A NUMERICAL INVARIANT FOR LINEAR REPRESENTATIONS OF FINITE GROUPS

NIKITA A. KARPENKO AND ZINOVY REICHSTEIN

Abstract. We study the notion of essential dimension for a linear representation of a finite group. In characteristic zero we relate it to the canonical dimension of certain products of Weil transfers of generalized Severi-Brauer varieties. We then proceed to compute the canonical dimension of a broad class of varieties of this type, extending earlier results of the first author. As a consequence, we prove analogues of classical theorems of R. Brauer and O. Schilling about the Schur index, where the Schur index of a representation is replaced by its essential dimension. In the last section we show that essential dimension of representations can behave in rather unexpected ways in the modular setting.

1. Introduction

Let K/k be a field extension, G be a finite group of exponent e, and ρ: G → GL_{n}(K) be a non-modular representation of G whose character takes values in k. (Here “nonmodular” means that char(k) does not divide |G|.) A theorem of Brauer says that if k contains a primitive eth root of unity ζ_{e} then ρ is defined over k, i.e., ρ is K-equivalent to a representation ρ′: G → GL_{n}(k); see, e.g. [30, §12.3]. If ζ_{e} ∉ k, we would like to know “how far” ρ is from being defined over k. In the case, where ρ is absolutely irreducible, a classical answer to this question is given by the Schur index of ρ, which is the smallest degree of a finite field extension l/k such that ρ is defined over l. Some background material on the Schur index and further references can be found in Section 2.

In this paper we introduce and study another numerical invariant, the essential dimension ed(ρ), which measures “how far” ρ is from being defined over k in a different way. Here ρ is not assumed to be irreducible; for the definition of ed(ρ), see Section 6. In Section 8 we show that the maximal value of ed(ρ), as ρ ranges over representations with a fixed character χ: G → k, which we denote by ed(χ), can be expressed as the canonical

Date: June 17, 2014. 2010 Mathematics Subject Classification. 14C25, 16K50, 20C05. Key words and phrases. Representations of finite groups, characters, Schur index, central simple al-

gebras, essential dimension, Severi-Brauer varieties, Weil transfer, Chow groups and motives, canonical dimension and incompressibility.

The first author acknowledges partial support of the French Agence Nationale de la Recherche (ANR) under reference ANR-12-BL01-0005; his work has been also partially supported by a start-up grant of the University of Alberta and a Discovery Grant from the National Science and Engineering Board of Canada.

The second author has been partially supported by a Discovery Grant from the National Science and Engineering Board of Canada.

dimension of a certain product of Weil transfers of generalized Severi-Brauer varieties. We use this to show that ed(ρ) ≤ |G|^{2}/4 for any n, k, and K/k in Section 9 and to prove a variant of a classical theorem of Brauer in Section 10. In Section 11 we compute the canonical dimension of a broad class of Weil transfers of generalized Severi-Brauer varieties, extending earlier results of the first author from [18] and [20]. This leads to a formula for the essential p-dimension of an irreducible character in terms of its decomposition into absolutely irreducible components; see Corollary 12.3. As an application we prove a variant of a classical theorem of Schilling in Section 13. In the last section we show that ed(ρ) can be unexpectedly large in the non-modular setting.

2. Notation and representation-theoretic preliminaries

Throughout this paper G will denote a finite group of exponent e, k a field, \overline{k} an algebraic closure of k, K and F field extensions of k, ζ_{d} a primitive dth root of unity, ρ a finite-dimensional representation of G, and χ a character of G. In this section we will assume that char(k) does not divide G.

2a. Characters and character values. A function χ: G → k is said to be a character of G, if χ is the character of some representation ρ: G → GL_{n}(K) for some field extension K/k.

If χ: G → \overline{k} is a character, and F/k is a field, we set

F (χ) := F (χ(g)) | g ∈ G) ⊂ F (ζ_{e}).

Since F (ζ_{e}) is an abelian extension of F, so is F (χ). Moreover, F (χ) is stable under automorphisms F (ζ_{e})/F, i.e., independent of the choice of the eth root of unity ζ_{e} in \overline{F}.

Lemma 2.1. (a) Let χ, χ′: G → \overline{k} be characters and F/k be a field extension. Then (a) every automorphism h ∈ Gal(F (χ)/F) leaves k(χ) invariant.

(b) If χ and χ′ are conjugate over F then they are conjugate over k.

(c) Suppose k is algebraically closed in F. Then the converse to part (b) also holds. That is, if χ, χ′ are conjugate over k then they are conjugate over F.

Proof. (a) It is enough to show that h(χ(g)) ⊂ F (χ) for every g ∈ G. Since the sequence of Galois groups

1 → Gal(F (ζ_{e})/F (χ)) → Gal(F (ζ_{e})/F) → Gal(F (χ)/F) → 1

is exact, h can be lifted to an element of Gal(F (ζ_{e})/F). By abuse of notation, we will continue to denote it by h. The eigenvalues of ρ(g) are of the form ζ_{e}^{i_{1}}, …, ζ_{e}^{i_{n}} for some integers i_{1}, …, i_{n}. The automorphism h sends ζ_{e} to another primitive eth root of unity ζ_{e}^{j} for some integer j. Then

h(χ(g)) = h(ζ_{e}^{i_{1}} + ⋯ + ζ_{e}^{i_{n}}) = ζ_{e}^{ji_{1}} + ⋯ + ζ_{e}^{ji_{n}} = χ(g^{j}) ∈ F (χ) ,

as desired.

(b) is an immediate consequence of (a).

(c) If k is algebraically closed in F, then the homomorphism

Gal(F (χ_{1})/F) → Gal(k(χ_{1})/k)

given by σ ↦ σ_{|k(χ)} is surjective; see [25, Theorem VI.1.12]. □

2b. The envelope of a representation. If ρ: G → GL_{n}(F) is a representation over some field F/k, we define the k-envelope Env_{k}(ρ) as the k-linear span of ρ(G).

Lemma 2.2. Env_{k}(s · ρ) is k-isomorphic Env_{k}(ρ) for any integer s ⩾ 1.

Proof. The diagonal embedding M_{n}(F) ↪ M_{n}(F) × ⋯ × M_{n}(F) (s times) induces an isomorphism between Env_{k}(ρ) and Env_{k}(s · ρ). □

Lemma 2.3. Assume the character χ of ρ: G → GL_{n}(F) is k-valued. Then the natural homomorphism Env_{k}(ρ) ⊗_{k} F → Env_{F} (ρ) is an isomorphism of F -algebras.

Proof. It suffices to show that if ρ(g_{1}), …, ρ(g_{r}) are linearly dependent over F for some elements g_{1}, …, g_{r} ∈ G, then they are linearly dependent over k. Indeed, suppose

a_{1}ρ(g_{1}) + ⋯ + a_{r}ρ(g_{r}) = 0

in M_{n}(F) for some a_{1}, …, a_{r} ∈ F, such that a_{i} ≠ 0 for some i. Then

tr((a_{1}ρ(g_{1}) + ⋯ + a_{r}ρ(g_{r})) · ρ(g)) = 0

for every g ∈ G, which simplifies to

a_{1}χ(g_{1}g) + ⋯ + a_{r}χ(g_{r}g) = 0.

The homogeneous linear system

x_{1}χ(g_{1}g) + ⋯ + x_{r}χ(g_{r}g) = 0

in variables x_{1}, …, x_{r} has coefficients in k and a non-trivial solution in F. Hence, it has a non-trivial solution b_{1}, …, b_{r} in k, and we get that

tr((b_{1}ρ(g_{1}) + ⋯ + b_{r}ρ(g_{r})) · ρ(g)) = 0

for every g ∈ G.

Note that Env_{k}(ρ) is, by definition, a homomorphic image of the group ring k[G]. Hence, Env_{k}(ρ) is semisimple and consequently, the trace form in Env_{k}(ρ) is non-degenerate. It follows that the elements ρ(g_{1}), …, ρ(g_{r}) are linearly dependent over k, as desired. □

2c. The Schur index. Suppose K/k is a field extension, and ρ_{1} : G → GL_{n}(K) is an absolutely irreducible representation with character χ_{1} : G → K. By taking F = \overline{K} in Lemma 2.3, one easily deduces that Env_{k(χ_{1})}(ρ_{1}) is a central simple algebra of degree n over k(χ_{1}). The index of this algebra is called the Schur index of ρ_{1}. We will denote it by m_{k}(ρ_{1}).

In the sequel we will need the following properties of the Schur index.

Theorem 2.4. Let K be a field, G be a finite group such that char(K) does not divide |G|, and ρ: G → GL_{n}(K) be an irreducible representation. Denote the character of ρ by χ.

(a) Over the algebraic closure \overline{K}, ρ decomposes as

(2.5)  ρ\overline{K} ≃ m(ρ_{1} ⊕ ⋯ ⊕ ρ_{r}),

where ρ_{1}, …, ρ_{r} are pairwise non-isomorphic irreducible representations of G defined over \overline{K}, and m is their common Schur index m_{K}(ρ_{1}) = ⋯ = m_{K}(ρ_{r}).

(b) Let χ_{i}: G → \overline{K} be the character of ρ_{i} for i = 1, …, r. Then K(χ_{1}) = ⋯ = K(χ_{r}) is an abelian extension of K of degree r. Moreover, Gal(K(χ_{1})/K) transitively permutes χ_{1}, …, χ_{r}.

(c) Conversely, every irreducible representation ρ_{1} : G → GL_{1}(\overline{K}) occurs as an irreducible component of a unique K-irreducible representation ρ: G → GL_{n}(K), as in (2.5).

(d) The center Z of Env_{K}(ρ) is K-isomorphic to K(χ_{1}) = K(χ_{2}) = ⋯ = K(χ_{r}). Env_{K}(ρ) is a central simple algebra over Z of index m.

(e) The multiplicity of ρ_{1} in any representation of G defined over K is a multiple of m_{K}(ρ_{1}). Consequently, m_{K}(ρ_{1}) divides m_{k}(ρ_{1}) for any field extension K/k.

(f) m divides dim(ρ_{1}) = ⋯ = dim(ρ_{r}).

Proof. See [12, Theorem 74.5] for parts (a)-(d), and [11, Corollary 74.8] for parts (e) and (f). □

As a consequence of Theorem 2.4, we obtain the following

Corollary 2.6. Let K/k be a field extension, ρ: G → GL_{n}(K) be a representation, whose character takes values in k, and

ρ = d_{1}ρ_{1} ⊕ ⋯ ⊕ d_{r}ρ_{r}

be the irreducible decomposition of ρ over the algebraic closure \overline{K}. Then ρ can be realized over k (i.e., ρ is K-equivalent to a representation ρ′: G → GL_{n}(k)) if and only if the Schur index m_{k}(ρ_{i}) divides d_{i} for every i = 1, …, r. □

3. Preliminaries on essential and canonical dimension

3a. Essential dimension. Let F : Fields_{k} → Sets be a covariant functor, where Fields_{k} is the category of field extensions of k and Sets is the category of sets. We think of the functor F as specifying the type of algebraic objects under consideration, F(K) as the set of algebraic objects of this type defined over K, and the morphism F(i): F(K) → F(L) associated to a field extension

(3.1)  k ⊂ K ↪^{i} L

as “base change”. For notational simplicity, we will write α_{L} ∈ F(L) instead of F(i)(α),

Given a field extension L/K, as in (3.1), an object α ∈ F(L) is said to descend to K if it lies in the image of F(i). The essential dimension ed(α) is defined as the minimal transcendence degree of K/k, where α descends to K. The essential dimension ed(F) of the functor F is the supremum of ed(α) taken over all α ∈ F(K) and all K.

Usually ed(α) < ∞ for every α ∈ F(K) and every K/k; see [7, Remark 2.7]. On the other hand, ed(F) = ∞ in many cases of interest; for example, see Proposition 14.1 below.

The essential dimension ed_{p}(α) of α at a prime integer p is defined as the minimal value of ed(α_{L′}), as L′ ranges over all finite field extensions L′/L such that p does not divide the degree [L′ : L]. The essential dimension ed_{p}(F) is then defined as the supremum of ed_{p}(α) as K ranges over all field extensions of k and α ranges over F(K).

For generalities on essential dimension, see [2, 27, 28, 7].

3b. Canonical dimension. An interesting example of a covariant functor Fields_{k} → Sets is the “detection functor” D_{X} associated to an algebraic k-variety X. For a field extension K/k, we define

D_{X}(K) := \begin{cases}a one-element set, if X has a K-point, and \\ ∅, otherwise. \end{cases}

If k ⊂ K ↪^{i} L then 0 ⩽ |D_{X}(K)| ⩽ |D_{X}(L)| ⩽ 1. Thus there is a unique morphism of sets D_{X}(K) → D_{X}(L), which we define to be D_{X}(i).

The essential dimension (respectively, the essential p-dimension) of the functor D_{X} is called the canonical dimension of X (respectively, the canonical p-dimension of X) and is denoted by cd(X) (respectively, cd_{p}(X)). If X is smooth and projective, then cd(X) (respectively, cd_{p}(X)) equals the minimal dimension of the image of a rational self-map X −−→ X (respectively, of a correspondence X ⇝ X of degree prime to p). In particular,

(3.2)  0 ⩽ cd_{p}(X) ⩽ cd(X) ⩽ dim(X)

for any prime p. If cd(X) = dim(X), we say that X is incompressible. If cd_{p}(X) = dim(X), we say that X is p-incompressible. For details on the notion of canonical dimension for algebraic varieties, we refer the reader to [27, §4].

We will say that smooth projective varieties X and Y defined over K are equivalent if there exist rational maps X −−→ Y and Y −−→ X. Similarly, we will say that X and Y are p-equivalent for a prime integer p, if there exist correspondences X ⇝ Y and Y ⇝ X of degree prime to p.

Lemma 3.3. (a) If X and Y are equivalent, then cd(X) = cd(Y) and cd_{p}(X) = cd_{p}(Y) for any p.

(b) If X and Y are p-equivalent for some p, then cd_{p}(X) = cd_{p}(Y).

Proof. (a) Let K/k be a field extension. By Nishimura’s lemma, X has a K-point if and only if so does Y; see [29, Proposition A.6]. Thus the detection functors D_{X} and D_{Y} are isomorphic, and cd(X) = ed(D_{X}) = ed(D_{Y}) = cd(Y). Similarly, cd_{p}(X) = cd_{p}(Y).

For a proof of part (b) see [23, Lemma 3.6 and Remark 3.7]. □

4. Balanced algebras

Let Z/k be a Galois field extension, and A be a central simple algebra over Z. Given α ∈ Gal(Z/k), we will denote the “conjugate” Z-algebra A ⊗_{Z} Z, where the tensor product is taken via α: Z → Z, by ^{α}A. We will say that A is balanced over k if ^{α}A is Brauer-equivalent to a tensor power of A for every α ∈ Gal(Z/k).

Note that A is balanced, if the Brauer class of A descends to k: ^{α}A is then isomorphic to A for any α. In this section we will consider another family of balanced algebras.

Let K/k be a field extension, ρ: G → GL_{n}(K) be an irreducible representation whose character χ is k-valued. Recall from Theorem 2.4 that Env_{k}(ρ) is a central simple algebra over Z ≃ k(χ_{1}) = ⋯ = k(χ_{n}).

Proposition 4.1. Env_{k}(ρ) is balanced over k.

Proof. Recall from [33, p. 14] that a cyclotomic algebra B/Z is a central simple algebra of the form

B = ⨁_{g∈Gal(Z(ζ)/Z)} Z(ζ)u_{g} ,

where ζ is a root of unity, Z(ζ) is a maximal subfield of B, and the basis elements u_{g} are subject to the relations

u_{g}x = g(x)u_{g} and u_{g}u_{h} = β(g, h)u_{gh} for every x ∈ Z(ζ) and g, h ∈ Gal(Z(ζ)/Z).

Here β: G × G → Z(ζ)^{∗} is a 2-cocycle whose values are powers of ζ. Following the notational conventions in [33], we will write B := (β, Z(ζ)/Z).

By [33, Corollary 3.11], Env_{k}(ρ) is Brauer-equivalent to some cyclotomic algebra B/Z, as above. Thus it suffices to show that every cyclotomic algebra is balanced over k, i.e., αB is Brauer-equivalent to a power of B over Z for every α ∈ Gal(Z/k).

By Theorem 2.4(d), Z is k-isomorphic to k(χ_{1}), which is, by definition a subfield of k(ζ_{e}), where e is the exponent of G. Thus there is a root of unity ϵ such that

Z(ζ) ⊂ k(ζ, ζ_{e}) = k(ϵ)

and both ζ and ζ_{e} are powers of ϵ. Note that k(ϵ)/k is an abelian extension, and the sequence of Galois groups

1 → Gal(k(ϵ)/Z) → Gal(k(ϵ)/k) → Gal(Z/k) → 1

is exact. In particular, every α ∈ Gal(Z/k) can be lifted to an element of Gal(k(ϵ)/k), which we will continue to denote by α. Then α(ϵ) = ϵ^{t} for some integer t. Since ζ is a power of ϵ, and each β(g, h) is a power of ζ, we have

(4.2)  α(β(g, h)) = β(g, h)^{t} for every g, h ∈ Gal(Z(ζ)/k).

We claim that ^{α}B is Brauer-equivalent to B^{⊗t} over Z. Indeed, since

B = (β, Z(ζ)/Z),

we have ^{α}B = (α(β), Z(ζ)/Z). By (4.2), ^{α}B = (α(β), Z(ζ)/Z) = (β^{t}, Z(ζ)/Z), and (β^{t}, Z(ζ)/Z) is Brauer-equivalent to B^{⊗t}, as desired. □

5. Generalized Severi-Brauer varieties and Weil transfers

Suppose Z/k is a finite Galois field extension and A is a central simple algebra over Z. For 1 ⩽ m ⩽ deg(A), we will denote by SB(A, m) the generalized Severi-Brauer variety (or equivalently, the twisted Grassmannian) of (m − 1)-dimensional subspaces in SB(A). The Weil transfer R_{Z/k}(SB(A, m)) is a smooth projective absolutely irreducible k-variety of dimension [Z : k]·m·(deg(A)−m). For generalities on the Weil transfer, see, e.g., [15].

Proposition 5.1. Let Z, k and A be as above, X := R_{Z/k}(SB(A, m)) for some 1 ⩽ m ⩽ deg(A), and K/k be a field extension.

(a) Write K_{Z} := K ⊗_{k} Z as a direct product K_{1} × ⋯ × K_{s}, where K_{1}/Z, …, K_{s}/Z are field extensions. Then X has a K-point if and only if the index of the central simple algebra A_{K_{i}} := A ⊗_{Z} K_{i} divides m for every i = 1, …, s.

(b) Assume that m divides ind(A), A is balanced and K = k(X) is the function field of X. Then K_{Z} = K ⊗_{k} Z is a field, and A ⊗_{k} K ≃ A ⊗_{Z} K_{Z} is a central simple algebra over K_{Z} of index m.

Proof. First note that A ⊗_{k} K ≃ A ⊗_{Z} K_{Z}.

(a) By the definition of the Weil transfer, X = R_{Z/k}(SB(A, m)) has a K-point if and only if SB(A, m) has a K_{Z}-point or equivalently, if and only if SB(A, m) has a K_{i}-point for every i = 1, …, s. On the other hand, by [4, Proposition 3], SB(A, m) has a K_{i}-point if and only if the index of A_{K_{i}} divides m.

(b) Since X is absolutely irreducible (see, e.g. [4, Lemma 3]), K_{Z} is Z-isomorphic to the function field of the Z-variety

X_{Z} := X ×_{Spec(k)} Spec(Z) = ∏_{α∈Gal(Z/k)} SB(^{α}A, m) ,

see [5, §2.8]. Set F := Z(SB(A, m)). By [31, Corollary 1],

ind(A ⊗_{Z} F) = m .

Since A is balanced, i.e., each algebra ^{α}A is a power of A, ind(^{α}A ⊗_{Z} F) divides m for every α ∈ Gal(Z/k). By [4, Proposition 3], each SB(^{α}A, m)_{F} is rational over F. Thus the natural projection of Z-varieties

X_{Z} = ∏_{α∈Gal(Z/k)} SB(^{α}A, m) → SB(A, m)

induces a purely transcendental extension of function fields F ↪ K_{Z}. Consequently,

ind(A ⊗_{Z} K_{Z}) = ind(A ⊗_{Z} F) = m ,

as claimed. □

6. The essential dimension of a representation

Let us now fix a finite group G and an arbitrary field k, and consider the covariant functor

Rep_{G,k} : Fields_{k} → Sets

defined by Rep_{G,k}(K) := {K-isomorphism classes of representations G → GL_{n}(K)} for every field K/k. Here n ⩾ 1 is allowed to vary.

The essential dimension ed(ρ) of a representation ρ: G → GL_{n}(K) is defined by viewing ρ as an object in Rep_{G,k}(K), as in Section 3. That is, ed(ρ) is the smallest transcendence degree of an intermediate field k ⊂ K_{0} ⊂ K such that ρ is K-equivalent to a representation ρ′: G → GL_{n}(K_{0}). To illustrate this notion, we include an example, where ed(ρ) is positive and two elementary lemmas.

Example 6.1. Let ℍ = (−1, −1) be the algebra of Hamiltonian quaternions over k = ℝ, i.e., the 4-dimensional ℝ-algebra given by two generators i, j, subject to relations, i^{2} = j^{2} = −1 and ij = −ji. The multiplicative subgroup G = {±1, ±i, ±j, ±ij} of ℍ is the quaternion group of order 8. Let K = ℝ(SB(ℍ)), where SB(ℍ) denotes the Severi-Brauer variety of ℍ. The representation ρ: G ↪ ℍ ↪ ℍ ⊗_{ℝ} K ≃ M_{2}(K) is easily seen to be absolutely irreducible. We claim that ed(ρ) = 1. Indeed, trdeg_{ℝ}(F) = 1, for any intermediate extension ℝ ⊂ F ⊂ K, unless F = ℝ. On the other hand, ρ cannot descend to ℝ, because Env_{ℝ}(ρ) = ℍ, and thus m_{ℝ}(ρ) = ind(ℍ) = 2 by Theorem 2.4(d). □

Lemma 6.2. Let G be a finite group, K/k be a field, ρ_{i}: G → GL_{n_{i}}(K) be representations of G over K (for i = 1, …, s) and ρ ≃ a_{1}ρ_{1} ⊕⋯⊕a_{s}ρ_{s}, where a_{1}, …, a_{s} ⩾ 1 are integers. Then ed(ρ) ⩽ ed(ρ_{1}) + ⋯ + ed(ρ_{s}).

Proof. Suppose ρ_{i} descends to an intermediate field k ⊂ K_{i} ⊂ K, where trdeg_{k}(K_{i}) = ed(ρ_{i}). Let K_{0} be the subfield of K generated by K_{1}, …, K_{s}. Then ρ descends to K_{0} and ed(ρ) ⩽ trdeg_{k}(K_{0}) ⩽ trdeg_{k}(K_{1}) + ⋯ + trdeg_{k}(K_{s}) = ed(ρ_{1}) + ⋯ + ed(ρ_{s}). □

Lemma 6.3. Let k ⊂ K be fields, G be a finite group, and ρ: G → GL_{n}(K) be a representation. Let k′ := k(χ) ⊂ K, where χ is the character of ρ. Then the essential dimension of ρ is the same, whether we consider it as an object on Rep_{K,k} or Rep_{K,k′}.

Proof. If ρ descends to an intermediate field k ⊂ F ⊂ K, then F automatically contains k′. Moreover, trdeg_{k}(F) = trdeg_{k′}(F). The rest is immediate from the definition. □

Lemma 6.4. Assume that char(k) does not divide |G| and the Schur index m_{k}(λ) equals 1 for every absolutely irreducible representation λ: G → GL_{n}(K), where K contains k. (In fact, it suffices to only consider K = \overline{k}.) Then ed(ρ) = 0 for any representation ρ: G → GL_{n}(L) over any field L/k. In other words, ed(Rep_{G,k}) = 0.

Proof. Let χ be the character of ρ and k′ := k(χ). By Theorem 2.4(e), m_{k′}(λ) = 1 for every absolutely irreducible representation λ: G → GL_{n}(K) of G. By Lemma 6.3 we may replace k by k′ = k(χ) and thus assume that χ is k-valued. Corollary 2.6 now tells us that ρ descends to k. □

Remark 6.5. The condition of Lemma 6.4 is always satisfied if char(k) > 0; see [12, Theorem 74.9]. This tells us that for non-modular representations the notion of essential dimension is only of interest when char(k) = 0. The situation is drastically different in the modular setting; see Section 14.

7. Irreducible characters

In view of Remark 6.5, we will now assume that char(k) = 0. In this setting there is a tight connection between representations and characters.

Lemma 7.1. Suppose F_{1}/k, F_{2}/k are field extensions, and

ρ_{1} : G → GL_{n}(F_{1}), ρ_{2} : G → GL_{n}(F_{2})

are representations of a finite group G, with the same character χ: G → k. Then the k-algebras Env_{k}(ρ_{1}) and Env_{k}(ρ_{2}) are isomorphic.

Proof. Let F/k be a field containing both F_{1} and F_{2}. Then ρ_{1} and ρ_{2} are equivalent over F, because they have the same character. Thus Env_{k}(ρ_{1}) and Env_{k}(ρ_{2}) are conjugate inside M_{n}(F). □

Given a representation ρ: G → GL_{n}(F), with a k-valued character χ: G → k, Lemma 7.1 tells us that, up to isomorphism, the k-algebra Env_{k}(ρ) depends only on χ and not on the specific choice of F and ρ. Thus we may denote this algebra by End_{k}(χ).

If ρ is absolutely irreducible (and the character χ is not necessarily k-valued), it is common to write m_{k}(χ) for the index of End_{k(χ)}(χ) instead of m_{k}(ρ).

Let χ: G → k be a character of G. Write

(7.2)  χ = ∑_{i=1}^{r} m_{i}χ_{i} ,

where χ_{1}, …, χ_{r}: G → \overline{k} are absolutely irreducible and m_{1}, …, m_{r} are non-negative integers. Since χ is k-valued, m_{i} = m_{j} whenever χ_{i} and χ_{j} are conjugate over k.

Lemma 7.3. Let χ = ∑_{i=1}^{r} m_{i}χ_{i}: G → k be a character of G, as in (7.2). Then the following are equivalent.

(a) χ is the character of a K-irreducible representation ρ: G → GL_{n}(K) for some field extension K/k.

(b) χ_{1}, …, χ_{r} form a single Gal(k(χ_{1})/k)-orbit and m_{1} = ⋯ = m_{r} divides m_{k}(χ_{1}) = ⋯ = m_{k}(χ_{r}).

Proof. (a) ⟹ (b): By Theorem 2.4(a) and (b), χ = m(χ_{1} + ⋯ + χ_{r}), where χ_{1}, …, χ_{r} are absolutely irreducible characters transitively permuted by Gal(K(χ_{1})/K), and m = m_{K}(χ_{1}) = ⋯ = m_{K}(χ_{r}). By Lemma 2.1(b), χ_{1}, …, χ_{r} are also transitively permuted by Gal(k(χ_{1})/k). Moreover, by Theorem 2.4(e), m divides m_{k}(χ_{1}) = ⋯ = m_{k}(χ_{r}).

(b) ⟹ (a): Let K be the function field of the Weil transfer variety R_{Z/k}(SB(A, m)), where A is the underlying division algebra and Z is the center of Env_{k}(χ). Since the variety R_{Z/k}(SB(A, m)) is absolutely irreducible, k is algebraically closed in K. Lemma 2.1(c) now tells us that χ_{1}, …, χ_{r} are conjugate over K. By Theorem 2.4(c) there exists an irreducible K-representation ρ whose character is m_{K}(χ_{1})(χ_{1} + ⋯ + χ_{r}). It remains to show that m_{K}(χ_{1}) = m. Indeed,

m_{K}(χ_{1}) = ind(Env_{K}(χ)) = ind(Env_{k}(χ) ⊗_{k} K) = m .

Here the first equality follows from Theorem 2.4(d), the second from Lemma 2.3, and the third from Proposition 5.1(b). □

We will say that a character χ: G → k is irreducible over k if it satisfies the equivalent conditions of Lemma 7.3.

8. The essential dimension of a character

In this section we will assume that char(k) = 0 and consider subfunctors

Rep_{χ} : Fields_{k} → Sets

of Rep_{G,k} given by

Rep_{χ}(K) := {K-isomorphism classes of representations ρ: G → GL_{n}(K) with character χ}

for every field K/k. Here χ: G → k is a fixed character and n = χ(1_{G}). The assumption that χ takes values in k is natural in view of Lemma 6.3, and the assumption that char(k) = 0 in view of Remark 6.5. Since any two K-representations with the same character are equivalent, Rep_{χ}(K) is either empty or has exactly one element. We will say that χ can be realized over K/k if Rep_{χ}(K) ≠ ∅. In particular, Rep_{χ} and Rep_{χ′} are isomorphic if and only if χ and χ′ can be realized over the same fields K/k.

Definition 8.1. Let χ: G → k be a character of a finite group G and p be a prime integer. We will refer to the essential dimension of Rep_{χ} as the essential dimension of χ and will denote this number by ed(χ). Similarly for the essential p-dimension:

ed(χ) := ed(Rep_{χ}) and ed_{p}(χ) := ed_{p}(Rep_{χ}).

We will say that characters χ and λ of G, are disjoint if they have no common absolutely irreducible components.

Lemma 8.2. (a) If the characters χ, λ: G → k are disjoint then Rep_{χ+λ} ≃ Rep_{χ}×Rep_{λ}.

(b) Suppose a character χ = ∑_{i=1}^{s} m_{i}χ_{i}: G → k is as in (7.2). Set χ′ := ∑_{i=1}^{r} m′_{i}χ_{i}, where m′_{i} is the greatest common divisor of m_{i} and m_{k}(χ_{i}). Then Rep_{χ} ≃ Rep_{χ′}.

Proof. Let K be a field extension of k.

(a) By Corollary 2.6, χ + λ can be realized over K if and only if both χ and λ can be realized over K.

(b) By Corollary 2.6

(i) χ can be realized over K if and only if

(ii) m_{K}(χ_{i}) divides m_{i}, for every i = 1, …, s. By Theorem 2.4(e), m_{K}(χ_{i}) divides m_{k}(χ). Thus (ii) is equivalent to

(iii) m_{K}(χ_{1}) = ⋯ = m_{K}(χ_{s}) divides m′. Applying Corollary 2.6 one more time, we see that (iii) is equivalent to

(iv) χ′ can be realized over K. In summary, χ can be realized over K if and only if χ′ can be realized over K, as desired.

□

As we observed above, Rep_{χ}(K) has at most one element for every field K/k. In other words, Rep_{χ} is a detection functor in the sense of [21] or [27, Section 4a]. We saw in Section 3b that to every algebraic variety X defined over k, we can associate the detection functor D_{X}, where D_{X}(K) is either empty or has exactly one element, depending on whether or not X has a K-point. Given a character χ: G → k, it is thus natural to ask if there exists a smooth projective k-variety X_{χ} such that the functors Rep_{χ} and D_{X_{χ}} are isomorphic. The rest of this section will be devoted to showing that this is, indeed, always the case. We begin by defining X_{χ}.

Definition 8.3. (a) Let G be a finite group and χ := m(χ_{1} + ⋯ + χ_{r}): G → k be an irreducible character of G, where χ_{1}, …, χ_{r} are Gal(k(χ_{1})/k)-conjugate absolutely irreducible characters, and m ⩾ 1 divides m_{k}(χ_{1}) = ⋯ = m_{k}(χ_{r}). We define the kvariety X_{χ} as the Weil transfer R_{Z/k}(SB(A_{χ}, m)), where Z is the center and A_{χ} is the underlying division algebra of Env_{k}(χ).

(b) More generally, suppose χ := λ_{1} + ⋯ + λ_{s}, where λ_{1}, …, λ_{s}: G → k are pairwise disjoint and irreducible over k. Then we define X_{χ} := X_{λ_{1}} ×_{k} ⋯ ×_{k} X_{λ_{r}}, where each X_{λ_{i}} is a Weil transfer of a generalized Severi-Brauer variety, as in part (a).

Theorem 8.4. Let G be a finite group and χ := λ_{1} + ⋯ + λ_{s} be a character, where

λ_{1}, …, λ_{s}: G → k

are pairwise disjoint and irreducible over k. Let X_{χ} be the k-variety, as in Definition 8.3. Then the functors Rep_{χ} and D_{X_{χ}} are isomorphic. Consequently ed(χ) = cd(X_{χ}) and ed_{p}(χ) = cd_{p}(X_{χ}) for any prime p.

Proof. In view of Lemma 8.2(a) we may assume that χ is irreducible over k, i.e., s = 1 and χ = λ_{1}. Write χ := m(χ_{1} + ⋯ + χ_{r}), where χ_{1}, …, χ_{r}: G → \overline{k} are the absolutely irreducible components of χ. Let K/k be a field extension. By Corollary 2.6 the following conditions are equivalent:

(i) Rep_{χ}(K) ≠ ∅, i.e., χ can be realized over K,

(ii) m_{K}(χ_{j}) divides m for j = 1, …, r. Note that while the characters χ_{1}, …, χ_{r} are conjugate over k, they may not be conjugate over K. Denote the orbits of Gal(\overline{K}/K)-action on χ_{1}, …, χ_{r} by O_{1}, …, O_{t}, and set μ_{i} := ∑_{χ_{j}∈O_{i}} χ_{j}, so that χ = m(μ_{1} + ⋯ + μ_{t}).

Denote the center of the central simple algebra Env_{k}(χ) by Z. Write K_{Z} := K ⊗_{k} Z as a direct product K_{1} × ⋯ × K_{s}, where K_{1}/Z, …, K_{s}/Z are field extensions, as in Proposition 5.1. By Lemma 2.3, (8.5) Env_{K}(χ) ≃ Env_{k}(χ)⊗_{Z} K ≃ Env_{k}(χ)⊗_{Z} K_{Z} ≃ (Env_{K}(χ)⊗_{Z} K_{1})×⋯×(Env_{K}(χ)⊗_{Z} K_{s}) ,

where ≃ denotes isomorphism of K-algebras. On the other hand, since μ_{1}, …, μ_{t} are K-valued characters,

(8.6)  Env_{K}(χ) ≃ Env_{K}(mμ_{1}) × ⋯ × Env_{K}(mμ_{t}) .

Suppose χ_{j} ∈ O_{i}. Then by Lemma 2.2 Env_{K}(mμ_{i}) ≃ Env_{K}(μ_{i}) ≃ Env_{K}(m_{K}(χ_{j})μ_{i}), and by Theorem 2.4(d), Env_{K}(m_{K}(χ_{j})μ_{i}) is a central simple algebra of index m_{K}(χ_{j}). Comparing (8.5) and (8.6), we conclude that s = t, and after renumbering K_{1}, …, K_{s}, we may assume that K_{i} is the center of Env_{K}(mμ_{i}). Thus (ii) is equivalent to

(iii) the index of Env_{K}(mμ_{i}) ≃ Env_{K}(χ) ⊗_{Z} K_{i} divides m for every i = 1, …, s. By Proposition 5.1(a), (iii) is equivalent to

(iv) X_{χ} has a K-point, i.e., D_{X_{χ}}(K) ≠ ∅. The equivalence of (i) and (iv) shows that the functors Rep_{χ} and D_{X_{χ}} are isomorphic. Now

ed(χ) =^{def} ed(Rep_{χ}) = ed(D_{X_{χ}}) =^{def} cd(X_{χ})

and similarly for the essential dimension at p. □

Remark 8.7. Theorem 8.4 can, in fact, be applied to an arbitrary k-valued character χ: G → k. Indeed, the character χ′ of Lemma 8.2(b) is a sum of pairwise disjoint k-irreducible characters. Thus Rep_{χ} ≃ Rep_{χ′} by Lemma 8.2, and Rep_{χ′} ≃ D_{X_{χ′}} by Theorem 8.4.

9. Upper bounds

In this section we will, once again, assume that char(k) = 0. Combining Theorem 8.4 with the inequality (3.2), we obtain the following

Corollary 9.1. Let G be a finite group and χ = m(χ_{1}+⋯+χ_{r}): G → k be an irreducible character over k, as in Section 7. Then ed(χ) ⩽ dim(X_{χ}) = rm(m_{k}(χ_{1}) − m). □

We are now in a position to prove upper bounds on the essential dimension of an arbitrary representation of a finite group G.

Proposition 9.2. Let ρ: G → GL_{n}(K) be a representation of a finite group G over a field K/k.

(a) If ρ is K-irreducible (but not necessarily absolutely irreducible), then ed(ρ) ⩽ n^{2}/4.

(b) If ρ is arbitrary, then ed(ρ) ⩽ |G|^{2}/4.

(c) ed(Rep_{G,k}) ≤ |G|^{2}/4 for any base field k. Here Rep_{G,k} is the functor defined at the beginning of Section 6.

Proof. (a) By Lemma 6.3 we may assume that the character χ of ρ is k-valued. By Lemma 7.3, χ is irreducible over k. Write χ = m(χ_{1} + ⋯ + χ_{r}), where m ≥ 1 divides m_{k}(χ_{1}) = ⋯ = m_{k}(χ_{r}). By Corollary 9.1

(9.3)  ed(ρ) ⩽ rm(m_{k}(χ_{1}) − m) ⩽ r\frac{m_{k}(χ_{1})^{2}}{4} .

Now recall that by Theorem 2.4(d), Env_{k}(ρ) is a central simple algebra of index m_{k}(χ_{1}) over a field Z such that [Z : k] = r. Thus

(9.4)  rm_{k}(χ_{1})^{2} ⩽ r dim_{Z}(Env_{k}(ρ)) = dim_{k}(Env_{k}(ρ)) = dim_{K}(Env_{K}(ρ)) ⩽ n^{2} .

Here the equality dim_{k}(Env_{k}(ρ)) = dim_{K}(Env_{K}(ρ)) follows from Lemma 2.3, and the inequality dim_{K}(Env_{K}(ρ)) ⩽ n^{2} follows from the fact that Env_{K}(ρ) is a K-subalgebra of M_{n}(K). Combining (9.3) and (9.4), we obtain ed(ρ) ⩽ n^{2}/4.

(b) Decompose ρ = a_{1}ρ_{1} ⊕ ⋯ ⊕ a_{s}ρ_{s} as a direct sum of distinct K-irreducibles. By Lemma 6.2, ed(ρ) ⩽ ed(ρ_{1}) + ⋯ + ed(ρ_{s}). Thus in view of part (b) we only need to show that

(9.5)  ∑_{i=1}^{s} dim(ρ_{i})^{2} ≤ |G|^{2} .

Over \overline{K}, ρ_{i} ≃ m_{i}(ρ_{i1} ⊕ ⋯ ⊕ ρ_{ir_{i}}), where the ρ_{ij} are distinct \overline{K}-irreducibles, and m_{i} is the common Schur index m_{K}(ρ_{i1}) = ⋯ = m_{K}(ρ_{ir_{i}}), as in Theorem 2.4(a). The representations ρ_{i1}, …, ρ_{ir_{i}} are conjugate over K and thus have the same dimension. By Theorem 2.4(c), the \overline{K}-irreducible representations ρ_{ij} are pairwise non-isomorphic, and by Theorem 2.4(f), m_{i} ⩽ dim(ρ_{i1}) = ⋯ = dim(ρ_{ir_{i}}). Thus

dim(ρ_{i}) = m_{i}r_{i} dim(ρ_{i1}) ⩽ r_{i} dim(ρ_{i1})^{2}

and

∑_{i=1}^{s} dim(ρ_{i})^{2} ⩽ ∑_{i=1}^{s} r_{i}^{2} dim(ρ_{i1})^{4} ⩽ (∑_{i=1}^{s} r_{i} dim(ρ_{i1})^{2})^{2} = (∑_{i=1}^{s} ∑_{j=1}^{r_{i}} dim(ρ_{ij})^{2})^{2} ⩽ |G|^{2}.

Here the last inequality follows from the fact that the sum of the squares of the dimensions of \overline{K}-irreducible representations is |G|; see, e.g., [30, Corollary 2.4.2(a)].

(c) is just a restatement of (b); see Section 3. □

10. A variant of a theorem of Brauer

A theorem of R. Brauer [6] asserts for every integer l ⩾ 1 there exists a number field k, a finite group G and an absolutely irreducible character χ_{1} such that the Schur index m_{k}(χ_{1}) = l. For alternative proofs of Brauer’s theorem, see [3] or [32].

In this section we will prove an analogous statement with the Schur index replaced by the essential dimension. Note however, that the analogy is not perfect, because the representation we construct will not be irreducible for any l ⩾ 2.

Proposition 10.1. For every integer l ⩾ 0 there exists a number field k, a finite group G, and a character χ: G → k such that ed(χ) = l.

Proof. The proposition is obvious for l = 0. Indeed, the trivial representation ρ: G → GL_{1}(k) has essential dimension 0 for any group G and any number field k. We may thus assume that l ⩾ 1.

The strategy of the proof is as follows. We will construct finite groups G_{1}, …, G_{l} and 2-dimensional absolutely irreducible characters χ_{i}: G_{i} → k, for a suitable number field k, such that the Brauer classes of the quaternion algebras A_{i} := Env_{k}(χ_{i}) are linearly independent over ℤ/2ℤ in Br(k). (Proving linear independence for these Brauer classes will be the most delicate part of the argument. We will defer it to Lemma 10.3.) We will view each χ_{i} as a character of G = G_{1} × ⋯ × G_{l} via the natural projection G → G_{i} and set χ := χ_{1} + ⋯ + χ_{r}: G → k. By Theorem 8.4

ed(χ) = cd(X_{χ}) ,

where X_{χ} := X_{χ_{1}} ×_{k} ⋯×_{k} X_{χ_{l}}, and X_{χ_{i}} is the 1-dimensional Severi-Brauer variety SB(A_{i}). Since the Brauer classes of A_{1}, …, A_{l} are linearly independent over ℤ/2ℤ in Br(k), [22, Theorem 2.1] tells us that cd(X_{χ}) = l, as desired.

We now proceed with the construction of k, G_{1}, …, G_{l} and χ_{1}, …, χ_{l}. Choose l distinct prime integers p_{1}, …, p_{l} ≡ 3 (mod 4) and let F = ℚ(ζ_{p_{1}}, …, ζ_{p_{l}}), where as usual, ζ_{p} denotes a primitive pth root of unity. The extension F/ℚ is Galois with

Gal(F/ℚ) = Γ_{1} × ⋯ × Γ_{l} ,

where Γ_{i} ≃ ℤ/(p−1)ℤ. Since p_{i} ≡ 3 (mod 4), Γ_{i} has a unique subgroup of order 2, which we will denote by Γ_{i}[2]. The non-trivial element of Γ_{i}[2] takes ζ_{p_{i}} to ζ_{p_{i}}^{−1} and preserves ζ_{p_{j}} for every j ≠ i. The elementary 2-group Γ_{1}[2] × ⋯ × Γ_{l}[2] is the Sylow 2-subgroup of Γ; we will denote it by Γ[2]. We now set

k := F^{Γ[2]} = ℚ(ζ_{p_{1}} + ζ_{p_{1}}^{−1}, …, ζ_{p_{l}} + ζ_{p_{l}}^{−1}) .

Let A_{i} be the quaternion algebra ((ζ_{p_{i}} −ζ_{p_{i}}^{−1})^{2}, −1) over k. That is, A_{i} is the 4-dimensional k-algebra generated by two elements, x and y, subject to the relations

x^{2} = (ζ_{p_{i}} − ζ_{p_{i}}^{−1})^{2}, y^{2} = −1 and xy = −yx.

Note that k(ζ_{p_{i}} − ζ_{p_{i}}^{−1}) = k(ζ_{p_{i}}) is a maximal subfield of A_{i}, and

G_{i} :=< ζ_{p_{i}} > ⋊ < y >≃ ℤ/p_{i}ℤ ⋊ ℤ/4ℤ

is a multiplicative subgroup of A_{i}^{∗}. Since k(ζ_{p_{i}}) splits A_{i} and k(ζ_{p_{i}}) ⊂ F, the inclusion G_{i} ↪ A_{i} gives rise to a 2-dimensional representation

ρ_{i}: G_{i} ↪ A_{i} ↪ M_{2}(F) .

By our construction A_{i} = Env_{k}(ρ_{i}) and the character χ_{i} of ρ_{i} is k-valued. Since ρ_{i} is faithful and G_{i} is a non-abelian group, this representation is absolutely irreducible. (Otherwise ρ_{i} would embed G as a subgroup of the abelian group GL_{1}(\overline{ℚ}) × GL_{1}(\overline{ℚ}), a contradiction.) Proposition 10.1 is now a consequence of Lemma 10.3 below. □

Remark 10.2. Proposition 10.1 implies that there exists a field extension K/k and a linear representation ρ: G → GL_{n}(K) such that ed(ρ) = l. Note however, that ρ is not the same as ρ_{1} × ⋯ × ρ_{l}: G → GL_{n}(F), even though they have the same character. Indeed, since F/k is a finite extension, ed(ρ_{1} × ⋯ × ρ_{l}) = 0. Under the isomorphism of functors Rep_{χ} ≃ D_{X_{χ}} from Theorem 8.4, ρ_{1} × ⋯ × ρ_{l} corresponds to an F -point of X_{χ}, while ρ corresponds to the generic point.

Lemma 10.3. Let p_{1}, …, p_{l} be distinct prime integers, each ≡ 3 (mod 4), and A_{i} be the quaternion algebra ((ζ_{p_{i}} −ζ_{p_{i}}^{−1})^{2}, −1) over k = ℚ(ζ_{p_{1}} +ζ_{p_{1}}^{−1}, …, ζ_{p_{l}} +ζ_{p_{l}}^{−1}). Then the classes of A_{1}, …, A_{l} are linearly independent over ℤ/2ℤ in Br(k).

Proof of Lemma 10.3. After renumbering p_{1}, …, p_{l}, it suffices to show that A_{1}⊗_{k}⋯⊗_{k}A_{s} is not split for any s = 1, …, l. Since [(a, c)] ⊗ [(b, c)] = [(ab, c)] in Br(k), we see that

[A_{1} ⊗_{k} ⋯ ⊗_{k} A_{s}] = [(T, −1)] .

Here [A] denote the Brauer class of a central simple algebra A/k, and

T := ∏_{i=1}^{s}(ζ_{p_{i}} − ζ_{p_{i}}^{−1})^{2} ∈ k .

Now recall that the quaternion algebra (T, −1) splits over k if and only if T is a norm in k(\sqrt{−1})/k; see, e.g., [24, Theorem 2.7]. Thus it suffices to show that T is not a norm in k(\sqrt{−1})/k. Assume the contrary: T = N_{k(\sqrt{−1})/k}(x) for some x ∈ k(\sqrt{−1}). Taking the norm in k/ℚ on both sides, we see that

N_{k/ℚ}(T) = N_{k/ℚ}(N_{k(\sqrt{−1})/k}(x)) = N_{k(\sqrt{−1})/ℚ}(x) = N_{ℚ(\sqrt{−1})/ℚ}(N_{k(\sqrt{−1})/ℚ(\sqrt{−1})}(x))

is a norm in ℚ(\sqrt{−1})/ℚ. Thus it suffices to prove the following

Claim: N_{k/ℚ}(T) is not a norm in ℚ(\sqrt{−1})/ℚ, i.e., is not the sum of two rational squares.

Our proof of this claim will be facilitated by the following diagram.

[figure: F; 2^{l}; =; ℚ(ζ_{p_{1}}, …, ζ_{p_{l}}); k; odd; =; ℚ(ζ_{p_{1}} + ζ_{p_{1}}^{−1}, …, ζ_{p_{l}} + ζ_{p_{l}}^{−1}); ℚ(ζ_{p_{i}}); 2; (ζ_{p_{i}} − ζ_{p_{i}}^{−1})^{2}; ∈; k_{i}; \frac{p_{i}−1}{2} , odd; =; ℚ(ζ_{p_{i}} + ζ_{p_{i}}^{−1}); ℚ]

We now proceed to compute N_{k/ℚ}(T):

N_{k/ℚ}(T) = ∏_{i=1}^{s} N_{k/ℚ}((ζ_{p_{i}} − ζ_{p_{i}}^{−1})^{2}) = ∏_{i=1}^{s} N_{k_{i}/ℚ}((ζ_{p_{i}} − ζ_{p_{i}}^{−1})^{2})^{[k:k_{i}]} ,

and since (ζ_{p_{i}} − ζ_{p_{i}}^{−1})^{2} = −N_{ℚ(ζ_{p_{i}})/k_{i}}(ζ_{p_{i}} − ζ_{p_{i}}^{−1}), for each i,

N_{k_{i}/ℚ}((ζ_{p_{i}} − ζ_{p_{i}}^{−1})^{2}) = N_{k_{i}/ℚ}(−N_{ℚ(ζ_{p_{i}})/k_{i}}(ζ_{p_{i}} − ζ_{p_{i}}^{−1})) =
(−1)^{[k_{i}:ℚ]}N_{ℚ(ζ_{p_{i}})/ℚ}(ζ_{p_{i}} − ζ_{p_{i}}^{−1}) = −p_{i}.

The last equality follows from the identities ∏_{j=1}^{p_{i}−1} ζ_{p_{i}}^{j} = 1 and ∏_{j=1}^{p_{i}−1}(1 − ζ_{p_{i}}^{j}) = p_{i}, where the latter is obtained by substituting t = 1 into ∏_{j=1}^{p_{i}−1}(t − ζ_{p_{i}}^{j}) = t^{p_{i}−1} + ⋯ + t + 1. In summary,

N_{k_{i}/ℚ}(T) = (−1)^{s} ∏_{i=1}^{s} p_{i}^{[k:k_{i}]} ∈ ℤ.

If s is odd then N_{k_{i}/ℚ}(T) < 0, and the claim is obvious. In the case where s is even, recall that by a classical theorem of Fermat, a positive integer n can be written as a sum of two rational squares if and only if it can be written as a sum of two integer squares if and only if every prime p which is ≡ 3 (mod 4) occurs to an even power in the prime decomposition of n. Since each p_{i} is ≡ 3 (mod 4) and each [k_{i} : k] is odd, we conclude that N_{k_{i}/ℚ}(T) cannot be written as a sum of two rational squares for any s = 1, …, l. This completes the proof of the claim and thus of Lemma 10.3 and Proposition 10.1. □

11. Computation of canonical p-dimension

This section aims to determine canonical p-dimension of a broad class of Weil transfers of generalized Severi-Brauer varieties. Here p is a fixed prime integer. The base field k is allowed to be of arbitrary characteristic.

Let Z/k be a finite Galois field extension (not necessarily abelian). We will work with Chow motives with coefficients in a finite field of p elements; see [13, §64]. For a motive M over Z, R_{Z/k}M is the motive over k given by the Weil transfer of M introduced in [15]. Although the coefficient ring is assumed to be ℤ in [15], and the results obtained there over ℤ do not formally imply similar results for other coefficients, the proofs go through for an arbitrary coefficient ring.

For any finite separable field extension K/k and a motive M over K, the corestriction of M is a well-defined motive over k; see [17].

Lemma 11.1. Let Z/k be an arbitrary finite Galois field extension and let M_{1}, …, M_{m} be m ⩾ 1 motives over Z. Then the motive R_{Z/k}(M_{1} ⊕ ⋯ ⊕ M_{m}) decomposes in a direct sum

R_{Z/k}(M_{1} ⊕ ⋯ ⊕ M_{m}) ≃ R_{Z/k}M_{1} ⊕ ⋯ ⊕ R_{Z/k}M_{m} ⊕ N,

where N is a direct sum of corestrictions to k of motives over fields K with k ⊊ K ⊂ Z.

Proof. For m = 1 the statement is void. For m = 2 use the same argument as in [18, Proof of Lemma 2.1]. For m ⩾ 3 argue by induction. □

Now recall from Section 3b that a k-variety X is called incompressible if cd(X) = dim(X) and p-incompressible if cd_{p}(X) = dim(X).

Theorem 11.2. Let p be a prime number, Z/k a finite Galois field extension of degree p^{r} for some r ⩾ 0, D a balanced central division Z-algebra of degree p^{n} for some n ⩾ 0, and X the generalized Severi-Brauer variety SB(D, p^{i}) of D for some i = 0, 1, …, n. Then the k-variety R_{Z/k}X, given by the Weil transfer of X, is p-incompressible.

Note that in the case, where Z/k is a quadratic Galois extension, D is balanced if the k-algebra given by the norm of D is Brauer-trivial; ^{α}D for α ≠ 1 is then opposite to D. In this special case Theorem 11.2 was proved in [18, Theorem 1.1].

Proof of Theorem 11.2. In the proof we will use Chow motives with coefficients in a finite field of p elements. Therefore the Krull-Schmidt principle holds for direct summands of motives of projective homogeneous varieties by [10] (see also [20]).

We will prove Theorem 11.2 by induction on r + n. The base case, where r + n = 0, is trivial. Moreover, in the case where r = 0 (and n is arbitrary), we have Z = k and thus R_{Z/k}X = X is p-incompressible by [20, Theorem 4.3]. Thus we may assume that r ⩾ 1 from now on.

If i = n, then X = Spec Z, R_{Z/k}X = Spec k, and the statement of Theorem 11.2 is trivial. We will thus assume that i ⩽ n − 1 and, in particular, that n ⩾ 1.

Let k′ be the function field of the variety R_{Z/k} SB(D, p^{n−1}). Set Z′ := k′ ⊗_{k} Z. By Proposition 5.1(b), the index of the central simple Z′-algebra D_{Z′} = D ⊗_{Z} Z′ = D ⊗_{k} k′ is p^{n−1}. Thus there exists a central division Z′-algebra D′ such that the algebra of (p × p)- matrices over D′ is isomorphic to D_{Z′}. Let X′ = SB(D′, p^{i}). By [14, Theorem 10.9 and Corollary 10.19] (see also [9]) and [20, Theorems 3.8 and 4.3], the motive of the variety X_{Z′} decomposes in a direct sum

M(X_{Z′}) ≃ U(X′) ⊕ U(X′)(p^{i+n−1}) ⊕ U(X′)(2p^{i+n−1}) ⊕ ⋯ ⊕ U(X′)((p − 1)p^{i+n−1}) ⊕ N,

where U(X′) is the upper motive of X′ and N is a direct sum of shifts of upper motives of the varieties SB(D′, p^{j}) with j < i. Therefore, by Lemma 11.1 and [15, Theorem 5.4], the motive of the variety (R_{Z/k}X)_{k′} ≃ R_{Z′/k′}(X_{Z′}) decomposes in a direct sum

(11.3) M(R_{Z/k}X)_{k′} ≃ R_{Z′/k′}U(X′) ⊕ R_{Z′/k′}U(X′)(p^{r+i+n−1})⊕

R_{Z′/k′}U(X′)(2p^{r+i+n−1}) ⊕ ⋯ ⊕ R_{Z′/k′}U(X′)((p − 1)p^{r+i+n−1}) ⊕ N ⊕ N′,

where now N is a direct sum of shifts of R_{Z′/k′}U(SB(D′, p^{j})) with j < i, and N′ is a direct sum of corestrictions of motives over fields K with k′ ⊊ K ⊂ Z′. By the induction hypothesis, the variety R_{Z′/k′}X′ is p-incompressible. By [16, Theorem 5.1], this means that no positive shift of the motive U(R_{Z′/k′}X′) is a direct summand of the motive of R_{Z′/k′}X′. It follows by [17] that R_{Z′/k′}U(X′) is a direct sum of U(R_{Z′/k′}X′), of U(R_{Z′/k′} SB(D′, p^{j})) with j < i, and of corestrictions of motives over fields K with k′ ⊊ K ⊂ Z′. Therefore we may exchange R_{Z′/k′} with U in (11.3) and get a decomposition of the form

(11.4) M(R_{Z/k}X)_{k′} ≃ U(R_{Z′/k′}X′) ⊕ U(R_{Z′/k′}X′)(p^{r+i+n−1})⊕

U(R_{Z′/k′}X′)(2p^{r+i+n−1}) ⊕ ⋯ ⊕ U(R_{Z′/k′}X′)((p − 1)p^{r+i+n−1}) ⊕ N ⊕ N′,

where N is now a direct sum of shifts of some U(R_{Z′/k′} SB(D′, p^{j})) with j < i, and N′ is a direct sum of corestrictions of motives over fields K with k′ ⊊ K ⊂ Z′. Note that the first p summands of decomposition (11.4) (that is, all but the last two) are shifts of an indecomposable motive; moreover, no shift of this motive is isomorphic to a summand of N or of N′. Since the variety R_{Z′/k′}X′ is p-incompressible, we have

dim U(R_{Z′/k′}X′) = dim R_{Z′/k′}X′ = [Z′ : k′] · dim X′ = p^{r} · p^{i}(p^{n−1} − p^{i}) .

(We refer the reader to [16, Theorem 5.1] for the definition of the dimension of the upper motive, as well as its relationship to the dimension and p-incompressibility of the corresponding variety). Note that the shifting number of the p-th summand in (11.4) plus dim R_{Z′/k}X′ equals dim R_{Z/k}X:

(p − 1)p^{r+i+n−1} + p^{r}p^{i}(p^{n−1} − p^{i}) = p^{r}p^{i}(p^{n} − p^{i}).

We want to show that the variety R_{Z/k}X is p-incompressible. In other words, we want to show that dim U(R_{Z/k}X) = dim R_{Z/k}X. Let l be the number of shifts of U(R_{Z′/k′}X′) contained in the complete decomposition of the motive U(R_{Z/k}X)_{k′}. Clearly, 1 ⩽ l ⩽ p and it suffices to show that l = p because in this case the p-th summand of (11.4) is contained in the complete decomposition of U(R_{Z/k}X)_{k′}.

The complete motivic decomposition of R_{Z/k}X contains several shifts of U(R_{Z/k}X). Let N be any of the remaining (indecomposable) summands. Then, by [17], N is either a shift of the upper motive U(R_{Z/k} SB(D, p^{j})) with some j < i or a corestriction to k of a motive over a field K with k ⊊ K ⊂ Z. It follows that the complete decomposition of N_{k′} does not contain any shift of U(R_{Z′/k′}X′). Therefore l divides p, that is, l = 1 or l = p, and we only need to show that l ≠ 1.

We claim that l > 1 provided that dim U(R_{Z/k}X) > dim U(R_{Z′/k′}X′). Indeed, by [19, Proposition 2.4], the complete decomposition of U(R_{Z/k}X)_{k′} contains as a summand the motive U(R_{Z′/k′}X′) shifted by the difference dim U(R_{Z/k}X)−dim U(R_{Z′/k′}X′). Therefore, in order to show that l ≠ 1 it is enough to show that

dim U(R_{Z/k}X) > dim U(R_{Z′/k′}X′).

We already know the precise value of the dimension on the right, so we only need to find a good enough lower bound on the dimension on the left. This will be given by dim U((R_{Z/k}X)k̃), where k̃/k is a degree p Galois field subextension of Z/k. We can determine the latter dimension using the induction hypothesis.

Indeed, since R_{Z/k}X ≃ R_{k̃/k}R_{Z/k̃}X, the variety (R_{Z/k}X)k̃ is isomorphic to

(R_{Z/k}X)k̃ ≃ ∏_{α̃∈Γ̃} α̃R_{Z/k̃}X ≃ R_{Z/k̃} ∏_{α̃∈Γ̃}^{α}X,

where Γ is the Galois group of Z/k, Γ̃ is the Galois group of k̃/k, and α ∈ Γ is a representative of α̃ ∈ Γ̃ (see [5, §2.8]). Since D is balanced, the product ∏_{α̃∈Γ̃} ^{α}X is equivalent to X. Hence, by Lemma 3.3 these varieties have the same canonical p-dimension (i.e., the dimensions of their upper motives coincide). The latter variety is p-incompressible by the induction hypothesis. Consequently,

dim U(R_{Z/k}X) ⩾ dim U((R_{Z/k}X)k̃) = dim R_{Z/k̃}X = p^{r−1} · p^{i}(p^{n} − p^{i}).

The lower bound p^{r−1} · p^{i}(p^{n} − p^{i}) on dim U(R_{Z/k}X) thus obtained is good enough for our purposes, because

p^{r−1} · p^{i}(p^{n} − p^{i}) > p^{r} · p^{i}(p^{n−1} − p^{i}) = dim U(R_{Z′/k′}X′).

This completes the proof of Theorem 11.2. □

The following example, due to A. Merkurjev, shows that Theorem 11.2 fails if D is not assumed to be balanced.

Example 11.5. Let L be a field containing a primitive 4-th root of unity. Let Z be the field Z := L(x, y, x′, y′) of rational functions over L in four variables x, y, x′, y′. Consider the degree 4 cyclic central division Z-algebras C := (x, y)_{4} and C′ := (x′, y′)_{4}. Let k ⊂ Z be the subfield Z^{α} of the elements in Z fixed under the L-automorphism α of Z exchanging x with x′ and y with y′. The field extension Z/k is then Galois of degree 2, and the algebra C′ is conjugate to C.

The index of the tensor product of Z-algebras C ⊗C′^{⊗2} is 8. Let D/Z be the underlying (unbalanced!) division algebra of degree 8. The subgroup of the Brauer group Br(Z) generated by the classes of D and ^{α}D = C′ ⊗ C^{⊗2} coincides with the subgroup generated by the classes of C and ^{α}C = C′. Therefore the varieties X_{1} := R_{Z/k} SB(D) and X_{2} := R_{Z/k} SB(C) are equivalent. Thus, by Lemma 3.3,

cd(X_{1}) = cd(X_{2}) ⩽ dim(X_{2}) < dim(X_{1})

and consequently, X_{1} is compressible (and in particular, 2-compressible).

12. Some consequences of Theorem 11.2

Theorem 11.2 makes it possible to determine the canonical p-dimension of the Weil transfer in the situation, where the degrees of Z/k and of D are not necessarily p-powers.

Corollary 12.1. Let Z/k be a finite Galois field extension and D a balanced central division Z-algebra. For any positive integer m dividing deg(D), one has

cd_{p} R_{Z/k} SB(D, m) = dim R_{Z/k′} SB(D′, m′) = [Z : k′] · m′(deg D′ − m′),

where m′ is the p-primary part of m (i.e., the highest power of p dividing m), D′ is the pprimary component of D, and k′ = Z^{Γ_{p}}, where Γ_{p} is a Sylow p-subgroup of Γ := Gal(Z/k) (so that [Z : k′] is the p-primary part of [Z : k]).

Proof. Since the degree [k′ : k] is prime to p, we have

cd_{p} R_{Z/k} SB(D, m) = cd_{p}(R_{Z/k} SB(D, m))_{k′};

see [26, Proposition 1.5(2)]. The k′-variety R_{Z/k} SB(D, m)_{k′} is isomorphic to a product of R_{Z/k′} SB(D, m) with several varieties of the form R_{Z/k′} SB(D̃, m) where D̃ ranges over a set of conjugates of D. Since D is balanced, these algebras D̃ are Brauer-equivalent to powers of D. Thus the product is equivalent to the k′-variety R_{Z/k′} SB(D, m). We conclude by Lemma 3.3 that cd_{p} R_{Z/k} SB(D, m) = cd_{p} R_{Z/k′} SB(D, m). In the sequel we will replace k by k′, so that the degree [Z : k] becomes a power of p.

We may now replace k by its p-special closure; see [13, Proposition 101.16]. This will not change the value of cd_{p}(X). In other words, we may assume that k is p-special. Under this assumption the algebras D and D′ become Brauer-equivalent and consequently, the k-varieties R_{Z/k} SB(D, m) and R_{Z/k} SB(D′, m′) become equivalent. By Lemma 3.3,

cd_{p} R_{Z/k} SB(D, m) = cd_{p} R_{Z/k} SB(D′, m′).

Since the Z-algebra D′ is balanced over k, Theorem 11.2 tells us that R_{Z/k} SB(D′, m′) is p-incompressible. That is,

cd_{p} R_{Z/k} SB(D′, m′) = dim(R_{Z/k} SB(D′, m′)) = [Z : k] · m′(deg D′ − m′) ,

and the corollary follows. □

Remark 12.2. Corollary 12.1 can be used to compute the p-canonical dimension of R_{Z/k} SB(D, j) for any j = 1, …, deg(D), even if j does not divide deg(D). Indeed, let m is the greatest common divisor of j and deg(D). Proposition 5.1(a) tells us that for any field extension K/k, R_{Z/k} SB(D, j) has a K-point if and only if R_{Z/k} SB(D, m) has a K-point. In other words, the detection functors for these two varieties are isomorphic. Consequently,

cd(R_{Z/k} SB(D, j)) = cd(R_{Z/k} SB(D, m)) and cd_{p}(R_{Z/k} SB(D, j)) = cd_{p}(R_{Z/k} SB(D, m)),

and the value of cd_{p}(R_{Z/k} SB(D, m)) is given by Corollary 12.1.

We now return to the setting of Sections 7–9. In particular, G is a finite group, and the base field k is of characteristic 0.

Corollary 12.3. Let χ = m(χ_{1} + ⋯ + χ_{r}): G → k be an irreducible k-valued character, where χ_{1}, …, χ_{r} are absolutely irreducible and conjugate over k, and m divides m_{k}(χ_{1}) = ⋯ = m_{k}(χ_{r}), as in Section 7.

(a) ed_{p}(χ) = r′m′(m_{k}(χ_{1})′ − m′). Here x′ denotes the p-primary part of x (i.e., the highest power of p dividing x) for any integer x ≥ 1.

(b) If r and m_{k}(χ_{1}) are powers of p, then ed_{p}(χ) = ed(χ) = dim(X_{χ}) = rm(m_{k}(χ_{1})−m). Here X_{χ} is as in Definition 8.3.

Proof. (a) Let D be the underlying division algebra and Z/k be the center of Env_{k}(χ). By Theorem 8.4, ed_{p}(χ) = cd_{p}(X_{χ}). By Proposition 4.1, D is balanced. The desired conclusion now follows from Corollary 12.1.

(b) Here r′ = r, m_{k}(χ_{1})′ = m_{k}(χ) and thus m′ = m. By part (a),

dim(X_{χ}) = rm(m_{k}(χ_{1}) − m) = ed_{p}(χ) ⩽ ed(χ) .

On the other hand, by Corollary 9.1, ed(χ) ⩽ rm(m_{k}(χ_{1}) − m), and part (b) follows. □

Remark 12.4. While a priori ed_{p}(χ) depends on k, G, and χ, Corollary 12.3(a) shows that, in fact, ed_{p}(χ) depends only on the integers r, m, and m_{k}(χ_{1}). (Here we are assuming that χ is irreducible.) We do not know if the same is true of ed(χ).

Remark 12.5. We do not know a formula for the canonical p-dimension of a product of Weil transfers of generalized Severi-Brauer varieties similar to Corollary 12.1, except for [22, Theorem 2.1], in the case, where Z = k and m = 1. (Note that this special case played a key role in the proof of Proposition 10.1.) For this reason we do not know how to generalize Corollary 12.3 to the case, where the character χ is not k-irreducible.

13. A variant of a theorem of Schilling

Let G be a p-group and χ_{1} be an absolutely irreducible character of G. It is well known that for any field k of characteristic 0, m_{k}(χ_{1}) = 1 if p is odd, and m_{k}(χ_{1}) = 1 or 2 if p = 2. Following C. Curtis and I. Reiner, we will attribute this theorem to O. Schilling; see [12, Theorem 74.15]. For further bibliographical references, see [33, Corollary 9.8].

In this section we will use Corollary 12.3 to prove the following analogous statement, with the Schur index replaced by the essential dimension.

Proposition 13.1. Let k be a field of characteristic 0, G be a p-group, and χ: G → k be an irreducible character over k.

(a) If p is odd then ed(χ) = 0.

(b) If p = 2 then ed_{2}(χ) = ed(χ) = 0 or 2^{l} for some integer l ⩾ 0.

(c) Moreover, every l ⩾ 0 in part (b) can occur with k = ℚ, for suitable choices of G and χ.

Proof. Write χ = m(χ_{1} + ⋯ + χ_{r}), where χ_{i}: G → \overline{k} are absolutely irreducible characters and m divides m_{k}(χ_{1}). If m = m_{k}(χ_{1}) then ed(χ) = 0 by Corollary 9.1.

(a) In particular, this will always be the case if p is odd. Indeed, by Schilling’s theorem, m_{k}(χ_{1}) = 1 and thus m = 1. (Also cf. Lemma 6.4.)

(b) By Schilling’s theorem, m_{k}(χ) = 1 or 2, and by the above argument, we may assume that m < m_{k}(χ_{1}). Thus the only case we need to consider is m_{k}(χ_{1}) = 2 and m = 1. By Theorem 2.4(b), r = [k(χ_{1}) : k]. Since k(χ_{1}) ⊂ k(ζ_{e}), where the exponent e of G is a power of 2, we see that r divides [k(ζ_{e}) : k], which is, once again, a power of 2. Thus we conclude that r is a power of 2. Corollary 12.3(b) now tells us that

(13.2)  ed_{2}(χ) = ed(χ) = rm(m_{k}(χ) − m) = r · 1 · (2 − 1) = r

is a power of 2, as claimed.

(c) Let s = 2^{l+2}, and σ ∈ Gal(ℚ(ζ_{s})/ℚ) be complex conjugation, and

F := ℚ(ζ_{s})^{σ} = ℚ(ζ_{s}) ∩ ℝ = ℚ(ζ_{s} + ζ_{s}^{−1}) .

Consider the quaternion algebra A = ((ζ_{s}−ζ_{s}^{−1})^{2}, −1) over F, i.e., the F -algebra generated by elements x and y, subject to the relations

x^{2} = (ζ_{s} − ζ_{s}^{−1})^{2}, y^{2} = −1 and xy = −yx.

Arguing as in the proof of Proposition 10.1, we see that F (ζ_{s} − ζ_{s}^{−1}) = ℚ(ζ_{s}) is a maximal subfield of A, ζ_{s} and y generate a a multiplicative subgroup of A of order 2s, which spans A as an F -vector space, and the inclusion G ↪ A gives rise to an absolutely irreducible 2-dimensional representation

ρ_{1} : G ↪ A ↪ GL_{2}(F) .

Denote the character of ρ_{1} by χ_{1} : G → F. Since F is generated by χ_{1}(ζ_{s}) = ζ_{s} + ζ_{s}^{−1} as a field extension of ℚ, we have ℚ(χ_{1}) = F. Thus χ_{1} has exactly

r = [F : ℚ] = \frac{1}{2}[ℚ(ζ_{s}) : ℚ] = 2^{l}

conjugates over ℚ, and χ = χ_{1} + ⋯ + χ_{r} is an irreducible character over ℚ.

Note that since s = 2^{l+2} ⩾ 4, (ζ_{s}−ζ_{s}^{−1})^{2} < 0, A⊗_{F} ℝ is ℝ-isomorphic to the Hamiltonian quaternion algebra ℍ = (−1, −1) and hence, is non-split. Thus ind(A) = 2. Since A = Env_{ℚ}(ρ), Theorem 2.4(d) tells us that m_{ℚ}(χ_{1}) = 2. Applying Corollary 12.3(b), as in (13.2), we conclude that ed_{2}(χ) = ed(χ) = r = 2^{l}, as desired. □

14. Essential dimension of modular representations

Let G be a finite group and Rep_{G,k} be the functor of representations defined at the beginning of Section 6. In the non-modular setting (where char(k) does not divide |G|), we know that

ed(Rep_{G,k}) is \begin{cases}0, if char(k) > 0, by Remark 6.5, and \\ ⩽ |G|^{2}/4, if char(k) = 0, by Proposition 9.2. \end{cases}

We shall now see that essential dimension of representations behaves very differently in the modular case.

Proposition 14.1. Let k be a field of characteristic p. Suppose a finite group G contains an elementary abelian subgroup E = ⟨g_{1}, g_{2}⟩ ≃ (ℤ/pℤ)^{2} of rank 2. Then ed(Rep_{G,k}) = ∞.

Proof. Let Sub_{ℙ^{1},k}: Fields_{k} → Sets be a covariant functor, given by

Sub_{ℙ^{1},k}(K) := {closed subvarieties of ℙ_{K}^{1}}.

Here subvarieties of ℙ_{K}^{1} are required to be reduced but not necessarily irreducible. Closed subvarieties X, Y ⊂ ℙ_{K}^{1} represent the same element in Sub_{ℙ^{1},k}(K) if X(\overline{K}) = Y (\overline{K}) in ℙ^{1}(\overline{K}). We will now consider the morphism of functors

V_{E,k}: Rep_{G,k} → Sub_{,ℙ^{1},k}

which associates to a representation ρ: G → GL_{n}(K), the rank variety V_{E}(ρ), as defined by J. Carlson [8]. Recall that in projective coordinates (x_{1} : x_{2}) on ℙ_{K}^{1}, the rank variety V_{E}(ρ) is given by

(14.2)  rank(A_{x_{1},x_{2}}) < \frac{(p − 1)n}{p} ,

where A = x_{1}(ρ(g_{1}) − I_{n}) + x_{2}(ρ(g_{2}) − I_{n}) ∈ M_{n}(K[x_{1}, x_{2}]), and I_{n} is the n × n identity matrix. In other words, condition (14.2) is equivalent to the vanishing of certain minors of A_{x_{1},x_{2}}. These minors are homogeneous polynomials in K[x_{1}, x_{2}], and V_{E}(ρ) is the reduced subvariety of ℙ_{K}^{1} they cut out. Note that the generators g_{1}, g_{2} of E are assumed to be fixed throughout. For details on this construction, see [8, Section 4] or [1, Section 5.8].

To make the rest of the proof more transparent, we will first consider the case, where G = E. In this case it follows from the work of Carlson that the functor V_{E,k} is surjective. That is, for any given field K/k, every reduced subvariety X ⊂ ℙ_{K}^{1} can be realized as the rank variety of a suitable representation ρ: E → GL_{n}(K); see [1, Corollary 5.9.2]. Thus ed(Rep_{G,k}) ⩾ ed(Sub_{ℙ^{1},k}); see [2, Lemma 1.9]. It now suffices to show that ed(Sub_{ℙ^{1},k}) = ∞. Let L/k be a field, a_{1}, …, a_{n} ∈ L, and X[n] be the union of the points

(14.3)  X_{1} = (1 : a_{1}), …, X_{n} = (1 : a_{n})

in ℙ^{1}. We view X[n] as an element of Sub_{ℙ^{1},k}(L).

Claim: Suppose X[n] descends to a subfield K ⊂ L. Then a_{i} is algebraic over K for every i = 1, …, n.

By the definition of the functor Sub_{ℙ^{1},k}, X[n] descends to K if X[n] can be cut out (set-theoretically) by homogeneous polynomials f_{1}, …, f_{s} ∈ K[x_{1}, x_{2}]. In other words, the points X_{1}, …, X_{n} in (14.3) are the only non-zero solutions, in the algebraic closure \overline{L}, of a system of homogeneous equations

f_{1}(x_{1}, x_{2}) = ⋯ = f_{s}(x_{1}, x_{2}) = 0

with coefficients in K. Since every solution of such a system can be found over \overline{K}, we have a_{1}, …, a_{n} ∈ \overline{K}. This proves the claim.

Taking a_{1}, …, a_{n} to be independent variables and L := k(a_{1}, …, a_{n}), we see that trdeg_{k}(K) = trdeg_{k}(L) = n and thus in this case ed(X[n]) = n. Therefore,

ed(Sub_{ℙ^{1},k}) ⩾ sup_{n⩾1} ed(X[n]) = ∞ .

This completes the proof of the proposition in the case where G = E.

We now proceed with the proof of Proposition 14.1 for a general group G. Denote the centralizer and the normalizer of E in G by C_{G}(E) and N_{G}(E), respectively. Then W_{G}(E) := N_{G}(E)/C_{G}(E) acts on ℙ_{k}^{1}. By the Quillen Stratification Theorem [1, Theorem 5.6.3] and [1, Corollary 5.9.2], there exists a closed W_{G}(E)-invariant k-subvariety B ⊊ ℙ_{k}^{1} with the following property. Every X ∈ Sub_{ℙ^{1},k}(L) satisfying conditions

(i) X is W_{G}(E)-invariant, and

(ii) no irreducible component of X is contained in B_{K}, lies in the image of V_{E,k}(L). (The subvariety B ⊂ ℙ_{k}^{1} comes from the cohomology of elementary abelian subgroups of G strictly contained in E; see the bottom of [1, p. 178].) Consequently, for any X ∈ Sub_{ℙ^{1},k}(L) satisfying (i) and (ii), ed(Rep_{G,k}) ⩾ ed(X).

In particular, let L := k(a_{1}, …, a_{n}), where a_{1}, …, a_{n} are independent variables over k, X_{1}, …, X_{n} ⊂ ℙ_{L}^{1} be as in (14.3), and

Z[n] := ⋃ w(X_{i}) ,

where the union is taken over all i = 1, …, n and all w ∈ W_{G}(E). Note that every point of the form w(X_{i}) corresponds to a dominant k-morphism Spec(L) → ℙ_{k}^{1} and hence, cannot lie in B_{K} for any proper subvariety B ⊂ ℙ_{k}^{1}. Thus Z[n] satisfies conditions (i) and (ii). We conclude that Z[n] lies in the image of V_{E,k}, and thus

(14.4)  ed(Rep_{G,k}) ⩾ ed(Z[n])

for every n ⩾ 1. On the other hand, the Claim above shows that if Z[n] descends to an intermediate field k ⊂ K ⊂ L then every a_{i} is algebraic over K. Hence, ed(Z[n]) = trdeg_{k}(L) = n, and (14.4) tells us that ed(Rep_{G,k}) = ∞, as desired. □

Acknowledgements. The authors are grateful to Alexander Merkurjev for contributing Example 11.5 and to Patrick Brosnan, Jon Carlson, Jerome Lefebvre, Julia Pevtsova, and Lior Silberman for stimulating discussions.

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Mathematical & Statistical Sciences, University of Alberta, Edmonton, CANADA

  E-mail address: karpenko at ualberta.ca, web page:  www.ualberta.ca/~karpenko

Department of Mathematics, University of British Columbia, Vancouver, CANADA

  E-mail address: reichst at math.ubc.ca, web page:  www.math.ubc.ca/~reichst