ON ISOTYPIES BETWEEN GALOIS CONJUGATE BLOCKS

RADHA KESSAR

Abstract. We show that between any pair of Galois conjugate blocks of a finite group, there is an isotypy with all signs positive.

1. Introduction

Let p be a prime number, let k be an algebraic closure of the field of p elements and let G be a finite group. Let

σ : k → k, (λ → λ^{p}, λ ∈ k)

be the Frobenius homomorphism of k and we write also σ : kG → kG for the ring automorphism induced by σ. This is defined by

σ(∑_{g∈G} α_{g}g) = ∑_{g∈G} α_{g}^{p}g.

The map σ is a ring automorphism but not a k-algebra automorphism. Since σ is a ring automorphism, σ induces a permutation of the blocks of kG. Here by a block of a ring A, we mean a primitive idempotent of the center of A. Blocks b and c of kG are said to be Galois conjugate if c = σ^{n}(b) for some natural number n.

Let (K, O, k) be a p-modular system and let ^{−} : O → k be the canonical quotient mapping. Let ^{−} : OG → kG be the induced O-algebra homomorphism of the group algebras. In particular, b → b̄ induces a bijection between the set of blocks of OG and the set of blocks of kG. Blocks b and c of OG are said to be Galois conjugate if b̄ and c̄ are Galois conjugate blocks of kG.

Our main result is the following.

Theorem 1.1. With the notation above, suppose that K contains a primitive |G|-th root of unity. Then any two Galois conjugate blocks of OG are isotypic.

As a corollary, we obtain a new proof of the following result of Cliff, Plesken and Weiss [7].

Corollary 1.2. Let G be a finite group and b be a block of kG. Then the center Z(kGb) of kGb has an 𝔽_{p}-form.

We recall that a finite dimensional k-algebra A is said to have an 𝔽_{p}-form if there exists an 𝔽_{p}-algebra A_{0} such that A ≅ k ⊗_{𝔽_{p}} A_{0} as k-algebra.

The notion of isotypic blocks (see Definition 2.2) is due to Michel Broué [6, Definition 4.3]. Isotypies between blocks are interesting as they are often the character theoretic shadow of a derived or Morita equivalence between the k-linear module categories of the corresponding block algebras. In [2], it was shown that there exist pairs of Galois conjugate blocks which are not derived equivalent as k-algebras. Thus the blocks studied in [2] provide examples of pairs of blocks which are isotypic and isomorphic as rings but not derived equivalent as k-algebras. However, it is conjectured that the number of Morita equivalence classes of algebras in any set of Galois conjugate blocks is bounded by a number which depends only on the defect of the blocks and which is independent of G. This conjecture is related to the Donovan finiteness conjectures in block theory [9].

For a non-negative integer d and a finite dimensional commutative kalgebra A, we will say that A occurs as the center of a d-block if there exists a finite group H and a block c of kH with defect d such that A ≅ Z(kHc) as k-algebras. Combining Corollary 1.2 with a theorem of Brauer and Feit [4] gives a proof of the following finiteness result. The result itself is known to experts, but as far as I am aware has not appeared before in the literature.

Corollary 1.3. Let d be a non-negative integer and set m := \frac{1}{4}p^{2d} + 1. Up to isomorphism there are at most p^{m^{3}} k-algebras that occur as centers of d-blocks.

The paper is divided into three sections. In Section 2, we set up notation and recall the definitions of perfect isometries and isotypies. Section 3 contains the proofs of Theorem 1.1, Corollary 1.2 and Corollary 1.3.

2. Notation and definitions

Throughout, G is a finite group, k is algebraically closed of characteristic p and (K, O, k) is a p-modular system such that K contains a primitive |G|-th root of unity. For an element g of G, we will denote by g_{p} the p-part of g and by g_{p′}-the p′-part of G. Denote by G_{p} the set of elements of G of p-power order and by G_{p′} the set of elements of G of p′-order. For a natural number n, let n_{p} denote the p-part n and n_{p′} denote the p′-part of n.

2.1. Generalized decomposition maps. Let b be a central idempotent of OG. Denote by CF(G, K) the K-space of K-valued class functions on G. Identifying K-valued class functions on G with their canonical K-linear extensions to K-valued functions on KG, denote by CF(G, b, K) the Ksubspace of class functions ϕ in CF(G, K) such that ϕ(gb) = ϕ(g) for all g ∈ G. Denote by CF_{p′}(G, K) (respectively CF_{p′}(G, b, K)) the K-subspaces of CF(G, K) (respectively CF(G, b, K)) of class functions which vanish on the p-singular classes of G; Irr(G, K) (respectively Irr(G, b, K)) the subset of CF(G, K) (respectively CF(G, b, K)) of irreducible K-valued characters of G; by IBr(G, K) (respectively IBr(G, b, K)) the set of irreducible Brauer characters of G (respectively irreducible Brauer characters of G in b). For ϕ ∈ IBr(G, K), let ϕ_{0} ∈ CF_{p′}(G, K) be defined by ϕ_{0}(y) = ϕ(y) if y ∈ G_{p′}. Denote by IBr(G, K)_{0} the set of class functions ϕ_{0}, for ϕ ∈ IBr(G, K) and by IBr(G, b, K)_{0} the subset of IBr(G, K)_{0} consisting of those ϕ_{0} for which ϕ ∈ IBr(G, b, K)

Let x be a p-element of G. The generalized decomposition map

d_{G}^{x} : CF(G, K) → CF_{p′}(C_{G}(x), K)

is defined by d_{G}^{x}(α)(y) = α(xy) for α ∈ CF(G, K), y ∈ C_{G}(x)_{p′}. If χ ∈ Irr(G, K), x ∈ G_{p}, then

d_{G}^{x}(χ) = ∑_{ϕ∈IBr(C_{G}(x),K)} δ_{(G,χ)}^{(x,ϕ)}ϕ_{0}

for uniquely determined elements δ_{(G,χ)}^{(x,ϕ)} of K; δ_{(G,χ)}^{(x,ϕ)} is called the generalized decomposition number associated to the triple (χ, ϕ, x).

Let e be a central idempotent of OC_{G}(x). The map

d_{G}^{(x,e)} : CF(G, K) → CF_{p′}(C_{G}(x), e, K)

is defined by d_{G}^{(x,e)}(α)(y) = α(xey) for α ∈ CF(G, K), y ∈ C_{G}(x)_{p′}.

2.2. Local structure of blocks. Let R denote one of the rings O or k. For Q a p-subgroup of G, let (RG)^{Q} denote the R-subalgebra of RG consisting of the elements of RG which are fixed by the conjugation action of Q. The Brauer homomorphism Br_{Q} : (RG)^{Q} → kC_{G}(Q) is the map defined by

Br_{Q}(∑_{g∈g} α_{g}g) = ∑_{g∈kC_{G}(Q)} ᾱ_{g}g.

Here if R = k, then ᾱ_{g} is to be interpreted as α_{g}. If b is a block of OG, then a b̄-Brauer pair is a pair (Q, ē), where Q is a p-subgroup of G and e is a block of OC_{G}(Q) such that Br_{Q}(b)ē = ē. Let (P, ē_{P}) be a maximal b-Brauer pair under the Alperin-Broué inclusion of Brauer pairs [1, Definition 1.1], and for each subgroup Q of P, let e_{Q} be the unique block of OC_{G}(Q) such that (Q, ē_{Q}) ≤ (P, ē_{P}).

Let F_{(P,ē_{P})}(G, b̄) denote the category whose objects are the subgroups of P and whose morphisms are defined as follows: For Q, R subgroups of P, the set of F_{(P,ē_{P})}(G, b̄) morphisms from Q to R is the set of those group homomorphisms φ : Q → R for which there exists an element g of G such that φ(x) = gxg^{−1} for x ∈ Q and such that (^{g}P, ^{g}ē_{Q}) ≤ (R, ē_{R}); composition of morphisms is the usual composition of functions.

2.3. Perfect isometries and isotypies. Let H be a finite group such that K contains a primitive |H|-th root of unity. Let b be a central idempotent of OG and c a central idempotent of OH.

Definition 2.1. A perfect isometry between b and c is a K-linear map

I : CF(G, b, K) → CF(H, c, K)

such that the following holds. For each χ ∈ Irr(G, b, K), there exists an ϵ_{χ} ∈ {±1} such that the map χ → ϵ_{χ}I(χ) is a bijection between Irr(G, b, K) and Irr(H, c, K) and such that setting

μ := ∑_{χ∈Irr(G,b,K)} χ × I(χ),

the class function on G × H which sends an element (x, y) of G × H to the element ∑_{χ∈Irr(G,b,K)} χ(x)I(χ)(y) of O the following holds:

(a) For each x ∈ G, y ∈ H, \frac{μ(x,y)}{|C_{G}(x)|} ∈ O.

(b) If x ∈ G, y ∈ H are such that exactly one of x and y is p-singular, then μ(x, y) = 0.

If I as in the above definition is a perfect isometry between b and c, then I induces by restriction a map

I_{p′} : CF_{p′}(G, b, K) → CF_{p′}(H, c, K).

Definition 2.2. Let b be a block of OG and c a block of OH. Then b and c are isotypic if the following conditions hold:

(a) There exists a p-group P and inclusions P ↪ G, P ↪ H such that identifying P with its image in G and in H, there exists a block e_{P} of OC_{G}(P) such that (P, ē_{P}) is a maximal b̄-Brauer pair and a block f_{P} of OC_{H}(P) such that (P, f̄_{P}) is a maximal c̄-Brauer pair and such that

F_{(P,ē_{P})}(G, b̄) = F_{(P,f̄_{P})}(H, c̄).

(b) For each cyclic subgroup Q of P, there exists a perfect isometry

I^{Q} : CF(C_{G}(Q), e_{Q}, K) → CF(C_{H}(Q), f_{Q}, K)

where e_{Q} (respectively f_{Q}) is the unique block of OC_{G}(Q) (respectively OC_{H}(Q)) with (Q, ē_{Q}) ≤ (P, ē_{P}) (respectively (Q, f̄_{Q}) ≤ (P, f̄_{P})) such that for every generator x of Q, we have

I_{p′}^{Q} ∘ d_{G}^{(x,e_{Q})} = d_{H}^{(x,f_{Q})} ∘ I^{{1}}.

3. Proofs.

Keep the notation and hypothesis of Theorem 1.1. In addition, let W (k) be the unique absolutely non-ramified complete discrete valuation ring having k as residue field, and identify W (k) with its image under the canonical injective homomorphism W (k) → O (see [12, Chapter 2 §5, Theorem 3 and Theorem 4]). There is a unique ring automorphism σ_{W (k)} : W (k) → W (k) such that \overline{σ_{W (k)}(η)} = σ(η̄) for all η ∈ W (k). Further, note that for any p′-root of unity η in K, η ∈ W (k) and σ_{W (k)}(η) = η^{p}.

Let K_{0} be the algebraic closure of ℚ in K. Choose a field automorphism σ_{K_{0}} : K_{0} → K_{0} such that if η ∈ K is any |G|-th root of unity, then σ_{K_{0}}(η) = η^{p} if the order of η is relatively prime to p and σ_{K_{0}}(η) = η if η is a power of p. Then, K_{0} ∩ W (k) contains a primitive |G|_{p′}-root of unity and σ_{K_{0}} and σ_{W (k)} coincide on ℚ[η] ∩ W (k) for any |G|_{p′}-root of unity η. Note that we are not claiming that σ_{0} extends to an automorphism of O which agrees with σ_{K_{0}} on restriction to ℚ[η] ∩ O for any |G|-th root of unity η.

Denote by σ_{W (k)} (respectively σ_{K_{0}}) the natural extension of σ_{W (k)} (respectively σ_{K_{0}}) to W (k)G (respectively K_{0}G).

Recall from [11, Chapter 3, Theorem 6.22 (ii)] that any block of OH, for H a finite group is an O-linear combination of p′-elements of H.

Lemma 3.1. Let η be a primitive |G|_{p′} root of unity in K and let H ≤ G. Let

c = ∑_{g∈H_{p′}} α_{g}g, α_{g} ∈ O

be a block of OH. Then, (i) α_{g} ∈ ℚ[η] ∩ W (k) for all g ∈ H_{p′}. (ii) σ_{W (k)}(c) is a block of OH, σ_{W (k)}(c) = σ_{K_{0}}(c) and

\overline{σ_{W (k)}(c)} = σ(c̄).

Proof. (i) By idempotent lifting, the canonical quotient map ^{−} : O → k induces a bijection between the set of central idempotents of OH and kH. Similarly, the restriction of ^{−} to W (k) induces a bijection between the set of central idempotents of W (k)H and kH. Since a central idempotent of W (k)H is a central idempotent of OH, it follows that OH and W (k)H have the same central idempotents. In particular, c ∈ W (k)H. So, α_{g} ∈ W (k) for all g ∈ H_{p′}. On the other hand, by [11, Chapter 3, Theorem 6.22], for g ∈ H_{p′}, α_{g} is a ℚ-linear combination of |g|-th roots of unity whence α_{g} ∈ ℚ[η]. This proves (i).

(ii) As shown above, the set of blocks of OH is the same as the set of blocks of W (k)H and σ_{W (k)} is an automorphism of W (k)H. This proves the first assertion. The others are immediate from (i).

Definition 3.2. For H a subgroup G, let

I^{H} : CF(H, K) → CF(H, K)

denote the K-linear map defined by

I^{H}(ϕ)(x) = ϕ(x_{p}x_{p′}^{p}), for ϕ ∈ CF(H, K), x ∈ G.

If an element ϕ ∈ CF(H, K) takes values in K_{0}, denote by σ(ϕ) the element of CF(H, K) which sends g ∈ H to σ_{K_{0}}(ϕ(g)). Similarly, if ϕ takes values in W (k), then σ_{W (k)}(ϕ) will denote the element of CF(H, K) which sends g ∈ H to σ_{W (k)}(ϕ(g)). We use the same conventions on K-valued class functions defined on H_{p′}.

Lemma 3.3. Let H be a subgroup of G and let c be a block of H.

(i) For any χ ∈ Irr(H, K),

I^{H}(χ) = σ_{K_{0}}(χ) ∈ Irr(H, K).

(ii) The map

χ → σ_{K_{0}}(χ)

is a bijection from Irr(H, c, K) to Irr(H, σ_{K_{0}}(c), K).

(iii) The restriction of I^{H} to CF(H, c, K) induces a perfect isometry between c and σ_{K_{0}}(c).

(iv) For any ϕ ∈ IBr(H, K) and any χ ∈ Irr(G, K),

I^{H}(ϕ_{0}) = σ_{K_{0}}(ϕ)_{0} ∈ IBr(H, K)_{0}

and

δ_{(H,σ_{K_{0}}(χ))}^{({1},σ_{K_{0}}(ϕ))} = δ_{(H,χ)}^{({1},ϕ)}.

Proof. (i) Let χ ∈ Irr(H, K), and let ρ : H → GL_{n}(K_{0}) be a representation affording χ. Such a ρ exists by Brauer’s splitting field theorem (see for example [11, Chapter 3, Theorem 4.11]). Denoting also by σ_{K_{0}} the automorphism (as abstract group) of GL_{n}(K_{0}) induced by σ_{K_{0}}, it follows that σ_{K_{0}} ∘ρ is an irreducible representation of H with character σ_{K_{0}}(χ). We show that I^{H}(χ) = σ(χ). Let h = h_{p}h_{p′} ∈ H. Since K_{0} contains all the eigen values of ρ(h), by replacing ρ with an equivalent representation if necessary, we may assume that ρ(h) is a diagonal matrix. Since h_{p} and h_{p′} are powers of h, it follows that ρ(h_{p}) is a diagonal matrix diag(ζ_{1}, ⋯, ζ_{n}), ρ(h_{p′}) is a diagonal matrix diag(η_{1}, ⋯, η_{n}), where each ζ_{i} is a |H|_{p} root of unity and each η_{j} is an |H|_{p′} root of unity, ρ(h)is the diagonal matrix diag(ζ_{1}η_{1}, ⋯, ζ_{n}η_{n}) and ρ(h_{p}h_{p′}^{p}) is the diagonal matrix diag(ζ_{1}η_{1}^{p}, ⋯, ζ_{n}η_{n}^{p}). Thus,

I^{H}(χ)(h) = ∑_{1≤i≤n} ζ_{i}η_{i}^{p} = σ_{K_{0}}( ∑_{1≤i≤n} ζ_{i}η_{i}) = σ(χ)(h).

(ii) Let χ ∈ Irr(H, K) and let

e_{χ} = \frac{χ(1)}{|H|} ∑_{h∈H} χ(h)h^{−1}

be the primitive central idempotent of KH corresponding to χ. Then,

e_{σ_{K_{0}}(χ)} = \frac{χ(1)}{|H|} ∑_{h∈H} σ_{K_{0}}(χ(h))h^{−1} = σ_{K_{0}}(e_{χ}).

From this it is immediate that if χ ∈ Irr(H, c, K) then σ_{K_{0}}(χ) ∈ Irr(H, σ_{K_{0}}(c), K). (iii) By (i) and (ii) it suffices to prove that the function

μ := ∑_{χ∈Irr(H,c,K)} χ × I^{H}(χ),

satisfies conditions (a) and (b) of Definition 1.1.

Set

ι := ∑_{χ∈Irr(H,K,c)} χ × χ

and let x, y ∈ H.

Then

μ(x, y) = ι(x, y_{p}y_{p′}^{p}).

Since the identity map on CF(H, c, K) is a perfect isometry with ι the corresponding (virtual) character of H × H, we see that μ(x, y) is divisible in O by both |C_{G}(x)| and by |C_{G}(y_{p}y_{p′}^{p})| and that μ(x, y) is 0 if exactly one of x and y_{p}y_{p′}^{p} is p-singular. But, clearly C_{G}(y_{p}y_{p′}^{p}) = C_{G}(y) and y_{p}y_{p′}^{p} is p-singular if and only if y is p-singular. This proves (iii).

(iv) Let ϕ ∈ IBr(H, K) and let τ : H → GL_{n}(k) be an irreducible representation of H affording the Brauer character ϕ. Then σ ∘τ is an irreducible representation of H, where again we denote by σ the induced automorphism of GL_{n}(k). Let ϕ′ be the Brauer character associated to σ∘τ and let h ∈ H_{p′}. Let Λ be the multiset of eigen values of τ(h). The multiset of eigen values of σ∘τ(h) is {λ^{p} : λ ∈ Λ}, hence ϕ′(h) = σ_{W (k)}(h). Further, since {λ^{p} : λ ∈ Λ} is also the multiset of eigen values of τ(h^{p}), I^{H}(ϕ_{0})(h) = ϕ(h^{p}) = σ_{W (k)}(h) for all h ∈ H_{p′}. Thus,

I^{H}(ϕ_{0}) = σ_{W (k)}(ϕ)_{0} = ϕ′ ∈ IBr(H, K).

Since ϕ takes values in ℤ[η] , with η a primitive |H|_{p′}-root of unity, σ_{W (k)}(ϕ)_{0} = σ_{K_{0}}(ϕ)_{0} whence I^{H}(ϕ_{0}) = σ_{K_{0}}(ϕ)_{0} ∈ IBr(H, K).

The compatibility of decomposition numbers is clear from the fact that for any χ ∈ Irr(H, K),

Res|_{H_{p′}} χ = ∑_{ϕ∈IBr(H,K)} δ_{H,χ}^{1,ϕ}ϕ

and that δ_{H,χ}^{1,ϕ} ∈ ℤ for all ϕ ∈ IBr(H, K).

Proof of Theorem 1.1. Let b be a block of OG. By Lemma 3.1 it suffices to show that b and σ_{K_{0}}(b) are isotypic. Let P a p-subgroup of G and e_{P} be a block of OC_{G}(P) such that (P, ē_{P}) is a maximal b̄-Brauer pair. For each Q ≤ P, let e_{Q} be the unique block of OC_{G}(Q) such that (Q, ē_{Q}) ≤ (P, ē_{P}). For any p-subgroup Q of G and any a ∈ (kG)^{Q}, σ(Br_{P} (a)) = Br_{P} (σ(a)). So, the map (Q, f) → (Q, σ(f)) is an isomorphism from the G-poset of b̄- Brauer pairs to the G-poset of σ(b̄)-Brauer pairs. By Lemma 3.1, σ(b̄) = \overline{σ_{K_{0}}(b)} and σ(f̄) = \overline{σ_{K_{0}}(f)} for any block f of kC_{G}(Q), Q a p-subgroup of G. Thus, (P, \overline{σ_{K_{0}}(e_{P})}) is a maximal \overline{σ_{K_{0}}(b)}-Brauer pair; for every subgroup Q of P, σ_{K_{0}}(e_{Q}) is the unique block of OC_{G}(Q) such that (Q, \overline{σ_{K_{0}}(e_{Q})}) ≤ (P, \overline{σ_{K_{0}}(e_{P})}), and F_{(P,\overline{σ_{K_{0}}(e_{P})})}(G, \overline{σ_{K_{0}}(b)}) = F_{(P,ē_{P})}(G, b̄).

We will use the maps I^{C_{G}(Q)} of Definition 3.2 to produce an isotypy between b and σ_{K_{0}}(b). For Q ≤ P, let I^{Q} be the restriction of I^{C_{G}(Q)} to CF(C_{G}(Q), e_{Q}, K). We will show that the maps I^{Q}, as Q runs over the cyclic subgroups of P defines an isotypy between b and c. So, let Q ≤ P be a cyclic group. By Lemma 3.3,

I^{Q} : CF(C_{G}(Q), e_{Q}, K) → CF(C_{G}(Q), σ_{K_{0}}(e_{Q}), K)

is a perfect isometry. It remains only to check the compatibility condition

(b) of Definition 2.2.

Set H = C_{G}(Q) and let Q = ⟨x⟩. We claim that

δ_{(G,χ)}^{(x,ϕ)} = δ_{(G,σ_{K_{0}}(χ))}^{(x,σ_{K_{0}}(ϕ))}

for all χ ∈ Irr(G, K) and all ϕ ∈ IBr(H, K).

Indeed, for τ ∈ Irr(H, K), let ζ_{(τ,x)} ∈ K be such that ρ(x) is the scalar matrix (ζ_{(τ,x)}, ⋯, ζ_{(τ,x)}) in any representation of H affording τ. Since the order of ζ_{(τ,x)} is a divisor of |G|_{p},

ζ_{(σ_{K_{0}}(τ),x)} = σ_{K_{0}}(ζ_{(τ,x)}) = ζ_{(τ,x)}.

Also, for χ ∈ Irr(G, K), ϕ ∈ IBr(H, K),

δ_{(G,χ)}^{(x,ϕ)} = ∑_{τ∈Irr(H,K)}⟨Res|_{H}χ, τ⟩ζ_{(τ,x)}δ_{(H,τ)}^{({1},ϕ)}.

The claim follows since for all τ ∈ Irr(H, K) and all ϕ ∈ IBr(H, K), ζ_{(σ_{K_{0}}(τ),x)} = ζ_{(τ,x)}, δ_{(H,σ_{K_{0}}τ))}^{({1},σ_{K_{0}}(ϕ))} = δ_{(H,τ)}^{({1},ϕ)} by Lemma 3.3 (iv) and ⟨Res|_{H}χ, τ⟩ζ_{(τ,x)} ∈ ℤ.

For χ ∈ Irr(G, K), ϕ ∈ IBr(H, K),

d_{G}^{(x,e_{Q})}(χ) = ∑_{ϕ∈IBr(H,e_{Q},K)} δ_{(G,χ)}^{(x,ϕ)}ϕ_{0},

The compatibility condition (b) of Definition 2.2 is easily seen to follow from the linearity of the maps I^{Q}, the claim and Lemma 3.3

I_{p′}^{Q} ∘ d_{G}^{(x,e_{Q})}(χ) = ∑_{ϕ∈IBr(H,e_{Q},K)} δ_{(G,χ)}^{(x,ϕ)}σ_{K_{0}}I^{Q}(ϕ_{0})

= ∑_{ϕ∈IBr(H,e_{Q},K)} δ_{(G,σ_{K_{0}}(χ))}^{(x,σ_{K_{0}}(ϕ))}I^{Q}(ϕ_{0})

= ∑_{ϕ∈IBr(H,e_{Q},K)} δ_{(G,σ_{K_{0}}(χ))}^{(x,σ_{K_{0}}(ϕ))}σ_{K_{0}}(ϕ)

= ∑_{ϕ′∈IBr(H,σ_{K_{0}}(e_{Q}),K)} δ_{(G,σ_{K_{0}}(χ))}^{(x,ϕ′)}ϕ′_{0}

= d_{G}^{(x,σ_{K_{0}}(e_{Q}))}I^{{1}}(χ).
□

Proof of Corollary 1.2. Let (K, O, k) be a p-modular system such that k is an algebraic closure of 𝔽_{p} and K contains a primitive |G|-th root of unity. By the theorem, there is a perfect isometry between OGb and OGσ_{K_{0}}(b). Hence, by [5, Théorème 1.4], there is an O-algebra isomorphism f : Z(OGσ_{K_{0}}(b)) → Z(OGb). This induces a k-algebra isomorphism f : Z(kGσ(b̄)) → Z(kGb̄). On the other hand, σ induces a ring isomorphism σ : Z(kGb̄) → Z(kGσ(b̄)) such that for all a ∈ Z(kGb̄) and all λ ∈ k, σ(λa) = λ^{p}σ(a). Thus, σ ∘ f : Z(kGσ(b)) → Z(kGσ(b)) is a ring automorphism which satisfies σ ∘ f(λa) = λ^{p}σ ∘ f(a) for all a ∈ Z(kGσ(b̄)) and all λ ∈ k. By [9, Lemma 2.1], the fixed points (Z(kGσ(b̄)))^{σ∘f} of Z(kGσ(b̄)) under σ ∘ f are an 𝔽_{p}-subspace of Z(kGσ(b̄)) such that Z(kGσ(b̄)) ≅ k ⊗_{𝔽_{p}} (Z(kGσ(b̄)))^{σ∘f} as k-vector spaces. Since σ ∘ f is a homomorphism of rings, (Z(kGσ(b̄)))^{σ∘f} is an 𝔽_{p}-algebra. Thus Z(kGσ(b̄)) and hence Z(kGb) has an 𝔽_{p}-form. □

Proof of Corollary 1.3. Let b be a block of kG with defect d. Since dim_{k}(Z(kGb)) = dim_{K}(Z(KGb)) = |Irr(G, b, K)|, by [4, Theorem 1] the k-dimension of Z(kGb) is bounded by m. By Corollary 1.2, Z(kGb) has a k-basis such that the multiplicative constants of Z(kGb) with respect to this basis are all in 𝔽_{p}. Thus there are at most p^{m^{3}} possibilities for the isomorphism type of Z(kGb). □

Remarks 3.4. (i) The proof of Theorem 1.1 can be readily adapted to prove that between any pair of Galois conjugate blocks there is a global isotypy in the sense of [3, 1.9]. It is not known whether there is a p-permutation equivalence (cf. [3, Definiton 1.3], [10, Definiton 1.3]) between any pair of Galois conjugate blocks.

(ii) By Lemma 3.3, the isometries between various blocks in Theorem 1.1 all appear without signs, which seems to render even more surprising the fact that Galois conjugate blocks need not be Morita equivalent [2].

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Institute of Mathematics, University of Aberdeen, Fraser Noble Building, King’s College, Aberdeen AB24 3UE, U.K.