THE ESSENTIAL DIMENSION OF THE NORMALIZER OF

A MAXIMAL TORUS IN THE PROJECTIVE LINEAR GROUP

AUREL MEYER^{†} AND ZINOVY REICHSTEIN^{††}

Abstract. Let p be a prime, k be a field of characteristic ≠ p and N be the normalizer of the maximal torus in the projective linear group PGL_{n}. We compute the exact value of the essential dimension ed_{k}(N; p) of N at p for every n ≥ 1.

Contents

1. Introduction 1 Acknowledgements 4 2. A general strategy 4 3. Representation-theoretic preliminaries 5 4. Subgroups of prime-to-p index 7 5. First reductions and proof of Theorem 1.1 parts (a) and (b) 9 6. Proof of Theorem 1.1 part (c): The upper bound 11 7. Theorem 1.1 part (c): The lower bound 12 8. Proof of Theorem 1.1 part (d) 15 References 17

1. Introduction

Let p be a prime, k be a field of characteristic ≠ p and N be the normalizer of a split maximal torus in the projective linear group PGL_{n}, for some integer n. The purpose of this paper is to compute the essential dimension ed_{k}(N; p) of N at p. For the definition of essential dimension of an algebraic group (and more generally, of a functor), we refer the reader to [Re_{2}], [BF], [BRV] or [Me]. As usual, if the reference to k is clear from the context, we will sometimes write ed in place of ed_{k}.

1991 Mathematics Subject Classification. 11E72, 20D15, 16K20.

Key words and phrases. Essential dimension, central simple algebra, character lattice, finite p-group, Galois cohomology.

† Aurel Meyer was partially supported by a University Graduate Fellowship at the University of British Columbia.

†† Z. Reichstein was partially supported by NSERC Discovery and Accelerator Supplement grants.

We begin by explaining why we are interested in the essential dimension of N. One of the central problems in the theory of essential dimension is to find the exact value of the essential dimension of the projective linear group PGL_{n} or equivalently, of the functor

H^{1}( ∗ , PGL_{n}): K ↦ { degree n central simple algebras A/K,

up to K-isomorphism },

where K is a field extension of k. This problem arises naturally in the theory of central simple algebras. To the best of our knowledge, it was first raised by C. Procesi, who showed (using different terminology) that ed(PGL_{n}) ≤ n^{2}; see [Pr, Theorem 2.1]. This problem, and the related question of computing the relative essential dimension ed(PGL_{n}; p) at a prime p, remain largely open. The best currently known lower bound,

ed(PGL_{p^{r}}; p) ≥ 2r

(cf. [Re_{1}, Theorem 16.1(b)] or [RY, Theorem 8.6]), falls far below the best known upper bound,

(1)  ed(PGL_{n}) ≤ \begin{cases}\frac{(n−1)(n−2)}{2}, for every odd n ≥ 5 and \\ n^{2} − 3n + 1, for every n ≥ 4; \end{cases}

see [LR], [LRRS, Theorem 1.1], [Le, Proposition 1.6] and [FF].

We remark that the primary decomposition theorem reduces the computation of ed(PGL_{n}; p) to the case where n is a power of p. That is, if n = p_{1}^{r_{1}} … p_{s}^{r_{s}} then ed(PGL_{n}; p_{i}) = ed(PGL_{p_{i}^{r_{i}}}; p_{i}). The computation of ed(PGL_{n}) also partially reduces to the prime power case, because

ed(PGL_{p_{i}^{r_{i}}}) ≤ ed(PGL_{n}) ≤ ed(PGL_{p_{1}^{r_{1}}} ) + … + ed(PGL_{p_{s}^{rs}})

for every i = 1, …, s; cf. [Re_{2}, Proposition 9.8].

It is important to note that the proofs of the upper bounds (1) are not based on a direct analysis of the functor H^{1}( ∗ , PGL_{n}). Instead, one works with the related functor

H^{1}( ∗ , N): K ↦ { K-isomorphism classes of pairs (A, L) },

where K is a field extension of k, A is a degree n central simple algebra over K, L is a maximal étale subalgebra of A, and N is the normalizer of a (split) maximal torus in PGL_{n}. This functor is often more accessible than H^{1}( ∗ , PGL_{n}) because many of the standard constructions in the theory of central simple algebras depend on the choice of a maximal subfield L in a given central simple algebra A/K. Projecting a pair (A, L) to the first component, we obtain a surjective morphism of functors H^{1}( ∗ , N) → H^{1}( ∗ , PGL_{n}). The surjectivity of this morphism (which is a special case of a more general result of T. Springer, see [Se_{2}, III.4.3, Lemma 6]) leads to the inequalities

(2)  ed(N) ≥ ed(PGL_{n}) and ed(N; p) ≥ ed(PGL_{n}; p) ;

see [Me, Proposition 1.3], [BF, Lemma 1.9] or [Re_{2}, Proposition 4.3]. The inequalities (1) were, in fact, proved as upper bounds on ed(N); see [LRRS] and [Le]. It is thus natural to try to determine the exact values of ed(N) and ed(N; p). In addition to being of independent interest, these numbers represent a limitation on the techniques used in [LRRS] and [Le]. This brings us to the main result of this paper.

Theorem 1.1. Let N the normalizer of a maximal torus in the projective linear group PGL_{n} defined over a field k with char(k) ≠ p. Then

(a) ed_{k}(N; p) = [n/p],  if n is not divisible by p.
(b) ed_{k}(N; p) = 2,  if n = p.
(c) ed_{k}(N; p) = n^{2}/p − n + 1,  if n = p^{r} for some r ≥ 2.
(d) ed_{k}(N; p) = p^{e}(n − p^{e}) − n + 1, in all other cases.

Here [n/p] denotes the integer part of n/p and p^{e} denotes the highest power of p dividing n.

In each part we will prove an upper bound and a lower bound on ed(N) separately, using rather different techniques. There is nothing about the methods we use that in any way guarantees that the lower bounds should match the upper bounds, thus yielding an exact value of ed(N; p). The fact that this happens, for any base field k of characteristic ≠ p, may be viewed as a lucky coincidence. We also remark that our proof of the upper bounds on ed_{k}(N; p) in part (c) and (d) does not use the assumption that char(k) ≠ p; these bounds are valid for every k.

As we mentioned above, the computation of ed(PGL_{n}; p), reduces to the case where n is a power of p. A quick glance at the statement of Theorem 1.1 shows that, the computation of ed(N; p) does not. On the other hand, the proof of part (c), where n = p^{r} and r ≥ 2, requires the most intricate arguments. Another reason for our special interest in part (c) is that it leads to a new upper bound on ed(PGL_{n}; p). More precisely, combining the upper bound in part (c) with (2), and remembering that the upper bound in part (c) is valid for any the ground field k, we obtain the following inequality.

Corollary 1.2. Let n = p^{r} be a prime power. Then

ed_{k}(PGL_{n}; p) ≤ p^{2r−1} − p^{r} + 1

for any field k and for any r ≥ 2. □

Corollary 1.2 fails for r = 1 because

(3)  ed_{k}(PGL_{p}; p) = 2,

see [Re_{2}, Corollary 5.7] or [RY, Lemma 8.5.7]. For r = 2, Corollary 1.2 is valid but is not optimal. Indeed, in this case L. H. Rowen and D. J. Saltman showed that, after a prime-to-p extension L/K, every degree p^{2} central simple algebra A/K becomes a (ℤ/pℤ)^{2}-crossed product; see [RS, Corollary 1.3]. The upper bound on the essential dimension of a crossed product given by [LRRS, Corollary 3.10] then yields the inequality

ed(PGL_{p^{2}}; p) ≤ p^{2} + 1 ,

which is stronger than Corollary 1.2 for any p ≥ 3. If r ≥ 3 we do not know how close the true value of ed(PGL_{p^{r}}; p) is to ed(N; p) = p^{2r−1} − p^{r} + 1; in this case Corollary 1.2 gives the best currently known upper bound on ed(PGL_{p^{r}}; p). We remark that, beyond the obvious inequality ed(PGL_{p^{r}}; p) ≤ ed(PGL_{p^{r}}), the relationship between ed(PGL_{p^{r}}; p) and ed(PGL_{p^{r}}) is quite mysterious as well.

A key ingredient in our proofs of the lower bounds in Theorem 1.1(c) and (d) is a recent theorem of N. A. Karpenko and A. S. Merkurjev [KM] on the essential dimension of a p-group, stated as Theorem 7.1 below. To the best of our knowledge, these results were not accessible by previously existing techniques. Corollary 1.2 and the other parts of Theorem 1.1 do not rely on the Karpenko-Merkurjev theorem.

Acknowledgements

The authors are grateful to A. S. Merkurjev and J.-P. Tignol for helpful comments.

2. A general strategy

Let G be an algebraic group defined over a field k. Recall that the action of G on an algebraic variety X defined over k is generically free if the stabilizer subgroup Stab_{G}(x) is trivial for x ∈ X(\overline{k}) in general position.

Remark 2.1. If G is a finite constant group and X is irreducible and smooth then the G-action on X is generically free if and only if it is faithful.

Indeed, the “only if” implication is obvious. Conversely, if the G-action on X is faithful then Stab_{G}(x) = {1} for any x outside of the closed subvariety ⋃_{1≠g∈G} X^{⟨g⟩}, whose dimension is ≤ dim(X). □

Remark 2.2. Suppose k′/k be a field extension of degree prime to p. Then essential dimension at p does not change if we replace k by k′; see [Me, Proposition 1.5(2)]. This happens in particular, if char(k) ≠ p and k′ is obtained from k by adjoining a primitive pth root of unity. Thus in the course of proving Theorem 1.1 we may assume without loss of generality that k contains a primitive pth root of unity.

In the sequel we will repeatedly encounter the following situation. Suppose we want to show that

(4)  ed_{k}(G) = ed_{k}(G; p) = d ,

where G is a linear algebraic group defined over k.

All such assertions will be proved by the following 2-step procedure.

(i) Construct a generically free linear representation of G of dimension d+dim(G) defined over k. This implies that ed_{k}(G) ≤ d; see [Re_{2}, Theorem 3.4] or [BF, Proposition 4.11].

(ii) Prove the lower bound ed_{k}(G; p) ≥ d.

Since clearly ed(G; p) ≤ ed(G), the desired equality (4) follows from (i) and (ii).

The group G will always be of the form G = D ⋊ F, where D is diagonalizable and F is finite. In the next section we will recall some known facts about representations of such groups. This will help us in carrying out step (i) and, in the most interesting cases, step (ii) as well, via the Karpenko-Merkurjev Theorem 7.1.

3. Representation-theoretic preliminaries

We will work over a ground field k which remains fixed throughout. Suppose that a linear algebraic k-group G contains a diagonalizable (over k) group D and the quotient G/D is a constant finite group F. Here by “diagonalizable over k” we mean that D is a subgroup of the split torus 𝔾_{m}^{d} defined over k or, equivalently, that every linear representation of D defined over k decomposes as a direct sum of 1-dimensional subrepresentations.

Denote the group of (multiplicative) characters of D by X(D). Note that since D is diagonalizable over k, every multiplicative character of D is defined over k. Consider a linear k-representation G → GL(V). Restricting this representation to D, we decompose V into a direct sum of 1-dimensional character spaces. Let Λ ⊂ X(D) be the set of characters (weights) of D which occur in this decomposition. Note that here |Λ| ≤ dim(V), and equality holds if and only if each character from Λ occurs in V with multiplicity 1. The finite group F acts on X(D) and Λ is invariant under this action. Moreover, if the G-action (and hence, the D-action) on V is generically free then Λ generates X(D) as an abelian group. In summary, we have proved the following lemma; cf. [Se_{1}, Section 8.1].

Lemma 3.1. Suppose every F -invariant generating set Λ of X(D) contains ≥ d elements. If G → GL(V) is a generically free k-representation of G then dim(V) ≥ d. □

As we explained in the previous section, we are interested in constructing low-dimensional generically free representations of G. In this section we will prove simple sufficient conditions for generic freeness for two particular families of representations.

Lemma 3.2. Let W be a faithful representation of F and V be a representation of G whose restriction to D is generically free. Then V × W is a generically free representation of G.

Here we view W as a representation of G via the natural projection G → G/D = F.

Proof. For w ∈ W (\overline{k}) in general position, Stab_{G}(w) = D; cf. Remark 2.1. Choosing v in general position in V (\overline{k}), we see that

Stab_{G}(v, w) = Stab_{G}(v) ∩ Stab_{G}(w) = Stab_{D}(v) = {1} .
□

From now on we will assume that G = D ⋊ F is the semidirect product of D and F. In this case, given an F -invariant generating set Λ ⊂ X(D), we can construct a linear (in fact, a monomial) k-representation V_{Λ} of G so that each character from Λ occurs in V_{Λ} exactly once. To do this, we associate a basis element v_{λ} to each λ ∈ Λ. The finite group F acts on

V_{Λ} = Span(v_{λ} | λ ∈ Λ)

by permuting these basis elements in the natural way, i.e., via

(5)  σ: v_{λ} ↦ v_{σ(λ)} .

for any σ ∈ F and any λ ∈ Λ. The diagonalizable group D-acts by the character λ on each 1-dimensional space Span(v_{λ}), i.e., via

(6)  t: v_{λ} ↦ λ(t)v_{λ}

for any t ∈ D and λ ∈ Λ. Extending (5) and (6) linearly to all of V_{Λ}, we obtain a linear representation G = D ⋊ F → GL(V_{Λ}). Note that by our construction dim(V_{Λ}) = |Λ|.

Our second criterion for generic freeness is a variant of [LR, Lemma 3.1] or [Le, Proposition 2.1]. For the sake of completeness we outline a characteristic-free proof.

Lemma 3.3. Let Λ be an F -invariant subset of X(D) and ϕ: ℤ[Λ] → X(D) be the natural morphism of ℤ[F]-modules, taking λ ∈ Λ to itself. Let V_{Λ} be the linear representation of G = D ⋊ F defined by (5) and (6), as above. The G-action on V_{Λ} is generically free if and only if

(a) Λ spans X(D) (or equivalently, ϕ is surjective) and

(b) the F -action on Ker(ϕ) is faithful.

Proof. Let U ≃ 𝔾_{m}^{n} be the diagonal subgroup of GL(V_{Λ}), in the basis e_{λ}, where λ ∈ Λ. Here n = |Λ| = dim(V_{Λ}). The G-action on V induces an F -equivariant morphism ρ: D → U, which is dual to ϕ under the usual (anti-equivalence) Diag between finitely generated abelian groups and diagonalizable algebraic groups. Applying Diag to the exact sequence

(0)  Ker(ϕ)  ℤ[Λ] ^{ϕ} X(D)  Coker(ϕ)  (0) ,

of finitely generated abelian ℤ[F]-modules we obtain an F -equivariant exact sequence

1 N D ^{ρ} U Q 1 ,

of diagonalizable groups, where U = Diag(ℤ[Λ]), N = Diag(Coker(ϕ)) and Q = Diag(Ker(ϕ)); cf. [Ja, I 5.6] or [DG, IV 1.1]. Since U is F -equivariantly isomorphic to a dense open subset of V, the G-action on V is generically free if and only if the G-action on U is generically free. On the other hand, the G-action on U is generically free if and only if (i) the D-action on U is generically free, and (ii) the F -action on Q is generically free.

It is now easy to see that (i) is equivalent to (a) and (ii) is equivalent to (b); cf. Remark 2.1. □

4. Subgroups of prime-to-p index

Our starting point is the following lemma.

Lemma 4.1. Let G′ be a closed subgroup of a smooth algebraic group G defined over k. Assume that the index [G : G′] is finite and prime to p. Then ed(G; p) = ed(G′; p).

In the case where G is finite a proof can be found in [Me, Proposition 4.10]; the argument below proceeds along similar lines.

Proof. Recall that if G is a linear algebraic group and H is a closed subgroup then

(7)  ed(G; p) ≥ ed(H; p) + dim(H) − dim(G) ;

for any prime p; see, [BRV, Lemma 2.2] or [Me, Corollary 4.3]. Since dim G′ = dim G, this yields ed(G; p) ≥ ed(G′; p).

To prove the opposite inequality, it suffices to show that for any field K/k the map H^{1}(K, G′) → H^{1}(K, G) induced by the inclusion G′ ⊂ G is p-surjective, i.e., that for every α ∈ H^{1}(K, G) there is a finite field extension L/K of degree prime to p such that α_{L} is in the image of H^{1}(L, G′) → H^{1}(L, G); see, e.g., [Me, Proposition 1.3].

Let X be a G-torsor over K and X/G the quotient by the action of G′. For a field L/K and an L-point Spec(L) → X/G′ we construct a G′-torsor Y as the pullback

Y  \overline{}X

[figure: Spec(L); X/G′]

Spec(K)

In this situation Y ×^{G′} G ≅ X_{L} as G-torsors. Thus we have the natural diagram

H^{1}(L, G′)  \overline{}H^{1}(L, G)
[Y] [X]_{L}

[X]
H^{1}(K, G)

where [X] and [Y] denote the classes of X and Y in H^{1}(K, G) and H^{1}(L, G′), respectively. It remains to show the existence of such an L-point, with the degree [L : K] prime to p.

Note that G/G′ is affine, since G and G′ are of the same dimension and hence G/G′ ≅ (G/G^{∘})/(G′/G^{∘}) = Spec k[G/G^{∘}]^{G′/G^{∘}} where G^{∘} is the connected component of G (and G′). Furthermore G/G′ is smooth; cf. [DG, III 3.2.7]. Let K_{s} be the separable closure of K. X being a G-torsor, we have X_{K_{s}} ≅ G_{K_{s}} and (X/G′)_{K_{s}} ≅ (G/G′)_{K_{s}} which implies that X/G′ is also affine, cf. [DG, III 3.5.6 d)]. Thus, K[X/G′] ⊗ K_{s} ≅ k[G/G′] ⊗ K_{s} is reduced and its dimension dim_{K} K[X/G′] = [G : G′] is not divisible by p by assumption.

Therefore K[X/G′] is étale or, equivalently, a product of separable field extensions of K

K[X/G′] = L_{1} × ⋯ × L_{r};

see, e.g., [Bo, V, Theorem 4]. For each L_{j} the projection K[X/G′] → L_{j} is an L_{j}-point of X/G′ and since

dim_{K} K[X/G′] = ∑_{j=1}^{r}[L_{j} : K] is prime to p,

one of the fields L_{j} must be of degree prime to p over K. We now take L = L_{j}. □

Corollary 4.2. Suppose k is a field of characteristic ≠ p. Then ed_{k}(S_{n}; p) = [n/p].

Proof. Let m = [n/p] and let D ≃ (ℤ/pℤ)^{m} be the subgroup generated by the disjoint p-cycles

σ_{1} = (1, …, p), …, σ_{m} = ((m − 1)p + 1, …, mp) .

The inequality ed(S_{n}; p) ≥ ed_{k}(D; p) ≥ [n/p] is well known; see, [BuR_{1}, Section 6], [BuR_{2}, Section 7], or [BF, Proposition 3.7].

To the best of our knowledge, the opposite inequality was first noticed by J.-P. Serre (private communication, May 2005) and independently by R. Lötscher [Lö]. The proof is quite easy; however, since it has not previously appeared in print, we reproduce it below.

The semi-direct product D ⋊ S_{m}, where S_{m} permutes σ_{1}, …, σ_{m}, embeds in S_{n} with index prime to p. By Lemma 4.1, ed_{k}(D ⋊ S_{m}; p) = ed_{k}(S_{n}; p) and it suffices to show that ed_{k}(D ⋊ S_{m}) ≤ [n/p]. As we mentioned in Section 2, in order to prove this, it is enough to construct a generically free m-dimensional representation of D ⋊ S_{m} defined over k. Moreover, by Remark 2.2 we may assume that ζ_{p} ∈ k, where ζ_{p} denotes a primitive root pth root of unity.

To construct a generically free m-dimensional representation of D ⋊ S_{m}, let σ_{1}^{∗}, …, σ_{m}^{∗} ⊂ X(D) be the “basis” of D dual to σ_{1}, …, σ_{m}. That is,

σ_{i}^{∗}(σ_{j}) = \begin{cases}ζ_{p}, if i = j and \\ 1, otherwise. \end{cases}

The S_{m}-invariant subset Λ = {σ_{1}^{∗}, …, σ_{m}^{∗}} of X(D) gives rise to the mdimensional k-representation V_{Λ} of D ⋊ S_{m}, as in Section 3. An easy application of Lemma 3.3 shows that this representation is generically free. □

5. First reductions and proof of Theorem 1.1 parts (a) and (b)

Let T ≃ 𝔾_{m}^{n}/Δ be the diagonal maximal torus in PGL_{n}, where Δ = 𝔾_{m} is diagonally embedded into 𝔾_{m}^{n}. Recall that the normalizer N of T is isomorphic to T ⋊S_{n}, where we identify S_{n} with the subgroup of permutation matrices in PGL_{n}.

Let P_{n} be a Sylow p-subgroup of S_{n}. Lemma 4.1 tells us that

ed_{k}(N; p) = ed_{k}(T ⋊ P_{n}; p) .

Note also that by Remark 2.2 we may assume without loss of generality that k contains a primitive pth root of unity.

Thus in order to prove Theorem 1.1 it suffices to establish the following proposition.

Proposition 5.1. Let T ≃ 𝔾_{m}^{n}/Δ, where Δ = 𝔾_{m} is diagonally embedded into 𝔾_{m}^{n}. Assume that k is of characteristic ≠ p, containing a primitive pth root of unity. Then

(a) ed_{k}(T ⋊ P_{n}) = ed_{k}(T ⋊ P_{n}; p) = [n/p],  if n is not divisible by p.
(b) ed_{k}(T ⋊ P_{n}) = ed_{k}(T ⋊ P_{n}; p) = 2,  if n = p.
(c) ed_{k}(T ⋊ P_{n}) = ed_{k}(T ⋊ P_{n}; p) = n^{2}/p − n + 1,  if n = p^{r} for some r ≥ 2.
(d) ed_{k}(T ⋊ P_{n}) = ed_{k}(T ⋊ P_{n}; p) = p^{e}(n − p^{e}) − n + 1, in all other cases.

Here P_{n} is a Sylow p-subgroup of S_{n}, [n/p] is the integer part of n/p and p^{e} is the highest power of p dividing n.

The assumption that k contains a primitive pth root of unity is only needed for the proof of the first equality in parts (a) and (b).

Our proof of each part of this proposition will be based on the strategy outlined in Section 2, with G = T ⋊P_{n}. Before we proceeding with the proof of the proposition, we recall that the character lattice X(T) is naturally isomorphic to

{(a_{1}, …, a_{n}) ∈ ℤ^{n} | a_{1} + ⋯ + a_{n} = 0} ,

where we identify the character

(t_{1}, …, t_{n}) → t_{1}^{a_{1}} … t_{n}^{a_{n}}

of T = 𝔾_{m}^{n}/Δ with (a_{1}, …, a_{n}) ∈ ℤ^{n}. Note that (t_{1}, …, t_{n}) is viewed as an element of 𝔾_{m}^{n} modulo the diagonal subgroup Δ, so the above character is well defined if and only if a_{1} + ⋯ + a_{n} = 0. An element σ of S_{n} (and in particular, of P_{n} ⊂ S_{n}) acts on a = (a_{1}, …, a_{n}) ∈ X(T) by naturally permuting a_{1}, …, a_{n}.

For notational convenience, we will denote by a_{i,j} the element of (a_{1}, …, a_{n}) ∈ X(T) such that a_{i} = 1, a_{j} = −1 and a_{h} = 0 for every h ≠ i, j.

We also recall that for n = p^{r} the Sylow p-subgroup P_{n} of S_{n} can be described inductively as the wreath product

P_{p^{r}} ≅ P_{p^{r−1}} ≀ ℤ/p ≅ (P_{p^{r−1}})^{p} ⋊ ℤ/p .

For general n, P_{n} is the direct product of certain P_{p^{r}}, see Section 8.

Proof of Proposition 5.1(a). Step (i): Since n is not divisible by p, we may assume that P_{n} is contained in S_{n−1}, where we identify S_{n−1} with the subgroup of S_{n} consisting of permutations σ ∈ S_{n} such that σ(1) = 1.

We will now construct a generically free linear representation V of T ⋊ S_{n−1} of dimension n−1+[n/p]. Restricting this representation to T ⋊P_{n}, we will obtain a generically free linear representation of dimension n−1+[n/p]. This will show that ed(T ⋊ P_{n}) ≤ [n/p].

To construct V, let Λ = {a_{1,i} | i = 2, …, n} and let W be a [n/p]- dimensional faithful linear representation of P_{n} constructed in the proof of Corollary 4.2 (we may adjoin pth roots of unity to k if needed, see Remark 2.2). Applying Lemma 3.2(b), we see that V = V_{Λ} × W is generically free.

Step (ii): Since the natural projection p: T ⋊ P_{n} → P_{n} has a section, so does the map p^{∗}: H^{1}(K, T ⋊ P_{n}) → H^{1}(K, P_{n}) of Galois cohomology sets. Hence, p^{∗} is surjective for every field K/k. This implies that

ed(T ⋊ P_{n}) ≥ ed(P_{n}; p) = [n/p] .

Here ed(P_{n}; p) = ed(S_{n}; p) by Lemma 4.1 and ed(S_{n}; p) = [n/p] by Corollary 4.2. □

Remark 5.2. We will now outline a different and perhaps more conceptual proof of the upper bound ed(N; p) ≤ [n/p] of Theorem 1.1(a). As we pointed out in the introduction, ed(N; p) is the essential dimension at p of the functor

H^{1}( ∗ , N): K ↦ { K-isomorphism classes of pairs (A, L) },

where A is a degree n central simple algebras over K, L is a maximal étale subalgebra of A. Similarly, ed(S_{n}; p) is the essential dimension at p of the functor

H^{1}( ∗ , S_{n}): K ↦ {K-isomorphism classes of n-dimensional étale algebras L/K }.

Let α: H^{1}( ∗ , S_{n}) → H^{1}( ∗ , N) be the map taking an n-dimensional étale algebra L/K to (End_{K}(L), L). Here we embed L in End_{K}(L) ≃ M_{n}(K) via the regular action of L on itself.

It is easy to see that, in the terminology of [Me, Section 1.3], α is psurjective. That is, for any class (A, L) in H^{1}(K, N) there exists a prime-to-p extension K′/K such that (A ⊗_{K} K′, L ⊗_{K} K′) lies in the image of α. In fact, any K′/K of degree prime-to-p which splits A will do; indeed, by the Skolem-Noether theorem, any two embeddings of L ⊗_{K} K′ into M_{n}(K′) are conjugate. By [Me, Proposition 1.3], we conclude that ed(N; p) ≤ ed(S_{n}; p). Combining this with Corollary 4.2 yields the desired inequality ed(N; p) ≤ [n/p]. □

Proof of Proposition 5.1(b). Here n = p and P_{n} ≃ ℤ/p is generated by the p-cycle (1, 2, …, n). We follow the strategy outlined in Section 2.

Step (i): To show that ed_{k}(T ⋊ P_{n}) ≤ 2, we will construct a generically free k-representation of T ⋊ P_{n} of dimension 2 + dim(T ⋊ P_{n}) = n + 1.

Let Λ = {a_{1,2}, …, a_{p−1,p}, a_{p,1}} and V = V_{Λ}×L, where L is a 1-dimensional faithful representation of P_{n} ≃ ℤ/p and T ⋊ P_{n} acts on L via the natural projection T ⋊P_{n} → P_{n}. Note that dim(V) = |Λ|+1 = n+1. Since Λ generates X(T), Lemma 3.2(b) tells us that V is a generically free representation of T ⋊ P_{n}.

Step (ii): Recall that ed_{k}(T ⋊ P_{n}; p) = ed_{k}(N; p) by Lemma 4.1. On the other hand, as we mentioned in the introduction,

ed_{k}(N; p) ≥ ed_{k}(PGL_{p}; p) = 2 ;

see (2) and (3). This completes the proof of Proposition 5.1(b) and of Theorem 1.1(b). □

6. Proof of Theorem 1.1 part (c): The upper bound

In the next two sections we will prove Proposition 5.1(c) and hence, Theorem 1.1(c). We will assume that n = p^{r} for some r ≥ 2 and follow the strategy of Section 2. In this section we will carry out Step (i). That is, we will construct construct a generically free representation V of T ⋊ P_{n} of dimension p^{2r−1}. This will show that ed(T ⋊ P_{n}) ≤ p^{2r−1} − p^{r} + 1. Our V will be of the form V_{Λ} for a particular P_{n}-invariant Λ ⊂ X(T), following the recipe of Section 3. Note that this construction (and thus the above inequality) will not require any assumption on the base field k.

For notational convenience, we will subdivide the integers 1, 2, …, p^{r} into p “big blocks” B_{1}, …, B_{p}, where each B_{i} consists of the p^{r−1} consecutive integers (i − 1)p^{r−1} + 1, (i − 1)p^{r−1} + 2, …, ip^{r−1}.

We define Λ ⊂ X(T) as the P_{n}-orbit of the element

a_{1,p^{r−1}+1} = (\underbrace{1, 0, …, 0}_{B_{1}}, \underbrace{−1, 0, …, 0}_{B_{2}}, \underbrace{0, 0, …, 0}_{B_{3}}, …, \underbrace{0, 0, …, 0}_{B_{p}})

in X(T). Thus, Λ consists of elements a_{α,β}, subject to the condition that if α lies in the big block B_{i} then β has to lie in B_{j}, where j − i ≡ 1 modulo p. There are p^{r} choices for α. Once α is chosen, there are exactly p^{r−1} further choices for β. Thus

|Λ| = p^{r} · p^{r−1} = p^{2r−1} .

As described in Section 3, we obtain a linear representation V_{Λ} of T ⋊ P_{n} of the desired dimension

dim(V_{Λ}) = |Λ| = p^{2r−1} .

It remains to prove that V_{Λ} is generically free. By Lemma 3.3 it suffices to show that

(i) Λ generates X(T) as an abelian group and

(ii) the P_{n} action on the kernel of the natural morphism ϕ: ℤ[Λ] → X(T) is faithful.

The elements a_{α,β} clearly generate X(T) as an abelian group, as α and β range over 1, 2, …, p^{r}. Thus in order to prove (i) it suffices to show that Span_{ℤ}(Λ) contains every element of this form. Suppose α lies in the big block B_{i} and β in B_{j}. If j − i ≡ 1 (mod p), then a_{α,β} lies in Λ and there is nothing to prove. If j − i ≡ 2 (mod p) then choose some γ ∈ B_{i+1} (where the subscript i + 1 should be viewed modulo p) and write

a_{α,β} = a_{α,γ} + a_{γ,β} .

Since both terms on the right are in Λ, we see that in this case a_{α,β} ∈ Span_{ℤ}(Λ). Using this argument recursively, we see that a_{α,β} also lies in Span_{ℤ}(Λ) if j − i ≡ 3, …, p (mod p), i.e., for all possible i and j. This proves (i).

To prove (ii), denote the kernel of ϕ by M. Since P_{n} is a finite p-group, every normal subgroup of P_{n} intersects the center of P_{n}, which we shall denote by Z_{n}. Thus it suffices to show that Z_{n} acts faithfully on M.

Recall that Z_{n} is the cyclic subgroup of P_{n} of order p generated by the product of disjoint p-cycles

σ_{1} · . . . · σ_{p^{r−1}} = (1 … p)(p + 1 … 2p) … (p^{r} − p + 1, …, p^{r}) .

Since |Z_{n}| = p, it either acts faithfully on M or it acts trivially, so we only need to check that the Z_{n}-action on M is non-trivial. Indeed, Z_{n} does not fix the non-zero element

a_{1,p^{r−1}+1} + a_{p^{r−1}+1,2p^{r−1}+1} + ⋯ + a_{(p−1)p^{r−1}+1,1} ∈ ℤ[Λ]

which lies in M. This completes the proof of the upper bound of Proposition 5.1 and Theorem 1.1(c). □

7. Theorem 1.1 part (c): The lower bound

In this section we will continue to assume that n = p^{r}. We will show that

(8)  ed(N; p) ≥ p^{2r−1} − p^{r} + 1 ,

thus completing the proof of Proposition 5.1(c) and Theorem 1.1(c). Let

(9)  q := p^{e}, where e ≥ 1 if p is odd and e ≥ 2 if p = 2.

be a power of p. The specific choice of e will not be important in the sequel; in particular, the reader may assume that q = p if p is odd and q = 4, if p = 2. Whatever e we choose, q = p^{e} will remain unchanged for the rest of this section.

We now recall that if k′/k is a field extension then

ed_{k}(N; p) ≥ ed_{k′}(N; p) ;

cf. [Me, Proposition 1.5(1)]. Thus for the purpose of proving (8) we may replace k by k′. In particular, we may assume that k′ contains a primitive qth root of unity.

Let T_{(q)} = μ_{q}^{n}/μ_{q} be the q-torsion subgroup of T = 𝔾_{m}^{n}/Δ. Applying the inequality (7) to G = T ⋊ P_{n} and its finite subgroup H = T_{(q)} ⋊ P_{n}, we obtain

ed(T ⋊ P_{n}; p) ≥ ed(T_{(q)} ⋊ P_{n}; p) − p^{r} + 1 .

Thus it suffices to show that

(10)  ed(T_{(q)} ⋊ P_{n}; p) ≥ p^{2r−1} .

The advantage of replacing T ⋊ P_{n} by T_{(q)} ⋊ P_{n} is that T_{(q)} ⋊ P_{n} is a finite p-group, so that we can apply the following recent result of Karpenko and Merkurjev [KM].

Theorem 7.1. Let G be a finite p-group and k be a field containing a primitive pth root of unity. Then ed_{k}(G; p) = ed_{k}(G) = the minimal value of dim(V), where V ranges over all faithful linear k-representations G → GL(V).

Now recall that we are assuming that k contains a primitive qth root of unity and hence, a primitive pth root of unity. Hence, Theorem 7.1 applies in our situation. That is, in order to prove (10) it suffices to show that T_{(q)} ⋊ P_{n} does not have a faithful linear representation of dimension < p^{2r−1}. Lemma 3.1 further reduces this representation-theoretic assertion to the combinatorial statement of Proposition 7.2 below.

Before stating Proposition 7.2 we recall that the character lattice of T_{(q)} ≃ μ_{q}^{n}/μ_{q} is

X_{n} := {(a_{1}, …, a_{n}) ∈ (ℤ/qℤ)^{n} | a_{1} + ⋯ + a_{n} = 0 in ℤ/qℤ },

where we identify the character

(t_{1}, …, t_{n}) → t_{1}^{a_{1}} … t_{n}^{a_{n}}

of T_{(q)} with (a_{1}, …, a_{n}) ∈ (ℤ/qℤ)^{n}. Here (t_{1}, …, t_{n}) stands for an element of μ_{q}^{n}, modulo the diagonally embedded μ_{q}, so the above character is well defined if and only if a_{1}+⋯+a_{n} = 0 in ℤ/qℤ. (This is completely analogous to our description of the character lattice of T in the previous section.) Note that X_{n} depends on the integer q = p^{e}, which we assume to be fixed throughout this section.

Proposition 7.2. Let n = p^{r} and P_{n} be a Sylow p-subgroup of S_{n}. If Λ is a P_{n}-invariant generating subset of X_{n} then |Λ| ≥ p^{2r−1} for any r ≥ 1.

Our proof of Proposition 7.2 will rely on the following special case of Nakayama’s Lemma [AM, Proposition 2.8].

Lemma 7.3. Let q = p^{e} be a prime power, M = (ℤ/qℤ)^{d} and Λ be a generating subset of M (as an abelian group). If we remove from Λ all elements that lie in pM, the remaining set, Λ\pM, will still generate M. □

Proof of Proposition 7.2. We argue by induction on r. For the base case, set r = 1. We need to show that |Λ| ≥ p. Assume the contrary. In this case P_{n} is a cyclic p-group, and every non-trivial orbit of P_{n} has exactly p elements. Hence, |Λ| < p is only possible if every element of Λ is fixed by P_{n}. Since we are assuming that Λ generates X_{n} as an abelian group, we conclude that P_{n} acts trivially on X_{n}. This can happen only if p = q = 2. Since these values are ruled out by our definition (9) of q, we have proved the proposition for r = 1.

In the previous section we subdivided the integers 1, 2, …, p^{r} into p “big blocks” B_{1}, …, B_{p}^{r−1} of length p. Now we will now work with “small blocks” b_{1}, …, b_{p^{r−1}}, where b_{j} consists of the p consecutive integers

(j − 1)p + 1, (j − 1)p + 2, …, jp .

We can identify P_{p^{r−1}} with the subgroup of P_{p^{r}} that permutes the small blocks b_{1}, …, b_{p^{r−1}} without changing the order of the elements in each block.

For the induction step, assume r ≥ 2 and consider the homomorphism Σ: X_{p^{r}} → X_{p^{r−1}} given by

(11)  a = (a_{1}, a_{2}, …, a_{p^{r}}) ↦ s = (s_{1}, …, s_{p^{r−1}}) ,

where s_{i} = a_{(i−1)p+1} + a_{(i−1)p+2} + … + a_{ip} is the sum of the entries of a in the ith small block b_{i}. Thus

(i) if Λ generates X_{p^{r}} then Σ(Λ) generates X_{p^{r−1}}.

(ii) if Λ is a P_{p^{r}}-invariant subset of X_{p^{r}} then Σ(Λ) is a P_{p^{r−1}}-invariant subset of X_{p^{r−1}}.

Let us remove from Σ(Λ) all elements which lie in pX_{p^{r−1}}. The resulting set, Σ(Λ) \ pX_{p^{r−1}}, is clearly P_{p^{r−1}}-invariant. By Lemma 7.3 this set generates X_{p^{r−1}}. Thus by the induction assumption |Σ(Λ) \ pX_{p^{r−1}}| ≥ p^{2r−3}.

We claim that the fiber of each element s = (s_{1}, …, s_{p^{r−1}}) in Σ(Λ)\pX_{p^{r−1}} has at least p^{2} elements in Λ. If we can show this, then we will be able to conclude that

|Λ| ≥ p^{2} · |Σ(Λ) \ pX_{p^{r−1}}| ≥ p^{2} · p^{2r−3} = p^{2r−1} ,

thus completing the proof of Proposition 7.2.

Let σ_{i} be the single p-cycle, cyclically permuting the elements in the small block b_{i}. To prove the claim, note that the subgroup

⟨σ_{i} | i = 1, …, p^{r−1}⟩ ≃ (ℤ/pℤ)^{p^{r−1}}

of P_{n} acts on each fiber of Σ.

To simplify the exposition in the argument to follow, we introduce the following bit of terminology. Let us say that a ∈ (ℤ/qℤ)^{n} is scalar in the small block b_{i} if all the entries of a in the block b_{i} are the same, i.e., if

a_{(i−1)p+1} = a_{(i−1)p+2} = ⋯ = a_{ip} .

We are now ready to prove the claim. Suppose a = (a_{1}, …, a_{p^{r}}) ∈ X_{p^{r}} lies in the preimage of s = (s_{1}, …, s_{p^{r−1}}), as in (11). If a is scalar in the small block b_{i} then clearly

s_{i} = a_{(i−1)p+1} + a_{(i−1)p+2} + ⋯ + a_{ip} ∈ pℤ/qℤ .

Since we are assuming that s lies in

Σ(Λ) \ pX_{p^{r−1}} ,

s must have at least two entries that are not divisible by p, say, s_{i} and s_{j}. (Recall that s_{1} + ⋯ + s_{p^{r}} = 0 in ℤ/qℤ, so s cannot have exactly one entry not divisible by p.) Thus a is non-scalar in the small blocks b_{i} and b_{j}. Consequently, the elements σ_{i}^{α}σ_{j}^{β}(a) are distinct, as α and β range between 0 and p − 1. All of these elements lie in the fiber of s under Σ. Therefore we conclude that this fiber contains at least p^{2} distinct elements. This completes the proof of the claim and thus of Proposition 7.2, Proposition 5.1(c) and Theorem 1.1(c). □

8. Proof of Theorem 1.1 part (d)

In this section we assume that n is divisible by p but is not a power of p. We will modify the arguments of the last two sections to show that

ed(T ⋊ P_{n}) = ed(T ⋊ P_{n}; p) = p^{e}(n − p^{e}) − n + 1 ,

where p^{e} is the highest power of p dividing n. This will complete the proof of Proposition 5.1 and thus of Theorem 1.1.

Write out the p-adic expansion

(12)  n = n_{1}p^{e_{1}} + n_{2}p^{e_{2}} + ... + n_{u}p^{e_{u}},

of n, where 1 ≤ e = e_{1} < e_{2} < ... < e_{u}, and 1 ≤ n_{i} < p for each i. Subdivide the integers 1, ..., n into n_{1} + ... + n_{u} blocks B_{j}^{i} of length p^{e_{i}}, for j ranging over 1, 2, ..., n_{i}. By our assumption there are at least two such blocks. The Sylow subgroup P_{n} is a direct product

P_{n} = (P_{p^{e_{1}}} )^{n_{1}} × ⋯ × (P_{p^{eu}})^{n_{u}}

where each P_{p^{e_{i}}} acts on one of the blocks B_{j}^{i}.

Once again we will use the strategy outlined in Section 2.

Step (i): We will construct a generically free representation of T ⋊ P_{n} of dimension p^{e_{1}}(n − p^{e_{1}}). This will prove the upper bound ed_{k}(T ⋊ P_{n}) ≤ p^{e_{1}}(n − p^{e_{1}}). Note that this construction (and thus the above inequality) do not require any assumption on the field k.

To construct this representation, let Λ ⊂ X(T) be the union of the P_{n}- orbits of the elements

a_{1,j+1} where j = p^{e_{1}}, ..., n_{1}p^{e_{1}}, n_{1}p^{e_{1}} + p^{e_{2}}, ....., n − p^{e_{u}}

i.e., the union of the P_{n}-orbits of elements of the form (1, 0 …, 0, −1, 0, …, 0), where 1 appears in the first position of the first block and −1 appears in the first position of one of the other blocks. For a_{α,β} in Λ there are p^{e_{1}} choices for α and n − p^{e_{1}} choices for β. Thus

dim(V_{Λ}) = |Λ| = p^{e_{1}}(n − p^{e_{1}}) .

It is not difficult to see that Λ generates X(T) as an abelian group. To conclude with Lemma 3.3 that V_{Λ} is a generically free representation of T ⋊ P_{n}, it remains to show that the P_{n}-action on the kernel of the natural morphism ϕ: ℤ[Λ] → X(T) is faithful when e_{1} ≥ 1. As in section 6 we only need to check that the center Z_{n} of P_{n} acts faithfully on the kernel. Let σ be a non trivial element of Z_{n} = (Z_{p^{e_{1}}} )^{n_{1}} × ⋯ × (Z_{p^{eu}})^{n_{u}}. We may assume that the first component of σ in the above direct product is non-trivial, and therefore σ permutes elements in the first block B_{1}^{1} cyclically. Note that B_{1}^{1} is of size at least p as e = e_{1} ≥ 1, and that we have at least 2 blocks. The second block is also of size ≥ p and if p = 2, at least of size 4 by (12). It follows from this that σ does not fix the non-zero element

a_{1,p^{e}+1} − a_{1,p^{e}+2} + a_{2,p^{e}+2} − a_{2,p^{e}+1}

which lies in the kernel of ϕ.

Step (ii): We now want to prove the lower bound, ed(T ⋊ P_{n}; p) ≥ p^{e_{1}}(n − p^{e_{1}}) − n + 1. Arguing as in Section 7 (and using the same notation, with q = p), it suffices to show that ed(T_{(p)} ⋊ P_{n}; p) ≥ p^{e_{1}}(n − p^{e_{1}}). By the Karpenko-Merkurjev theorem 7.1 this is equivalent to showing that every faithful representation of T_{(p)} ⋊ P_{n} has dimension ≥ p^{e_{1}}(n − p^{e_{1}}). By Lemma 3.1 it now suffices to prove the following lemma.

Lemma 8.1. Let n be a positive integer, P_{n} be the Sylow subgroup of S_{n}, p^{e} be the highest power of p dividing n, and

X_{n} := {(a_{1}, …, a_{n}) ∈ (ℤ/pℤ)^{n} | a_{1} + ⋯ + a_{n} = 0 in ℤ/pℤ }.

Then every P_{n}-invariant generating subset of X_{n} has at least p^{e}(n − p^{e}) elements.

In the statement of the lemma we allow e = 0, to facilitate the induction argument. For the purpose of proving the lower bound in Proposition 5.1(d) we only need this lemma for e ≥ 1.

Proof. Once again, we consider the p-adic expansion (12) of n, with 0 ≤ e_{1} < e_{2} < ... < e_{u} and 1 ≤ n_{i} < p. We may assume that n is not a power of p, since otherwise the lemma is vacuous.

We will argue by induction on e = e_{1}. For the base case, let e_{1} = 0. Here the lemma is obvious: since X_{n} has rank n − 1, every generating set (P_{n}-invariant or not) has to have at least n − 1 elements.

For the induction step, we may suppose e = e_{1} ≥ 1; in particular, n is divisible by p. Define Σ : X_{n} → X_{n/p} by sending (a_{1}, ...., a_{n}) to (s_{1}, ..., s_{n/p}), where

s_{j} = a_{(j−1)p+1} + ⋯ + a_{jp}

for j = 1, …, n/p. Arguing as in Section 7 we see that Σ(Λ) \ pX_{n/p} is a (P_{p^{e_{1}−1}})^{n_{1}} × ⋯ × (P_{p^{eu−1}})^{n_{u}}-invariant generating subset of X_{n/p} and that every

s ∈ Σ(Λ) \ pX_{n/p}

has at least p^{2} preimages in Λ. By the induction assumption,

|Σ(Λ) \ pX_{n/p}| ≥ p^{e−1}(\frac{n}{p} − p^{e−1})

and thus

|Λ| ≥ p^{2} · p^{e−1}(\frac{n}{p} − p^{e−1}) = p^{e}(n − p^{e})

This completes the proof of Lemma 8.1 and thus of parts (d) of Proposition 5.1 and of Theorem 1.1. □

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