BLOCKS INEQUIVALENT TO THEIR FROBENIUS TWISTS

DAVID BENSON, RADHA KESSAR

Abstract. Let k be an algebraically closed field of characteristic p. We give a general method for producing examples of blocks B of finite group algebras that are not Morita equivalent as k-algebras to the Frobenius twist B^{(p)}. Our method produces non-nilpotent blocks having one simple module and elementary abelian defect group. These also provide the first known examples of blocks where there is a perfect isotypy at the level of ordinary characters with all the signs positive, but no derived equivalence between the blocks. We do not know of any examples of blocks B that are not Morita

equivalent to the second Frobenius twist B^{(p^{2})}.

1. Introduction

Let p be a prime number, let k be an algebraically closed field of characteristic p, and let q = p^{a} be a power of p. If A is a k-algebra, we can define the Frobenius twist of A, denoted A^{(q)}, as follows. The underlying ring is the same, but we endow it with a new action of the scalars in k via the Frobenius map on k: for λ ∈ k and x ∈ A, the new action is given by λ · x = λ^{\frac{1}{q}} x.

Definition 1.1. We say that a k-algebra A is defined over 𝔽_{q} if there is a k-vector space basis of A such that all the structure constants lie in 𝔽_{q}.

Clearly an algebra A defined over 𝔽_{q} satisfies A ≅ A^{(q)}. Conversely, we have the following.

Lemma 1.2 (Kessar [9, Lemma 2.1]). If A is finite dimensional over k, then A ≅ A^{(q)} as k-algebras if and only if A is defined over 𝔽_{q}.

It follows from this lemma that a finite dimensional algebra A is Morita equivalent to A^{(q)} as a k-algebra if and only if the basic algebra of A is defined over 𝔽_{q}.

We define several invariants of a finite dimensional k-algebra A as follows.

Definition 1.3. The Frobenius number of A is the least a such that A^{(p^{a})} is isomorphic to A. Then A^{(p^{d})} is isomorphic to A if and only if d is divisible by a. The Morita Frobenius number is the least a such that A^{(p^{a})} is Morita equivalent to A, and the derived Frobenius number is the least a such that A^{(p^{a})} is derived equivalent to A.

In general, if A has only one simple module, then its Frobenius number and its Morita Frobenius number coincide, but the derived Frobenius number may be smaller. However, if A is a symmetric algebra and A has only one simple module, then by a result of Roggenkamp and Zimmermann (see Theorem 2.11 of Rouquier and Zimmermann [13]) the derived Frobenius number of A is equal to its Morita Frobenius number.

Example 1.4. Let k = 𝔽̄_{2} and let ω and ω̄ be the primitive cube roots of unity in k, so that 𝔽_{4} = {0, 1, ω, ω̄}. Let

A = k⟨X, Y ⟩/(X^{2} = Y^{2} = 0, XY = ωY X).

At first sight, it looks as though the field of definition for this algebra should be 𝔽_{4}. But A^{(2)} is isomorphic to A via the map that swaps X and Y, so in fact A is defined over 𝔽_{2}. Indeed, if we write U = X + Y and V = ωX + ω̄Y then

A = k⟨U, V ⟩/(U^{2} = UV = V^{2}, V U = 0).

So the Frobenius number for this algebra is equal to one.

However, if P_{1} and P_{2} are representatives of the two isomorphism classes of projective indecomposable A-modules then

B = End_{A}(P_{1} ⊕ P_{2} ⊕ P_{2})

is an algebra Morita equivalent to A, but with Frobenius number equal to two.

For blocks of a finite group, then there is another way to realise the Frobenius twist, via Galois conjugation inside the group algebra. Namely, if G is a finite group, we write σ: kG → kG for the map induced by the Frobenius automorphism of k. This is defined by

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

The map σ is an isomorphism of rings, but not of k-algebras. Since the blocks of kG are in one to one correspondence with primitive idempotents of the centre of kG, σ permutes the blocks of kG. We say that blocks related by powers of σ are Galois conjugate. For any block B of kG, σ defines an isomorphism of k-algebras between B^{(p)} and σ(B) as k-algebras. In particular, for any block B of a group algebra kG, the Frobenius twist B^{(p)} can be regarded as a Galois conjugate block of the same group algebra kG via this isomorphism.

For example, if B is the principal block, then B = σ(B), and so by Lemma 1.2 the principal block is always defined over 𝔽_{p}. So where should we look for examples of blocks B whose basic algebras are not defined over 𝔽_{p}?

Non-principal blocks of simple and almost simple groups appear not to give examples (we have not rigorously checked this, but it should not be difficult). Blocks of defect zero and blocks with cyclic defect groups can never provide examples. The same is known to hold for tame blocks, with the exception of blocks with quaternion defect groups and two isomorphism classes of simple modules. Here the problem is that it is not known which of the algebras occurring in Erdmann’s description of the basic algebras of such blocks actually do occur.

So it is natural to look at nonprincipal blocks of soluble groups that cannot be Morita equivalent to a principal block. Examples of such blocks whose basic algebra is a quantum complete intersection have recently been studied by Benson and Green [2] and by Holloway and Kessar [6]. Our goal in this paper is to provide a method for constructing blocks B of soluble groups along the lines of [2, 6] such that B is not Morita equivalent as a k-algebra to B^{(p)}.

In [10] it was shown that between any pair of Galois conjugate blocks there is a perfect isometry (isotypy) with all signs positive. Still, there is no reason to suppose that the module categories of Galois conjugate blocks are equivalent as k-linear categories. On the other hand, since all numerical invariants of pairs of such blocks match up, it is difficult to tell the corresponding module categories apart. Our analysis involves a careful study of commutation matrices and their Galois conjugation properties for this purpose. Our main theorem is as follows.

Theorem 1.5. For each prime p, there exists a soluble group G and a non-principal block B of kG with the following properties:

(i) B is not Morita equivalent as a k-algebra to B^{(p)}.

(ii) B has elementary abelian defect groups and abelian inertial quotient.

(iii) B has only one isomorphism class of simple modules, and its basic algebra is a quantum complete intersection.

Using Theorem 2.11 of [13], we deduce the following.

Corollary 1.6. A block B described in Theorem 1.5 is not derived equivalent as a k-algebra to B^{(p)}.

We mention, though, that all the blocks we construct have the property that B ≅ B^{(p^{2})}, so the Morita Frobenius number and the derived Frobenius number are exactly two. We have failed to construct blocks with larger Morita Frobenius number, and we do not know whether they exist.

The interest in these definitions comes in part from the observation made in [9] about Donovan’s conjecture. Conjecture 1.7 (Donovan’s conjecture [1, Conjecture M]). Given a finite p-group D, there are only a finite number of Morita equivalence classes of blocks of finite groups whose defect groups are isomorphic to D.

There is a weaker conjecture of Donovan, which is as follows.^{1} Conjecture 1.8. Given a finite p-group D, there is a bound on the entries in the Cartan matrices of blocks of finite groups whose defect groups are isomorphic to D.

The interest in fields of definition of blocks comes in part from the observation made in [9] which explains that the gap between the conjectures of Donovan and Brauer can be explained by the relationship between Galois conjugate blocks. Conjecture 1.9 (Kessar [9]). Given a finite p-group D, there is a bound on the Morita Frobenius numbers of blocks with defect group D. Theorem 1.10 (Kessar [9]). Conjectures 1.8 and 1.9 are together equivalent to Donovan’s conjecture 1.7.

In the next section we review the results of [2, 6]. In section 3, we study the combinatorics of commutation matrices, and give the construction of the examples.

2. Quantum complete intersections

Definition 2.1. A square matrix q = (q_{i,j})_{1≤i,j≤r}, q_{i,j} ∈ k, is called a commutation matrix over k if

q_{i,j}q_{j,i} = q_{i,i} = 1 for all 1 ≤ i, j ≤ r.

The quantum symmetric algebra k_{q}[X_{1}, …, X_{r}] is defined to be the quotient of the free (tensor) algebra by the commutation relations given by the matrix q:

k_{q}[X_{1}, …, X_{r}] := k⟨X_{1}, …, X_{r}⟩/(X_{i}X_{j} − q_{i,j}X_{j}X_{i}).

1Brauer [3, Problem 22] conjectured that the entries in the Cartan matrices are bounded by |D|, but Landrock [11] found a counterexample.

The quantum complete intersection algebra A_{q}[X_{1}, …, X_{r}] is defined to be

A_{q}[X_{1}, …, X_{r}] := k_{q}[X_{1}, …, X_{r}]/(X_{1}^{p}, …, X_{r}^{p}).

Definition 2.2. Let A = A_{q}[X_{1}, …, X_{r}] be a quantum complete intersection and let B be a subset of J(A), where J(A) denotes the Jacobson radical of A. We say that B is a special generating set of A if the image of B under the natural surjection of J(A) onto J(A)/J^{2}(A) is a k-basis of J(A)/J^{2}(A) and such that for all Y, Z ∈ B

Y^{p} = 0 and Y Z = λ_{Y Z}ZY for some λ_{Y Z} ∈ k.

Before proceeding, we note that for a quantum complete intersection A_{q}[X_{1}, …, X_{r}] the set of monomials {X_{1}^{i_{1}} … X_{r}^{i_{r}} | 0 ≤ i_{j} ≤ p − 1, 1 ≤ j ≤ r} is a k-basis of A_{q}, and omitting the monomial with (i_{1}, …, i_{r}) = (0, …, 0) gives a basis for J(A_{q}).

Lemma 2.3. Let A := A_{q}[X_{1}, …, X_{r}] be a quantum complete intersection and let B be a special generating set of A_{q}.

(i) Suppose that i, j with 1 ≤ i, j ≤ r are such that the coefficient of X_{i} and the coefficent of X_{j} in some element of B are both non-zero. Then q_{i,ℓ} = q_{j,ℓ} for all ℓ with 1 ≤ ℓ ≤ r, and in particular q_{i,j} = 1.

(ii) There is an ordering Y_{1}, …, Y_{r} of the elements of B such that Y_{i}Y_{j} = q_{i,j}Y_{j}Y_{i} for all i, j with 1 ≤ j ≤ r.

Proof. (i) Let Y ∈ B be such that the coefficient, say a of X_{i} and the coefficient, say b of X_{j} in Y are both non-zero. Since B is a special generating set, Y^{p} = 0. On the other hand, the coefficient of X_{i}^{p−1}X_{j} in Y^{p} is

a^{p−1}b(1 + q_{i,j} + ⋯ + q_{i,j}^{p−1}).

Since a^{p−1}b ≠ 0, it follows that 1 + q_{i,j} + ⋯ + q_{i,j}^{p−1} = 0. Multiplying by 1 − q_{i,j}, we see that 1 − q_{i,j}^{p} = 0, and so q_{i,j}^{p} = 1. Since we are in characteristic p, this implies that q_{i,j} = 1.

Now suppose, if possible, that ℓ ∈ {1, …, r} is such that q_{i,ℓ} ≠ q_{j,ℓ}. Since B generates A as algebra, the coefficient of X_{ℓ} in Z is non-zero for some Z ∈ B. Let λ ∈ k be such that Y Z = λZY. Since at least one of q_{i,ℓ} and q_{j,ℓ} is different from 1, and since the coefficients of both X_{i} and X_{j} in Y are non-zero, by the preceding argument it follows that the coefficient of X_{ℓ} in Y is zero. We claim that the coefficients of both X_{i} and X_{j} in Z are also zero. Indeed, suppose that the coefficient of X_{i} in Z is non-zero, so that in particular q_{i,ℓ} = 1; then q_{j,ℓ} ≠ 1, and so the coefficient of X_{j} in Z is zero. Then, since q_{i,j} = 1, by comparing coefficients of X_{i}X_{j} in Y Z and in ZY, it follows that λ = 1. But now comparing coefficients of X_{j}X_{ℓ} in Y Z and ZY respectively gives that q_{j,ℓ} = 1, a contradiction. This proves the claim. Now comparing coefficients of X_{i}X_{ℓ} in Y Z and in ZY, and then comparing coefficients of X_{j}X_{ℓ} in Y Z and in ZY, we get that q_{i,ℓ} = λ = q_{j,ℓ}.

(ii) Consider the partition {X_{1}, …, X_{r}} = ⋃_{t} I_{t} where X_{i} and X_{j} belong to the same I_{t} if and only if q_{i,ℓ} = q_{j,ℓ} for all ℓ with 1 ≤ ℓ ≤ r (and in particular q_{i,j} = 1). Then by part (i) above, we get a corresponding partition B = ⋃_{t} B_{t}, where Y ∈ B_{t} if and only if for some X_{i} ∈ I_{t} the coefficient of X_{i} in Y is non-zero. Since the elements of B are linearly independent, we obtain |B_{t}| ≤ |I_{t}| for each t. Since |B| = r, we deduce that |B_{t}| = |I_{t}| for each t. Now suppose Y ∈ I_{s} and Z ∈ I_{t}. Let X_{i} appear with non-zero coefficient in Y and X_{j} appear with non-zero coefficient in Z. Then clearly Y Z = q_{i,j}ZY. □

The following is an immediate consequence of the above lemma.

Proposition 2.4. Let q and q′ be commutation matrices over k of degree r and set A = A_{q}[X_{1}, …, X_{r}], A′ = A_{q′}[X_{1}, …, X_{r′}]. Then A and A′ are isomorphic as k-algebras if and only if r = r′ and

q′ = xqx^{−1}

for some r × r permutation matrix x.

Proof. One direction is obvious. For the other direction, suppose that A and A′ are isomorphic as k-algebras. Since r is the k-dimension of J(A)/J^{2}(A), it follows that r = r′. The isomorphism gives us a special generating set B = {Y_{1}, …, Y_{r}} for A such that Y_{i}Y_{j} = q′_{ij}Y_{j}Y_{i} for all i, j with 1 ≤ i, j ≤ r. Now the preceding lemma can be used to construct a suitable permutation matrix x. □

3. Quantum complete intersections and blocks with one simple module

Let L be an abelian p′-group and consider a central extension

1 → Z → H → L → 1

of L by a cyclic p′-group Z = ⟨z⟩. Let N be an elementary abelian pgroup on which L acts faithfully. Then N is naturally an 𝔽_{p}L-module. Let ϕ be the character of kL on the extension k ⊗_{𝔽_{p}} N. Since L is an abelian p′-group and k is algebraically closed, ϕ decomposes as a sum of one dimensional kL-modules,

ϕ = ⨁_{1}^{r} ϕ_{i},  (3.1)

where r is the rank of N.

Inflate the action of L on N to an action of H with Z acting trivially, and let G = N ⋊ H be the semidirect product. The blocks of kG are in one to one correspondence with the characters of Z. Let χ be a faithful irreducible k-character of Z and let

b = \frac{1}{|Z|} ∑_{i=1}^{|Z|} χ(z^{−i})z^{i}

be the correponding central idempotent in kG. Then b is a block idempotent of kG. Since N is in the kernel of every simple kG-module, the simple kGb-modules are in one to one correspondence with the simple constituents of Ind_{Z}^{H} χ. In particular kGb has one isomorphism type of simple module if and only if Ind_{Z}^{H} χ is a direct sum of isomorphic irreducibles. This happens if and only if kHb ≅ Mat_{d}(k), where d is the dimension of the corresponding irreducible, and in this case, |L| = dim_{k} Mat_{d}(k) = d^{2} is a square.^{2} In this situation the basic algebra of kGb is a quantum complete intersection.

Theorem 3.2. Suppose that kHb ≅ Mat_{d}(k). Then (i) The map

L → Hom(L, k^{×})  l ↦ (t ↦ χ([l̃, t̃])),  l, t ∈ L

(where l̃ and t̃ are any lifts of l and t respectively in H) is an isomorphism.

(ii) For each i, 1 ≤ i ≤ r let m_{i} be the element of L corresponding through (i) to the character ϕ_{i} appearing in equation (3.1). For 1 ≤ i, j ≤ r, set q_{i,j} = χ[m̃_{j}, m̃_{i}] and set q = (q_{i,j}). Then the basic algebra of kGb is isomorphic to A_{q}[X_{1}, …, X_{r}] .

Proof. This follows from [6, Theorem 4.2] and the proof thereof. □

Remark 3.3. Since ϕ is obtained by extending scalars from a representation defined over 𝔽_{p}, it follows that the set of characters ϕ_{i} occurring as summands in equation (3.1) is closed under the action of the Frobenius map. In other words, there is a permutation σ of the indices 1, …, r such that for all z ∈ Z we have ϕ_{σ(i)}(z) = ϕ_{i}(z)^{p}. Since q_{i,j} = χ[m̃_{j}, m̃_{i}], we have

q_{σ(i),j} = q_{i,σ(j)} = q_{i,j}^{p},  (3.4)

2There is a considerable literature [4, 5, 7, 8, 12, 14] on finite groups G with a faithful irreducible character of degree \sqrt{|G/Z(G)|}, which is the largest possible degree. It is known that such groups are soluble [7, 12], and that every finite soluble group embeds in G/Z(G) for some such group G [5]. It would be interesting to make a more general examination of the blocks with one simple module and abelian defect group that these give rise to.

and hence

q_{σ(i),σ(j)} = q_{i,j}^{p^{2}}.

It follows that kGb^{(p^{2})} is Morita equivalent to kGb, and hence isomorphic, as the single simple module has dimension d in each case. It is sometimes true that kGb^{(p)} ≅ kGb, and sometimes not. In the next section, we develop a method for determining whether this is so for any particular example.

4. Combinatorics

In this section, we describe a graphical method for telling whether two commutation matrices of the sort arising in Section 3 are equivalent by a permutation matrix. In the next section we look at some explicit examples.

Let (q_{i,j})_{1≤i,j≤r} be a commutation matrix, together with a permutation σ of the indices 1, …, r such that (3.4) holds. Let t be the number of orbits of σ on 1, …, r. Then we can form a reduced commutation matrix ζ = (ζ_{i,j})_{1≤i,j≤t} by choosing one index from each σ-orbit and only using the corresponding entries from the matrix q. The remaining entries are determined by (3.4).

Example 4.1. Let ε be an element of order p + 1 in 𝔽_{p^{2}}^{×}, and consider the commutation matrix

q = (\begin{matrix}1 & 1 & ε & ε^{−1} \\ 1 & 1 & ε^{−1} & ε \\ ε^{−1} & ε & 1 & 1 \\ ε & ε^{−1} & 1 & 1 \end{matrix}) .

Then we have

ζ = (\begin{matrix}1 & ε \\ ε^{−1} & 1 \end{matrix}) .

More complicated examples will appear in the next section.

We can display the information in more compact form as follows. Let 𝔽_{p^{a}} be the smallest splitting field of ϕ, and choose once and for all a generator ε for the multiplicative group 𝔽_{p^{a}}^{×}. Then the q_{i,j} and the ζ_{i,j} are powers of ε. Either exactly one of ζ_{i,j} and ζ_{j,i} is of the form ε^{s} with 0 < s ≤ \frac{p^{a}−1}{2}, or ζ_{i,j} = ζ_{j,i} = 1. Draw a labelled directed graph Γ whose vertices are the σ-orbits on 1, …, r, and where there is a directed edge from i to j with label ζ_{i,j} if ζ_{i,j} = ε^{s} with 0 < s ≤ \frac{p^{a}−1}{2}. Then Γ completely captures the information in the matrix q. In the example above, Γ is just · ⟶^{ε} · ; this notation is clearly much more compact than the matrix q.

If there is an isomorphism of labelled directed graphs Γ → Γ′ then the corresponding commutation matrices q and q′ are conjugate by a permutation matrix. But we also have to take account of the fact that we made a choice of representatives for the σ-orbits. We account for this as follows. Choose a vertex of Γ. Then we may “twist this vertex” by replacing the label on each edge incident with the vertex by the pth power of that label. If the power of ε now falls outside the proper range, then we reverse the edge and invert the label. This process amounts to replacing the representative of the corresponding orbit by its image under σ, and so it gives another graph for the same conjugation matrix.

Theorem 4.2. Given two labelled directed graphs Γ and Γ′, the following are equivalent:

(i) Γ is isomorphic to a graph obtained from Γ′ by a series of twists of vertices.

(ii) The commutation matrices q and q′ are conjugate by a permutation matrix.

(iii) The corresponding quantum complete intersections A_{q} and A_{q′} are isomorphic.

(iv) A_{q} and A_{q′} are Morita equivalent.

Proof. This is just a combinatorial restatement of Proposition 2.4; the equivalence with (iv) follows from the fact that basic algebras are isomorphic if and only if they are Morita equivalent. □

The problem of whether a block of the form considered here is equivalent to its Frobenius twist can now be reformulated. From the graph Γ, we can obtain a new graph Γ^{(p)} by replacing the label on each edge by its pth power, and then reversing edges and inverting their labels when the power of ε is not in the proper range. Then we have to determine whether Γ^{(p)} can be obtained from Γ by a series of twists of vertices.

5. Examples

The examples in this section all begin by letting p be a prime, and k be an algebraically closed field of characteristic p. Let ℓ ≥ 3 be a divisor of p + 1, and let ε be a primitive ℓth root of unity in 𝔽_{p^{2}} ⊆ k. Note that ε^{p} = ε^{−1} ≠ ε. So if we use powers of ε as labels, when we apply a twist to a vertex of the labelled directed graph Γ, it amounts to reversing the orientation of all the edges incident with that vertex.

The examples of smallest p-rank are constructed as follows.

Example 5.1. Let ℓ ≥ 7, let L = ⟨u, v⟩ ≅ ℤ/ℓ×ℤ/ℓ, an abelian group of order ℓ^{2}, and let H be the central extension

1 → Z → H → L → 1

with Z = ⟨z⟩ ≅ ℤ/ℓ and [ũ, ṽ] = z, where ũ and ṽ are lifts of u and v to H. Let χ: Z → k^{×} be the character defined by χ(z) = ε, and let b = \frac{1}{ℓ} ∑_{i=1}^{ℓ} χ(z^{−i})z^{i} be the corresponding central idempotent in kH.

Let N ≅ 𝔽_{p^{2}} × 𝔽_{p^{2}} × 𝔽_{p^{2}} be an elementary abelian p-group of rank six, and let L act on N in such a way that u acts by multiplication by (1, ε, ε^{2}) and v acts by multiplication by (ε, 1, ε^{3}). Inflate to an action of H on N, and let G = N ⋊ H, a group of order p^{6}.ℓ^{3}. The character ϕ of (3.1) is then given by

ϕ = ϕ_{u} ⊕ ϕ_{u^{−1}} ⊕ ϕ_{v^{−1}} ⊕ ϕ_{v} ⊕ ϕ_{u^{3}v^{−2}} ⊕ ϕ_{u^{−3}v^{2}}.

Here, the notation is taken so that a character ϕ_{x} corresponds to the element x ∈ L via Theorem 3.2 (i).

We regard b as a central idempotent in kG, and we examine the block kGb. By Theorem 3.2 (ii), its basic algebra is isomorphic to A_{q}[X_{1}, …, X_{6}], where

q = (\begin{matrix}1 & 1 & ε & ε^{−1} & ε^{2} & ε^{−2} \\ 1 & 1 & ε^{−1} & ε & ε^{−2} & ε^{2} \\ ε^{−1} & ε & 1 & 1 & ε^{−3} & ε^{3} \\ ε & ε^{−1} & 1 & 1 & ε^{3} & ε^{−3} \\ ε^{−2} & ε^{2} & ε^{3} & ε^{−3} & 1 & 1 \\ ε^{2} & ε^{−2} & ε^{−3} & ε^{3} & 1 & 1 \end{matrix}) .

Picking representatives for the orbits of σ, we obtain the reduced commutation matrix

ζ = (\begin{matrix}1 & ε & ε^{2} \\ ε^{−1} & 1 & ε^{−3} \\ ε^{−2} & ε^{3} & 1 \end{matrix}) .

So the corresponding graph Γ is

\xymatrix{· \ar[rr]^{ε} \ar[dr]_{ε^{2}} &  & · \\  & · \ar[ur]_{ε^{3}}}

The graph Γ^{(p)} is

\xymatrix{· &  & · \ar[ll]_{ε} \ar[dl]^{ε^{3}} \\  & · \ar[ul]^{ε^{2}}}

Since ℓ ≥ 7, the set {ε, ε^{2}, ε^{3}, ε^{−1}, ε^{−2}, ε^{−3}} contains no repetitions. It is then easy to see that it is impossible to get from Γ to Γ^{(p)} by a series of twists of vertices. For example, twisting the left hand vertex of Γ gives the graph

\xymatrix{· &  & · \ar[ll]_{ε} \\  & · \ar[ul]^{ε^{2}} \ar[ur]_{ε^{3}}}

and a sequence of twists will always reverse an even number of arrows. It follows, using Theorem 4.2, that kGb is not Morita equivalent to the block σ(kGb) ≅ kGb^{(p)} corresponding to the idempotent σ(b) = \frac{1}{ℓ} ∑_{i=1}^{ℓ} χ(z^{i})z^{i}.

For any p ≥ 7, we may take ℓ = p + 1 and obtain an example as above. For small primes, we need to go to larger rank because we do not have so many distinct ℓth roots of unity.

Example 5.2. Let p be any prime and let ℓ = p + 1. Let L = ⟨s, t, u, v⟩ ≅ (ℤ/ℓ)^{4}, an abelian group of order ℓ^{4}, and let H be the central extension

1 → Z → H → L → 1

with Z = ⟨z⟩ and [s̃, t̃] = [ũ, ṽ] = z, [s̃, ũ] = [s̃, ṽ] = [t̃, ũ] = [t̃, ṽ] = 1. Let χ: Z → k^{×} be the character defined by χ(z) = ε, and let b = \frac{1}{ℓ} ∑_{i=1}^{ℓ} χ(z^{−i})z^{i} be the corresponding central idempotent in kH.

Let L act on N ≅ (𝔽_{p^{2}})^{5}, an elementary abelian p-group of rank ten, in such a way that s, t, u, v act as multiplication by (1, ε, ε, ε, ε), (ε, 1, 1, 1, ε^{−3}), (1, 1, ε, ε, ε^{−2}) and (1, ε, 1, ε, ε^{2}) respectively. Inflate to an action of H on N, and let G = N ⋊ H, a group of order p^{10}.ℓ^{5}. The character ϕ of 3.1 is given by

ϕ = ϕ_{1} ⊕ ϕ_{1}^{−1}

where

ϕ_{1} = ϕ_{s} ⊕ ϕ_{t^{−1}u} ⊕ ϕ_{t^{−1}v^{−1}} ⊕ ϕ_{t^{−1}uv^{−1}} ⊕ ϕ_{s^{−3}t^{−1}u^{2}v^{2}}.

We regard b as a central idempotent in kG, and we examine the block kGb. By Theorem 3.2 (ii), its basic algebra is isomorphic to A_{q}[X_{1}, …, X_{10}], where

ζ = (\begin{matrix}1 & ε & ε & ε & ε \\ ε^{−1} & 1 & ε & ε & ε \\ ε^{−1} & ε^{−1} & 1 & ε^{−1} & ε \\ ε^{−1} & ε^{−1} & ε & 1 & ε^{−1} \\ ε^{−1} & ε^{−1} & ε^{−1} & ε & 1 \end{matrix})

and q is the 10 × 10 matrix obtained by replacing each entry α by a 2 × 2 block ( \begin{matrix}α & α^{−1} \\ α^{−1} & α \end{matrix} ). Since every label on Γ is ε, we omit the labels and just draw the directed graph

\xymatrix{ & 1 \ar[rr] \ar[drrr] \ar[ddr] \ar[dl] &  & 2 \ar[dr] \ar[ddl] \ar[dlll] \\ 5 \ar[drr] &  &  &  & 3 \ar[llll] \\  &  & 4 \ar[urr]}

The graph Γ^{(p)} is just the opposite graph of this. In order to ascertain whether it is possible to apply a series of twists of vertices of Γ^{(p)} to obtain a graph isomorphic to Γ, we make the following observation. Given a directed complete graph and a vertex, we may apply a series of twists to obtain a graph where that vertex is a source; in other words, all the edges incident with that vertex are oriented to point away from it. Moreover, the graph obtained this way is unique. So we just need to take each vertex in turn and apply twists to make that vertex a source, and check whether the resulting graph is isomorphic to Γ. This is easy to check, because Γ has the property that removing the source and all the edges incident with it leaves a directed graph which still has a source. This is not true of any of the five graphs obtained from Γ^{(p)} by applying twists to make a vertex into a source. Here are these five graphs.

[figure: ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·]

It now follows, using Theorem 4.2, that kGb is not Morita equivalent to the block σ(kGb) ≅ kGb^{(p)} corresponding to the idempotent σ(b) = \frac{1}{ℓ} ∑_{i=1}^{ℓ} χ(z^{i})z^{i}. This completes the proof of Theorem 1.5.

6. Some questions

Question 6.1. Given a p-group D, is there a bound for the Morita Frobenius number of blocks with D as defect group?

Question 6.2. Is there a bound for the Morita Frobenius number of blocks, independent of the defect group?

Question 6.3. Is every block of a finite group Morita equivalent to its second Frobenius twist? Is at least every block with abelian defect groups Morita equivalent to its second Frobenius twist?

Question 6.4. The twelve-fold covering group 12M_{22} of the Mathieu group M_{22} gives rise to an interesting example where we have not been able to ascertain the field of definition. Choose a prime p congruent to 5 modulo 12 but not dividing |M_{22}| (for example p = 17). We let H = 12M_{22} act on an elementary abelian group N of order p^{22} via the permutation representation of the quotient group M_{22}, and we let G be the split extension N ⋊ H. Then the p-blocks correspond to the irreducible characters of Z(H), so G has exactly four faithful p-blocks, corresponding to the faithful irreducible characters of the cyclic group of order twelve. These are obviously isomorphic in pairs, since there is an automorphism of M_{22} that inverts the centre and preserves the permutation module. But the Frobenius twist of a block corresponds to the fifth power of the character of Z(H) (by the choice of p), and it is not clear whether this is Morita equivalent to the original block. What is the field of definition of the faithful blocks of G, and what is the field of definition of the basic algebras of these blocks? Is there a general method for deciding problems like this?

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Department of Mathematics, University of Aberdeen, King’s College, Aberdeen AB24 3UE, Scotland

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