α_{5}, α_{6}, α_{7}}, and then take Γ := W (A_{6})^{(7)} = W (E_{7})^{(7)} = W (E_{8})^{(7)}. Then Γ sends the highest root α_{E_{7}} to α_{E_{7}} + α_{i}, and sends α_{8} to α_{8} + α_{i} for any i ∈ {1, 3, 4, 5, 6, 7}. So Λ_{E_{7}} := Γ · α_{E_{7}} generates an index 2 sublattice in E_{7}, and Λ_{E_{8}} := Γ·α_{8} generates an index 2 sublattice in E_{8}. So in both cases the symmetric 7-rank is equal to the rank.

5. Loose ends

Given a pair (G, p) of a split simple algebraic group and a prime p, which does not appear in Tables I, II, or III, we have ed(N; p) = ed(W; p), by Theorem 1.9. For completeness, we will compute ed(W; p) for all Weyl groups of irreducible root systems at all primes p. Then we will consider the individual cases of pairs (G, p) which have an entry in the Tables, but the (K) column contains an “n”; in other words the cases not covered by Corollory 1.11.

5.1. Essential p-dimension of the Weyl groups. In Table IV we list all values ed(W (R); p) for primes p. Though not directly related to this paper, we also we list value of ed(W (R)) = ed(W (R); 0) when it is known, assuming char k ≠ 2 (see Remark 5.2 for further discussion).

R  p = 0 p = 2 p = 3 p = 5 p = 7 p ≥ 11
A_{n}  ?? ⌊\frac{n+1}{2}⌋  ⌊(n + 1)/p⌋
B_{n}  n  n  ⌊n/p⌋
C_{n}  n  n  ⌊n/p⌋
D_{n} (n odd) n − 1 n − 1  ⌊n/p⌋
D_{n} (n even) n  n  ⌊n/p⌋
E_{6}  ??  4  3  1  0  0
E_{7}  7  7  3  1  1  0
E_{8}  8  8  4  2  1  0
F_{4}  4  4  2  0  0  0
G_{2}  2  2  1  0  0  0

Table IV. Essential dimensions of Weyl groups, ed(W; p)

We know ed(S_{n}; p) is the floor of n/p, by [MR09, Cor. 4.2]. So for p ≠ 2, this gives values in the A_{n}, B_{n}, C_{n}, and D_{n} cases. For p = 2, notice W (B_{n}) = W (C_{n}) contains (ℤ/2)^{n}, and hence has essential 2-dimension at least n (and therefore equal to n). For W (D_{n}), we have ed((ℤ/2)^{n−1} ⋊ S_{n}; 2) is n − 1 for n odd (has a faithful n − 1 dimensional representation, and contains (ℤ/2)^{n−1}, which has essential 2-dimension n − 1), and ed(W (D_{n}); 2) = n for n even (see [MR10, Prop. 5.2]).

For the exceptional groups, only the primes 2 and 3 require explanation, and we leave the F_{4} and G_{2} cases to the reader. At p = 2, we have that W (D_{8}) ⊂ W (E_{8}), and there is a faithful 8-dimensional representation of W (E_{8}), so ed(W (E_{8}); 2) = 8. We have W (D_{5})^{(2)} ≅ W (E_{6})^{(2)}, and so ed(W (E_{6}); 2) = 4.

The hardest case is E_{7}. We have that W (E_{7}) contains W (D_{6}), so 6 ≤ ed(W (E_{7}); 2) ≤ 7. Let Γ = W (E_{7})^{(2)}, and one can check (for example, with a computer algebra program), that the centre of Γ is isomorphic to (ℤ/2)^{3}, and that the intersection of the centre with [Γ, Γ] is the group (ℤ/2)^{2}. So we can apply [MR10, Theorem 1.2] to see that ed(Γ; 2) is odd. Therefore, it is ed(W (E_{7}); 2) = 7.

For p = 3, one sees that W (E_{6})^{(3)} ≅ S_{9}^{(3)}, and hence ed(W (E_{7}); 3) = ed(W (E_{6}); 3) = ⌊9/3⌋ = 3. And W (E_{8})^{(3)} ≅ ℤ/3 × W (E_{6})^{(3)}, so ed(W (E_{8}); 3) = 4.

For the absolute essential dimension, that is ed(W; 0) := ed(W), notice that for D_{n} for n even, B_{n}, C_{n}, E_{8}, E_{7}, F_{4}, and G_{2}, we have that ed(W (R); 2) ≤ ed(W) ≤ rank(R) = ed(W (R); 2). We also know that, for n odd, ed(W (D_{n})) = n − 1, by [FF08, Theorem 5.4].

For the standard representation of W (E_{6}), negation is not in the Weyl group, so ed(W (E_{6})) ≤ 6 − 1 ([BF03, Cor. 6.18]). But the best lower bound we have is given by ed(W (E_{6}); 2), and hence 4 ≤ ed(W (E_{6})) ≤ 5.

Remark 5.2. Computing ed(W (A_{n})) is a well-known open problem, and was one of the motivations behind the definition of essential dimension in [BR97]. The values ed(W (A_{n})) have only been computed for small values of n. For n = 1, 2, 3, 4, and 5 the values are 1,1,2,2, and 3, respectively (assuming char k = 0; see [BR97]). Recently Duncan has shown ed(W (A_{6})) = 4 (see [Du10], which also contains the most recent bounds for larger n). The only other case where ed(W (R)) is unknown for an irreducible root system is ed(W (E_{6})), which could be 4 or 5.

5.3. Cases when (K_{Γ}) is not satisfied. The following are the split simple algebraic groups which appear in Table I, II, or III, and have an “n” in the (K) column. In these cases, the best known upper bound for ed(N; p) comes from Theorem 1.9. The best lower bound comes from either Theorem 1.9, or ed(G; p) ≤ ed(N; p). In some cases the upper and lower bounds do not match, and hence the essential p-dimension of the normalizer is still not known exactly.

5.4. Case SL_{n}. Here Theorem 1.9 gives us ⌊n/p⌋ ≤ ed(N; p) ≤ 1 + ⌊n/p⌋.

5.5. Case P GL_{p}. The upper bound from Theorem 1.9 is ed(N; p) ≤ 2, and we also know that ed(P GL_{p}; p) = 2 (see [RY00, Lemma 8.5.7]), which gives a lower bound. Hence ed(N; p) = 2 in this case.

5.6. Case SL_{4}/μ_{2}. The upper bound from Theorem 1.9 gives ed(N; 2) ≤ 6−3+2 = 5. We also know that ed(SL_{4}/μ_{2}; 2) = 4, where the lower bound follows from the nontriviality of a cohomological invariant defined on SL_{4}/μ_{2}-torsors (see [RST06]). So 4 ≤ ed(N; p) ≤ 5.

5.7. Case SO_{2n}. For n odd, the upper bound from Theorem 1.9 gives ed(N; 2) ≤ 2n−n+(n−1) = 2n−1, and we know that ed(SO_{2n}; 2) = 2n−1. So ed(N; 2) = 2n−1. But for n even, ed(W; 2) = n, so we only know that 2n − 1 ≤ ed(N; 2) ≤ 2n.

5.8. Case SO_{2n+1}. The upper bound of Theorem 1.9 gives ed(N; 2) ≤ 2n−n+n = 2n, and we know that ed(SO_{2n+1}; 2) = 2n, and hence ed(N; 2) = 2n.

5.9. Case Sp_{2n}. From Theorem 1.9 we have that n ≤ ed(N; 2) ≤ 2n − n + n = 2n.

5.10. Case G_{2}. For p = 2, the upper bound from Theorem 1.9 gives ed(N; 2) ≤ 4 − 2 + 2 = 4, and we know that ed(G_{2}; 2) = 3. So 3 ≤ ed(N; 2) ≤ 4. For p = 3, Theorem 1.9 gives us 1 ≤ ed(N; 3) ≤ 3 − 2 + 1 = 2.

Remark 5.11. The only case where the upper and lower bounds differ by more than 1 is G = Sp_{2n}. The only cases where ed(G; p) ≤ ed(N; p) is not an improvement to the lower bound given in Theorem 1.9, are G = SL_{n} and G = Sp_{2n}, both of which have trivial essential dimension.

Remark 5.12. One can now check that for the cases considered in this section, even if we chose a different minimal p-generating Γ-invariant Λ ⊂ T̂, it could never satisfy (K_{Γ}). This isn’t obvious a priori; for example, in Section 3 we chose a subset Λ ⊂ A_{8}^{+3} which generates the lattice, is W (A_{8})^{(3)}-invariant, is size 30 (and hence minimal), and then we verified that Λ satisfied (K_{Γ}). Yet there exists another subset Λ′ ⊂ A_{8}^{+3} which is generating, Γ-invariant, and size 30, which does not satsify (K_{Γ}).

Acknowledgements

I would like to thank Zinovy Reichstein for several useful discussions, as well as Nicole Lemire and Philippe Gille for their comments. I’d also like to thank Aurel Meyer and Roland Lötscher for their corrections and comments.

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