Some Consequences of

the Karpenko-Merkurjev Theorem

To Andrei Alexandrovich Suslin

on the occasion of his 60th birthday

Aurel Meyer^{1}, ZinovyReichstein^{2}

Received: December 12, 2008 Revised: May 1, 2009

Abstract. We use a recent theorem of N. A. Karpenko and A. S. Merkurjev to settle several questions in the theory of essential dimension.

2010 Mathematics Subject Classification: 20D15, 20C15, 20G15 Keywords and Phrases: Essential dimension, linear representation, p-group, algebraic torus

1. Introduction

Let k be a field, Fields_{k} be the category of field extensions K/k, Sets be the category of sets, and F: Fields_{k} → Sets be a covariant functor. Given a tower of field extensions k ⊂ K ⊂ L, we will denote the image of a ∈ F (K) under the natural map F (K) → F (L) by a_{L}. Conversely, if b ∈ F (L) lies in the image of this map, we will say that b descends to K. Given a field extension K/k and b ∈ F (L), the essential dimension ed_{k}(b) of b is defined as the minimal transcendence degree trdeg_{k}(K), as K ranges over all intermediate subfields k ⊂ K ⊂ L such that b descends to K. Informally speaking, this is the minimal number of parameters one needs to define b. The essential dimension ed_{k}(F) of the functor F is the maximal value of ed_{k}(b), as L ranges over all field extensions of k and b ranges over F (L). Informally speaking, this is the minimal number of parameters required to define any object of F. The essential dimension ed_{k}(b; p) at a prime p is defined as the minimum of ed_{k}(b_{L′}), taken over all finite field extensions L′/L such that the degree [L′ : L] is prime to p. The essential dimension of ed_{k}(F; p) of F at a prime p is the supremum of ed_{k}(b; p) taken over all b ∈ F (L) and over all field extensions L/k. An important example where the above notions lead to a rich theory is the nonabelian cohomology functor F_{G} = H^{1}(∗, G), sending a field K/k to the set H^{1}(K, G) of isomorphism classes of G-torsors over Spec(K), in the fppf topology. Here G is an algebraic group defined over k. The essential dimension of this functor can be thought of as a numerical measure of complexity of G-torsors over fields or, alternatively, as the minimal number of parameters required to define a versal G-torsor. In the case where G is a finite (constant) group defined over k, which will be the main focus of this paper, ed_{k}(G) is the minimal number of parameters required to describe all G-Galois extensions. For details on the notion of essential dimension of a finite group we refer the reader to [BuR], [Re] or [JLY, Chapter 8], on the notion of essential dimension of a functor to [BF] or [BRV_{2}] and on essential dimension at a prime p to [Me]. N. Karpenko and A. Merkurjev [KM] recently proved the following formula for the essential dimension of a (finite) p-group.

1Partially supported by a University Graduate Fellowship at the University of British Columbia.

2Partially supported by NSERC Discovery and Accelerator Supplement grants.

Documenta Mathematica · Extra Volume Suslin (2010) 445–457

Theorem 1.1. Let G be a p-group and k be a field of characteristic ≠ p containing a primitive pth root of unity. Then

ed_{k}(G; p) = ed_{k}(G) = min dim(V) ,

where the minimum is taken over all faithful k-representations G ↪ GL(V).

The purpose of this paper is to explore some of the consequences of this theorem. The following notation will be used throughout. We will fix a prime p and a base field k such that

(1)  char(k) ≠ p and k contains ζ,

where ζ is a primitive pth root of unity if p ≥ 3 and a primitive 4th root of unity if p = 2. For a finite group H, we will denote the intersection of the kernels of all multiplicative characters χ: H → k^{∗} by H′. In particular, if k contains an eth root of unity, where e is the exponent of H, then H′ = [H, H] is the commutator subgroup of H. All p-groups in this paper will be assumed to be finite. Given a p-group G, we set C(G) to be the center of G and

(2)  C(G)_{p} := {g ∈ C(G) | g^{p} = 1}

to be the p-torsion subgroup of C(G). We will view C(G)_{p} and its subgroups as 𝔽_{p}-vector spaces, and write “dim_{𝔽_{p}}” for their dimensions. We further set

(3)  K_{i} := ⋂_{[G:H]=p^{i}} H′ and C_{i} := K_{i} ∩ C(G)_{p} .

for every i ≥ 0, K_{−1} := G and C_{−1} := K_{−1} ∩ C(G)_{p} = C(G)_{p}. Our first main result is following theorem. Part (b) may be viewed as a variant of Theorem 1.1.

Consequences of the Karpenko-Merkurjev Theorem 447

Theorem 1.2. Let G be a p-group, k be a base field satisfying (1) and ρ: G ↪ GL(V) be a faithful linear k-representation of G. Then

(a) ρ has minimal dimension among the faithful linear representations of G defined over k if and only if for every i ≥ 0 the irreducible decomposition of ρ has exactly

dim_{𝔽_{p}} C_{i−1} − dim_{𝔽_{p}} C_{i}

irreducible components of dimension p^{i}, each with multiplicity 1.

(b) ed_{k}(G; p) = ed_{k}(G) = ∑_{i=0}^{∞}(dim_{𝔽_{p}} C_{i−1} − dim_{𝔽_{p}} C_{i})p^{i}.

Note that K_{i} = C_{i} = {1} for large i (say, if p^{i} ≥ |G|), so only finitely many terms in the above infinite sum are non-zero. We also remark that the minimal number of irreducible components in a faithful representations of a finite group (but not necessarily a p-group) was studied in [Ta, Na], see also [Lo, Section 4]. We will prove Theorem 1.2 in section 2; the rest of the paper will be devoted to its applications. The main results we will obtain are summarized below.

Classification of p-groups of essential dimension ≤ p.

Theorem 1.3. Let p be a prime, k be as in (1) and G be a p-group such that G′ ≠ {1}. Then the following conditions are equivalent.

(a) ed_{k}(G) ≤ p,

(b) ed_{k}(G) = p,

(c) The center C(G) is cyclic and G has a subgroup H of index p such that H′ = {1}.

Note that the assumption that G′ ≠ {1} is harmless. Indeed, if G′ = {1} then by Theorem 1.2(b) ed_{k}(G) = rank (G); cf. also [BuR, Theorem 6.1] or [BF, section 3].

Essential dimension of p-groups of nilpotency class 2.

Theorem 1.4. Let G be a p-group of exponent e and k be a field of characteristic ≠ p containing a primitive e-th root of unity. Suppose the commutator subgroup [G, G] is central in G. Then

(a) ed_{k}(G; p) = ed_{k}(G) ≤ rank C(G) + rank [G, G](p^{⌊m/2⌋} − 1), where p^{m} is the order of G/C(G).

(b) Moreover, if [G, G] is cyclic then |G/C(G)| is a complete square and equality holds in (a). That is, in this case

ed_{k}(G; p) = ed_{k}(G) = \sqrt{|G/C(G)|} + rank C(G) − 1 .

Essential dimension of a quotient group. C. U. Jensen, A. Ledet and N. Yui asked if ed_{k}(G) ≥ ed_{k}(G/N) for every finite group G and normal subgroup N ◁ G; see [JLY, p. 204]. The following theorem shows that this inequality is false in general.

Theorem 1.5. Let p be a prime and k be a field of characteristic ≠ p containing a primitive pth root of unity. For every real number λ > 0 there exists a p-group G and a central subgroup H of G such that ed_{k}(G/H) > λ ed_{k}(G).

Essential dimension of SL_{n}(ℤ). G. Favi and M. Florence [FF] showed that ed_{k}(GL_{n}(ℤ)) = n for every n ≥ 1 and ed_{k}(SL_{n}(ℤ)) = n − 1 for every odd n. For details, including the definitions of ed_{k}(GL_{n}(ℤ)) and ed_{k}(SL_{n}(ℤ)), see Section 5. For even n Favi and Florence showed that ed_{k}(SL_{n}(ℤ)) = n − 1 or n and left the exact value of ed_{k}(SL_{n}(ℤ)) as an open question. In this paper we will answer this question as follows.

Theorem 1.6. Suppose k is a field of characteristic ≠ 2. Then

ed_{k}(SL_{n}(ℤ); 2) = ed_{k}(SL_{n}(ℤ)) = \begin{cases}n − 1, if n is odd, \\ n, if n is even \end{cases}

for any n ≥ 3.

Acknowledgement. Theorems 1.4(b) and 1.5 first appeared in the unpublished preprint [BRV_{1}] by P. Brosnan, the second author and A. Vistoli. We thank P. Brosnan and A. Vistoli for allowing us to include them in this paper. Theorem 1.4(b) was, in fact, a precursor to Theorem 1.1; the techniques used in [BRV_{1}] were subsequently strengthened and refined by Karpenko and Merkurjev [KM] to prove Theorem 1.1. The proof of Theorem 1.4(b) in Section 4 may thus be viewed as a result of reverse engineering. We include it here because it naturally fits into the framework of this paper, because Theorem 1.4(b) is used in a crucial way in [BRV_{2}], and because a proof of this result has not previously appeared in print. We are also grateful to R. Lötscher for pointing out and helping us correct an inaccuracy in the proof of Lemma 2.1.

2. Proof of Theorem 1.2

Throughout this section we assume k to be as in (1). An important role in the proof will be played by the p-torsion subgroup C(G)_{p} of the center of G and by the descending sequences

K_{−1} = G ⊃ K_{0} ⊃ K_{1} ⊃ K_{2} ⊃ … and
C_{−1} = C(G)_{p} ⊃ C_{0} ⊃ C_{1} ⊃ C_{2} ⊃ …

of characteristic subgroups of G defined in (3). To simplify the notation, we will write C for C_{−1} = C(G)_{p} for the rest of this section. We will repeatedly use the well-known fact that

(4) A normal subgroup N of G is trivial if and only if N ∩ C is trivial.

We begin with three elementary lemmas.

Lemma 2.1. K_{i} = ⋂_{dim(ρ)≤p^{i}} ker(ρ), where the intersection is taken over all irreducible representations ρ of G of dimension ≤ p^{i}.

Proof. Let j ≤ i. Recall that every irreducible representation ρ of G of dimension p^{j} is induced from a 1-dimensional representation χ of a subgroup H ⊂ G of index p^{j}; see [LG-P, (II.4)] for p ≥ 3 (cf. also [Vo]) and [LG-P, (IV.2)] for p = 2. (Note that our assumption (1) on the base field k is crucial here. In the case where k = ℂ a more direct proof can be found in [Se, Section 8.5]). Thus ker(ρ) = ker(ind_{H}^{G} χ) = ⋂_{g∈G} g ker(χ)g^{−1}, and since each g ker(χ)g^{−1} contains (gHg^{−1})′, we see that ker(ρ) ⊃ K_{j} ⊃ K_{i}. The opposite inclusion is proved in a similar manner. □

Lemma 2.2. Let ρ: G → GL(V) an irreducible representation of a p-group G. Then

(a) ρ(C) consists of scalar matrices. In other words, the restriction of ρ to C decomposes as χ ⊕ … ⊕ χ (dim(V) times), for some multiplicative character χ: C → 𝔾_{m}. We will refer to χ as the character associated to ρ.

(b) C_{i} = ⋂_{dim(ψ)≤p^{i}} ker(χ_{ψ}), where the intersection is taken over all irreducible G-representations ψ of dimension ≤ p^{i} and χ_{ψ}: C → 𝔾_{m} denotes the character associated to ψ. In particular, if dim(ρ) ≤ p^{i} then χ_{ρ} vanishes on C_{i}.

Proof. (a) follows from Schur’s lemma. (b) By Lemma 2.1

C_{i} = C ∩ ⋂_{dim(ψ)≤p^{i}} ker(ψ) = ⋂_{dim(ψ)≤p^{i}} (C ∩ ker(ψ)) = ⋂_{dim(ψ)≤p^{i}} ker(χ_{ψ}) .

□

Lemma 2.3. Let G be a p-group and ρ = ρ_{1} ⊕ … ⊕ ρ_{m} be the direct sum of the irreducible representations ρ_{i}: G → GL(V_{i}). Let χ_{i} := χ_{ρ_{i}}: C → 𝔾_{m} be the character associated to ρ_{i}.

(a) ρ is faithful if and only if χ_{1}, …, χ_{m} span C^{∗} as an 𝔽_{p}-vector space.

(b) Moreover, if ρ is of minimal dimension among the faithful representations of G then χ_{1}, …, χ_{m} form an 𝔽_{p}-basis of C^{∗}.

Proof. (a) By (4), Ker(ρ) is trivial if and only if Ker(ρ) ∩ C = ∩_{i=1}^{m} Ker(χ_{i}) is trivial. On the other hand, ∩_{i=1}^{m} Ker(χ_{i}) is trivial if and only if χ_{1}, …, χ_{m} span C^{∗}.

(b) Assume the contrary, say χ_{m} is a linear combination of χ_{1}, …, χ_{m−1}. Then part (a) tells us that ρ_{1} ⊕ … ⊕ ρ_{m−1} is a faithful representation of G, contradicting the minimality of dim(ρ). □

We are now ready to proceed with the proof of Theorem 1.2. Part (b) is an immediate consequence of part (a) and Theorem 1.1. We will thus focus on proving part (a). In the sequel for each i ≥ 0 we will set

δ_{i} := dim_{𝔽_{p}} C_{i−1} − dim_{𝔽_{p}} C_{i}

and

Δ_{i} := δ_{0} + δ_{1} + ⋯ + δ_{i} = dim_{𝔽_{p}} C − dim_{𝔽_{p}} C_{i} ,

where the last equality follows from C_{−1} = C. Our proof will proceed in two steps. In Step 1 we will construct a faithful representation μ of G such that for every i ≥ 0 exactly δ_{i} irreducible components of μ have dimension p^{i}. In Step 2 we will show that dim(ρ) ≥ dim(μ) for any other faithful representation ρ of G, and moreover equality holds if and only if ρ has exactly δ_{i} irreducible components of dimension p^{i}, for every i ≥ 0. Step 1: We begin by constructing μ. By definition,

C = C_{−1} ⊃ C_{0} ⊃ C_{1} ⊃ …,

where the inclusions are not necessarily strict. Dualizing this flag of 𝔽_{p}-vector spaces, we obtain a flag

(0) = (C^{∗})_{−1} ⊂ (C^{∗})_{0} ⊂ (C^{∗})_{1} ⊂ …

of 𝔽_{p}-subspaces of C^{∗}, where

(C^{∗})_{i} := {χ ∈ C^{∗} | χ is trivial on C_{i}} ≃ (C/C_{i})^{∗}.

Let Ass(C) ⊂ C^{∗} be the set of characters of C associated to irreducible representations of G, and let Ass_{i}(C) be the set of characters associated to irreducible representations of dimension p^{i}. Lemma 2.2(b) tells us that

Ass_{0}(C) ∪ Ass_{1}(C) ∪ ⋯ ∪ Ass_{i}(C) spans (C^{∗})_{i}

for every i ≥ 0. Hence, we can choose a basis χ_{1}, …, χ_{Δ_{0}} of (C^{∗})_{0} from Ass_{0}(C), then complete it to a basis χ_{1}, …, χ_{Δ_{1}} of (C^{∗})_{1} by choosing the last Δ_{1} − Δ_{0} characters from Ass_{1}(C), then complete this basis of (C^{∗})_{1} to a basis of (C^{∗})_{2} by choosing Δ_{2} − Δ_{1} additional characters from Ass_{2}(C), etc. We stop when C_{i} = (0), i.e., Δ_{i} = dim_{𝔽_{p}} C. By the definition of Ass_{i}(C), each χ_{j} is the associated character of some irreducible representation μ_{j} of G. By our construction

μ = μ_{1} ⊕ ⋯ ⊕ μ_{dim_{𝔽p} (C)}

has the desired properties. Indeed, since χ_{1}, …, χ_{dim_{𝔽p} (C)} form a basis of C^{∗}, Lemma 2.3 tells us that μ is faithful. On the other hand, by our construction exactly

δ_{i} − δ_{i−1} = dim_{𝔽_{p}} C_{i}^{∗} − dim_{𝔽_{p}} C_{i−1}^{∗} = dim_{𝔽_{p}} C_{i−1} − dim_{𝔽_{p}} C_{i}

of the characters χ_{1}, …, χ_{c} come from Ass_{i}(C). Equivalently, exactly

dim_{𝔽_{p}} C_{i−1} − dim_{𝔽_{p}} C

of the irreducible representations μ_{1}, …, μ_{c} are of dimension p^{i}.

Step 2: Let ρ: G → GL(V) be a faithful linear representation of G of the smallest possible dimension,

ρ = ρ_{1} ⊕ … ⊕ ρ_{c}

be its irreducible decomposition, and χ_{i}: C → 𝔾_{m} be the character associated to ρ_{i}. By Lemma 2.3(b), χ_{1}, …, χ_{c} form a basis of C^{∗}. In particular, c = dim_{𝔽_{p}} C and at most dim_{𝔽_{p}} C−dim_{𝔽_{p}} C_{i} of the characters χ_{1}, …, χ_{c} can vanish on C_{i}. On the other hand, by Lemma 2.2(b) every representation of dimension ≤ p^{i} vanishes on C_{i}. Thus if exactly d_{i} of the irreducible representations ρ_{1}, …, ρ_{c} have dimension p^{i} then

d_{0} + d_{1} + d_{2} + … + d_{i} ≤ dim_{𝔽_{p}} C − dim_{𝔽_{p}} C_{i}

for every i ≥ 0. For i ≥ 0, set D_{i} := d_{0} + ⋯ + d_{i} = number of representations of dimension ≤ p^{i} among ρ_{1}, …, ρ_{c}. We can now write the above inequality as

(5)  D_{i} ≤ Δ_{i} for every i ≥ 0.

Our goal is to show that dim(ρ) ≥ dim(μ) and that equality holds if and only if exactly δ_{i} of the irreducible representations ρ_{1}, …, ρ_{dim_{𝔽p} (C)} have dimension p^{i}. The last condition translates into d_{i} = δ_{i} for every i ≥ 0, which is, in turn equivalent to D_{i} = Δ_{i} for every i ≥ 0. Indeed, setting D_{−1} := 0 and Δ_{−1} := 0, we have,

dim(ρ) − dim(μ) = ∑_{i=0}^{∞}(d_{i} − δ_{i})p^{i} = ∑_{i=0}^{∞}(D_{i} − Δ_{i})p^{i} − ∑_{i=0}^{∞}(D_{i−1} − Δ_{i−1})p^{i}

= ∑_{i=0}^{∞}(D_{i} − Δ_{i})(p^{i} − p^{i+1}) ≥ 0 ,

where the last inequality follows from (5). Moreover, equality holds if and only if D_{i} = Δ_{i} for every i ≥ 0, as claimed. This completes the proof of Step 2 and thus of Theorem 1.2. □

3. Proof of Theorem 1.3

Since K_{0} = G′ is a non-trivial normal subgroup of G, we see that K_{0} ∩ C(G) and thus C_{0} = K_{0} ∩ C(G)_{p} is non-trivial. This means that in the summation formula of Theorem 1.2(b) at least one of the terms

(dim_{𝔽_{p}} C_{i−1} − dim_{𝔽_{p}} C_{i})p^{i}

with i ≥ 1 will be non-zero. Hence, ed_{k}(G) ≥ p; this shows that (a) and (b) are equivalent. Moreover, equality holds if and only if (i) dim_{𝔽_{p}} C_{−1} = 1, (ii) dim_{𝔽_{p}} C_{0} = 1 and (iii) C_{1} is trivial. Since we are assuming K_{0} = G′ ≠ {1} and hence, C_{0} = K_{0} ∩ C(G)_{p} ≠ {1} by (4), (ii) follows from (i) and thus can be dropped. It now suffices to prove that (i) and (iii) are equivalent to condition (c) of the theorem. Since C_{−1} = C(G)_{p}, (i) is equivalent to C(G) being cyclic. On the other hand, (iii) means that

(6)  K_{1} = ⋂_{[G:H]=p} H′

intersects C(G)_{p} trivially. Since K_{1} is a normal subgroup of G, (4) tells us that (iii) holds if and only if K_{1} = {1}. It remains to show that K_{1} = {1} if and only if H′ = {1} for some subgroup H of G of index p. One direction is obvious: if H′ = {1} for some H of index p then the intersection (6) is trivial. To prove the converse, assume the contrary: the intersection (6) is trivial but H′ ≠ {1} for every subgroup H of index p. Since every such H is normal in G (and so is H′), (4) tells us that that H′ ≠ {1} if and only if H′ ∩ C(G) ≠ {1}. Since C(G) is cyclic, the latter condition is equivalent to C(G)_{p} ⊂ H′. Thus

C(G)_{p} ⊂ K_{1} = ⋂_{[G:H]=p} H′ ,

contradicting our assumption that K_{1} = {1}. To sum up, we have shown that (c) is equivalent to conditions (i) and (iii) above, and that these conditions are in turn, equivalent to (a) (or to (b)). This completes the proof of Theorem 1.3.

Remark 3.1. p-groups that have a faithful representation of degree p over a field k, satisfying (1) are described in [LG-P, II.4, III.4, IV.2]; see also [Vo]. Combining this description with Theorem 1.1 yields the following variant of Theorem 1.3. Let k be a field satisfying (1) and G be a p-group such that G′ ≠ {1}. Then the following conditions are equivalent:

(a) ed_{k}(G) ≤ p,

(b) ed_{k}(G) = p,

(c) G is isomorphic to a subgroup of ℤ/p^{α} ≀ ℤ/p = (ℤ/p^{α})^{p} ⋊ ℤ/p, for some α ≥ 1 such that k contains a primitive root of unity of degree p^{α}. □

4. Proof of Theorems 1.4 and 1.5

Proof of Theorem 1.4. Since the commutator K_{0} = [G, G] is central, C_{0} = K_{0} ∩ C(G)_{p} is of dimension rank [G, G] and the p^{0} term in the formula of Theorem 1.2 is (rank C(G) − rank [G, G]). Let Q = G/C(G) which is abelian by assumption. Let h_{1}, ..., h_{s} be generators of [G, G], where s = rank [G, G], so that

[G, G] = ℤ/p^{e_{1}}h_{1} ⊕ ⋯ ⊕ ℤ/p^{e_{1}}h_{1} ,

written additively. For g_{1}, g_{2} ∈ G the commutator can then be expressed as

[g_{1}, g_{2}] = β_{1}(g_{1}, g_{2})h_{1} + … + β_{s}(g_{1}, g_{2})h_{s} .

Note that each β_{i}(g_{1}, g_{2}) depends on g_{1}, g_{2} only modulo the center C(G). Thus each β_{i} descends to a skew-symmetric bilinear form

Q × Q → ℤ/p^{e_{i}}

which, by a slight abuse of notation, we will continue to denote by β_{i}. Let p^{m} be the order of Q. For each form β_{i} there is an isotropic subgroup Q_{i} of Q of order at least p^{⌊(m+1)/2⌋} (or equivalently, of index at most p^{⌊m/2⌋} in Q); see [AT, Corollary 3]. Pulling these isotropic subgroups back to G, we obtain subgroups G_{1}, …, G_{s} of G of index ≤ p^{⌊m/2⌋} with the property that G′_{i} = [G_{i}, G_{i}] lies in the subgroup of C(G) generated by h_{1}, …, h_{i−1}, h_{i+1}, …, h_{s}. In particular, G′_{1} ∩ ⋯ ∩ G′_{s} = {1}. Thus, all K_{i} (and hence, all C_{i}) in (3) are trivial for i ≥ ⌊m/2⌋, and Theorem 1.2 tells us that

ed_{k}(G) = dim_{𝔽_{p}} C_{−1} − dim_{𝔽_{p}} C_{0} + ∑_{j=1}^{⌊m/2⌋}(dim_{𝔽_{p}} C_{j−1} − dim_{𝔽_{p}} C_{j})p^{j} ≤

dim_{𝔽_{p}} C_{−1} − dim_{𝔽_{p}} C_{0} + ∑_{j=1}^{⌊m/2⌋}(dim_{𝔽_{p}} C_{j−1} − dim_{𝔽_{p}} C_{j}) · p^{⌊m/2⌋} =

rank C(G) + rank [G, G](p^{⌊m/2⌋} − 1) .

(b) In general, the skew-symmetric bilinear forms β_{i} may be degenerate. However, if [G, G] is cyclic, i.e., s = 1, then we have only one form, β_{1}, which is easily seen to be non-degenerate. For notational simplicity, we will write β instead of β_{1}. To see that β is non-degenerate, suppose \overline{g} := g (modulo C(G)) lies in the kernel of β for some g ∈ G. Then by definition

β(g, g_{1}) = gg_{1}g^{−1}g_{1}^{−1} = 1

for every g_{1} ∈ G. Hence, g is central in G, i.e., \overline{g} = 1 in Q = G/C(G), as claimed. We conclude that the order of Q = G/C(G) is a perfect square, say p^{2i}, and Q contains a maximal isotropic subgroup I ⊂ Q of order p^{i} = \sqrt{|G/C(G)|}; see [AT, Corollary 4]. The preimage of I in G is a maximal abelian subgroup of index p^{i}. Consequently, K_{0} = [G, G], K_{1}, …, K_{i−1} are all of rank 1 and K_{i} is trivial, where p^{i} = \sqrt{|G/C(G)|}. Moreover, since all of these groups lie in [G, G] and hence, are central, we have C_{i} = (K_{i})_{p} and thus

dim_{𝔽_{p}}(C_{0}) = dim_{𝔽_{p}}(C_{1}) = … = dim_{𝔽_{p}}(C_{i−1}) = 1 and dim_{𝔽_{p}}(C_{i}) = 0 .

Specializing the formula of Theorem 1.4 to this situation, we obtain part (b). □

Proof of Theorem 1.5. Let Γ be the non-abelian group of order p^{3} given by generators x, y, z and relations x^{p} = y^{p} = z^{p} = [x, z] = [y, z] = 1, [x, y] = z. Choose a multiplicative character χ: H → k^{∗} of the subgroup A = ⟨x, z⟩ ≃ (ℤ/pℤ)^{2} which is non-trivial on the center ⟨z⟩ of Γ and consider the p-dimensional induced representation Ind_{A}^{Γ}(χ). Since the center ⟨z⟩ of Γ does not lie in the kernel of Ind_{A}^{Γ}(χ), we conclude that Ind_{A}^{Γ}(χ) is faithful. Thus we have constructed a faithful p-dimensional representation of Γ defined over k. Consequently

(7)  ed_{k}(Γ) ≤ p .

Taking the direct sum of n copies of this representation, we obtain a faithful representation of Γ^{n} of dimension np. Thus for any n ≥ 1 we have

(8)  ed_{k} Γ^{n} ≤ np .

(We remark that both (7) and (8) are in fact equalities. Indeed, if ζ_{p^{2}} is a primitive root of unity of degree p^{2} then

ed_{k}(Γ) ≥ ed_{k(ζ_{p^{2}} )}(Γ) = \sqrt{p^{2}} + 1 − 1 = p ,

where the middle equality follows from Theorem 1.4(b). Hence, we have ed_{k}(Γ) = p. Moreover, by [KM, Theorem 5.1], ed_{k} Γ^{n} = n · ed_{k}(Γ) = np. However, we will only need the upper bound (8) in the sequel.) The center of Γ is ⟨z⟩; denote it by C. The center of Γ^{n} is then isomorphic to C^{n}. Let H_{n} be the subgroup of C^{n} consisting of n-tuples (c_{1}, …, c_{n}) such that c_{1} ⋯ c_{n} = 1. The center C(Γ^{n}/H_{n}) of Γ^{n}/H_{n} is clearly cyclic of order p (it is generated by the class of the element (z, 1, …, 1) modulo H_{n}), and the commutator [Γ^{n}/H_{n}, Γ^{n}/H_{n}] is central. Hence,

(9)  ed_{k}(Γ^{n}/H_{n}) ≥ ed_{k(ζ^{2})}(Γ^{n}/H_{n}) = \sqrt{p^{2n}} + 1 − 1 = p^{n} ,

where the middle equality follows from Theorem 1.4(b). Setting G = Γ^{n} and H = H_{n}, and comparing (8) with (9), we see that the desired inequality ed_{k}(G/H) > λ ed_{k}(G) holds for suitably large n. □

5. Proof of Theorem 1.6

Recall that the essential dimension of the group GL_{n}(ℤ) over a field k, or ed_{k}(GL_{n}(ℤ)) for short, is defined as the essential dimension of this functor

H^{1}(∗, GL_{n}(ℤ)): K → {K-isomorphism classes of n-dimensional K-tori} ,

where K/k is a field extension. Similarly ed_{k}(SL_{n}(ℤ)) is defined as the essential dimension of the functor

H^{1}(∗, SL_{n}(ℤ)): K → {K-isomorphism classes of n-dimensional K-tori with ϕ_{T} ⊂ SL_{n}(ℤ) } ,

where ϕ_{T} : Gal(K) → GL_{n}(ℤ) is the natural representation of the Galois group of K on the character lattice of T. The essential dimensions ed_{k}(GL_{n}(ℤ); p) and ed_{k}(SL_{n}(ℤ); p) are respectively the essential dimensions of the above functors at a prime p. G. Favi and M. Florence [FF] showed that for Γ = GL_{n}(ℤ) or SL_{n}(ℤ),

(10)  ed_{k}(Γ) = max{ed_{k}(F)|F finite subgroup of Γ}.

From this they deduced that

ed_{k}(GL_{n}(ℤ)) = n, and ed_{k}(SL_{n}(ℤ)) = \begin{cases}n − 1, if n is odd, \\ n − 1 or n, if n is even. \end{cases}

For details, see [FF, Theorem 5.4]. Favi and Florence also proved that ed_{k}(SL_{2}(ℤ)) = 1 if k contains a primitive 12th root of unity and asked whether ed_{k}(SL_{n}(ℤ)) = n − 1 or n in the case where n ≥ 4 is even; see [FF, Remark 5.5]. In this section we will prove Theorem 1.6 which shows that the answer is always n. A minor modification of the arguments in [FF] shows that (10) holds also for essential dimension at a prime p:

(11)  ed_{k}(Γ; p) = max{ed_{k}(F; p)|F a finite subgroup of Γ},

where Γ = GL_{n}(ℤ) or SL_{n}(ℤ). The finite groups F that Florence and Favi used to find the essential dimension of GL_{n}(ℤ) and SL_{n}(ℤ) (n odd) are (ℤ/2ℤ)^{n} and (ℤ/2ℤ)^{n−1} respectively. Thus ed_{k}(GL_{n}(ℤ); 2) = ed_{k}(GL_{n}(ℤ)) = n for every n ≥ 1 and ed_{k}(SL_{n}(ℤ); 2) = ed_{k}(SL_{n}(ℤ)) = n − 1 if n is odd. Our proof of Theorem 1.6 will rely on part (b) of the following easy corollary of Theorem 1.2.

Corollary 5.1. Let G be a p-group, and k be as in (1).

(a) If C(G)_{p} ⊂ K_{i} then ed_{k}(G) is divisible by p^{i+1}.

(b) If C(G)_{p} ⊂ G′ then ed_{k}(G) is divisible by p.

(c) If C(G)_{p} ⊂ G^{(i)}, where G^{(i)} denotes the ith derived subgroup of G, then ed_{k}(G) is divisible by p^{i}.

Proof. (a) C(G)_{p} ⊂ K_{i} implies C_{−1} = C_{0} = ⋯ = C_{i}. Hence, in the formula of Theorem 1.2(b) the p^{0}, p^{1}, …, p^{i} terms appear with coefficient 0. All other terms are divisible by p^{i+1}, and part (a) follows.

(b) is an immediate consequence of (a), since K_{0} = G′.

(c) By [H, Theorem V.18.6] G^{(i)} is contained in the kernel of every p^{i−1}- dimensional representation of G. Lemma 2.1 now tells us that G^{(i)} ⊂ K_{i−1} and part (c) follows from part (a). □

Proof of Theorem 1.6. We assume that n = 2d ≥ 4 is even. To prove Theorem 1.6 it suffices to find a 2-subgroup F of SL_{n}(ℤ) of essential dimension n. Diagonal matrices and permutation matrices generate a subgroup of GL_{n}(ℤ) isomorphic to μ_{2}^{n} ⋊S_{n}. The determinant function restricts to a homomorphism

det: μ_{2}^{n} ⋊ S_{n} → μ_{2}

sending ((ϵ_{1}, …, ϵ_{n}), τ)) ∈ μ_{2}^{n} ⋊ S_{n} to the product ϵ_{1}ϵ_{2} ⋯ ϵ_{n} · sign(τ). Let P_{n} be a Sylow 2-subgroup of S_{n} and F_{n} be the kernel of det: μ_{2}^{n} ⋊ P_{n} → μ_{2}. By construction F_{n} is a finite 2-group contained in SL_{n}(ℤ). Theorem 1.6 is now a consequence of the following proposition.

Proposition 5.2. If char(k) ≠ 2 then ed_{k}(F_{2d}) = 2d for any d ≥ 2.

To prove the proposition, let

D_{2d} = {diag(ϵ_{1}, …, ϵ_{2d}) | each ϵ_{i} = ±1 and ϵ_{1}ϵ_{2} ⋯ ϵ_{2d} = 1}

be the subgroup of “diagonal” matrices contained in F_{2d}. Since D_{2d} ≃ μ_{2}^{2d−1} has essential dimension 2d − 1, we see that ed_{k}(F_{2d}) ≥ ed_{k}(D_{2d}) = 2d − 1. On the other hand the inclusion F_{2d} ⊂ SL_{2d}(ℤ) gives rise to a 2d-dimensional representation of F_{2d}, which remains faithful over any field k of characteristic ≠ 2. Hence, ed_{k}(F_{2d}) ≤ 2d. We thus conclude that

(12)  ed_{k}(F_{2d}) = 2d − 1 or 2d.

Using elementary group theory, one easily checks that

(13)  C(F_{2d}) ⊂ [F_{2d}, F_{2d}] ⊂ F′_{2d} .

Thus, if k′ ⊃ k is a field as in (1), ed_{k′}(F_{2d}) is even by Corollary 5.1; since ed_{k}(F_{2d}) ≥ ed_{k′}(F_{2d}), (12) now tells us that ed_{k}(F_{2d}) = 2d. This completes the proof of Proposition 5.2 and thus of Theorem 1.6. □

Remark 5.3. The assumption that d ≥ 2 is essential in the proof of the inclusion (13). In fact, F_{2} ≃ ℤ/4ℤ, so (13) fails for d = 1.

Remark 5.4. Note that for any integers m, n ≥ 2, F_{m+n} contains the direct product F_{m} × F_{n}. Thus

ed_{k}(F_{m+n}) ≥ ed_{k}(F_{m} × F_{n}) = ed_{k}(F_{m}) + ed_{k}(F_{n}) ,

where the last equality follows from [KM, Theorem 5.1]. Thus Proposition 5.2 only needs to be proved for d = 2 and 3 (or equivalently, n = 4 and 6); all other cases are easily deduced from these by applying the above inequality recursively, with m = 4. In particular, the group-theoretic inclusion (13) only needs to be checked for d = 2 and 3. Somewhat to our surprise, this reduction does not appear to simplify the proof of Proposition 5.2 presented above to any significant degree.

Remark 5.5. It is interesting to note that while the value of ed_{k}(SL_{2}(ℤ)) depends on the base field k (see [FF, Remark 5.5]), for n ≥ 3, the value of ed_{k}(SL_{n}(ℤ)) does not (as long as char(k) ≠ 2).

References

[AT] A. Amitsur, J.-P. Tignol, Symplectic modules, Israel J. Math. 54 (1986), 267-290.

[BF] G. Berhuy, G. Favi, Essential dimension: a functorial point of view (after A. Merkurjev), Doc. Math. 8 (2003), 279-330.

[BRV_{1}] P. Brosnan, Z. Reichstein, A. Vistoli, Essential dimension and algebraic

     stacks, 2007, arXiv:math/0701903v1 [math.AG].

[BRV_{2}] P. Brosnan, Z. Reichstein, A. Vistoli, Essential dimension, spinor groups and quadratic forms, 10 pages, Annals of Math. 171 (2010), no. 1, 533–544.

[BuR] J. Buhler, Z. Reichstein, On the essential dimension of a finite group, Compositio Math. 106 (1997), no. 2, 159-179.

[FF] G. Favi, M. Florence, Tori and essential dimension, J. Algebra, 319 (2008), no. 9, 3885-3900.

[H] B. Huppert, Endliche Gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, 134 Springer-Verlag, 1967.

[JLY] C. U. Jensen, A. Ledet, and N. Yui, Generic polynomials, Mathematical Sciences Research Institute Publications, vol. 45, Cambridge University Press, Cambridge, 2002, Constructive aspects of the inverse Galois problem.

[KM] N. A. Karpenko and A. S. Merkurjev, Essential dimension of finite pgroups, Inventiones Math., 172, no. 3 (2008), pp. 491-508.

[LG-P] C. R. Leedham-Green, W. Plesken, Some remarks on Sylow subgroups

of general linear groups, Math. Z. 191 (1986), no. 4, 529–535.

[Lo] R. Lötscher, Application of multihomogeneous covariants to the essential

dimension of finite groups, http://arxiv.org/PS_cache/arxiv/pdf/
0811/0811.3852v1.pdf

[Na] T. Nakayama, Finite groups with faithful irreducible and directly indecomposable modular representations, Proc. Japan Acad. 23 (1947), no. 3, 22–25.

[Me] A. Merkurjev, Essential dimension, in Quadratic forms – algebra, arithmetic, and geometry (R. Baeza, W.K. Chan, D.W. Hoffmann, and R. Schulze-Pillot, eds.), Contemporary Mathematics 493 (2009), 299–326.

[Re] Z. Reichstein, On the notion of essential dimension for algebraic groups, Transform. Groups 5 (2000), no. 3, 265–304.

[Se] J.-P. Serre, Linear representations of finite groups, Graduate Texts in Mathematics, 42. Springer-Verlag, New York–Heidelberg, 1977.

[Ta] M. Tazawa, Über die isomorphe Darstellung der endliehen Gruppe, Tôhoku Math. J. 47 (1940), 87–93.

[Vo] R. T. Vol’vacev, Sylow p-subgroups of the full linear group (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 27 (1963), 1031–1054.

Aurel Meyer Department of Mathematics University of British Columbia Vancouver Canada, V6T 1Z2 Zinovy Reichstein Department of Mathematics University of British Columbia Vancouver Canada, V6T 1Z2 458

Documenta Mathematica · Extra Volume Suslin (2010)