BLOCKS WITH NORMAL ABELIAN DEFECT AND ABELIAN p′ INERTIAL QUOTIENT

DAVID BENSON, RADHA KESSAR, AND MARKUS LINCKELMANN

Abstract. Let k be an algebraically closed field of characteristic p, and let O be either k or its ring of Witt vectors W (k). Let G a finite group and B a block of OG with normal abelian defect group and abelian p′ inertial quotient. We show that B is isomorphic to its second Frobenius twist. This is motivated by the fact that bounding Frobenius numbers is one of the key steps towards Donovan’s conjecture. For O = k, we give an explicit description of the basic algebra of B as a quiver with relations. It is a quantised version of the group algebra of the semidirect product P ⋊ L.

1. Introduction

The purpose of this paper is to bound the Frobenius numbers and to give a structure theorem for blocks of finite groups with normal abelian defect groups and abelian p′ inertial quotients. This extends the results of Benson and Green [2], Holloway and Kessar [7], Benson and Kessar [3].

We show that these blocks are isomorphic to their second Frobenius twist. By [8], bounding Frobenius numbers is a key step towards Donovan’s conjecture; see for instance [4], [5]. We obtain further a complete description of the basic algebra of such a block over a field by means of quiver with relations.

Our main theorems are as as follows. Let k be an algebraically closed field of characteristic p and let W (k) be the ring of Witt vectors over k. Let O ∈ {k, W (k)}. For q a power of p, the Frobenius automorphism λ ↦ λ^{q} of the field k lifts uniquely to an automorphism of the ring W (k), and we denote its inverse in both cases by μ ↦ μ^{\frac{1}{q}}.

Recall from [3] that for an O-algebra A, the Frobenius twist A^{(q)} is the O-algebra which equals A as a ring, and where scalar multiplication is twisted via the Frobenius map; that is, for λ ∈ O, and a ∈ A, the action on A^{(q)} is given by λ · a = λ^{\frac{1}{q}} a.

Theorem 1.1. Let P be a finite abelian p-group, L an abelian p′-subgroup of Aut(P) and α ∈ H^{2}(L, O^{×}). The twisted group algebra O_{α}(P ⋊ L) is isomorphic to its second Frobenius twist O_{α}(P ⋊ L)^{(p^{2})}.

Theorem 1.1 is proved in Section 2.

Theorem 1.2. Let P be a finite abelian p-group, L an abelian p′-subgroup of Aut(P) and α ∈ H^{2}(L, k^{×}) and 𝔄 the basic algebra of the twisted group algebra k_{α}(P ⋊L). Then k_{α}(P ⋊L) is a matrix algebra over 𝔄 and 𝔄 has an explicit presentation as a quantised version of the group algebra of the semidirect product P ⋊ L.

2010 Mathematics Subject Classification. 20C20, 20J06.

Key words and phrases. Finite groups, block theory, abelian defect, Frobenius twist.

The explicit generators and relations for 𝔄 are given in Theorem 4.14.

By a theorem of Külshammer [9], any block of a finite group algebra over O with a normal defect group is isomorphic to a matrix algebra of a twisted group algebra of the semi-direct product of the defect group of the block with the inertial quotient of the block. Combining [9] with the two results above yields the following.

Corollary 1.3. Let B be a block of a finite group algebra over O with a normal defect group P and abelian inertial quotient L. Then B is isomorphic to its second Frobenius twist and if O = k, then B is a matrix algebra over a quantised version of the group algebra of the semidirect product P ⋊ L.

Remark 1.4. It seems unclear whether the same bound holds for strong Frobenius numbers, introduced by Eaton and Livesey in [5]. One issue is that we do not have a sufficiently explicit description of the automorphism φ of P ⋊ L constructed in Lemma 2.3 below.

Acknowledgement. This material is based on work supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2018 semester. The first author thanks City, University of London for its hospitality during part of the preparation of this paper. The third author acknowledges support from EPSRC grant EP/M02525X/1.

2. Proof of Theorem 1.1.

For L a group, ϕ ∈ Aut(L) and α : L × L → O^{×}, denote by ^{ϕ}α : L × L → O^{×} the map defined by ^{ϕ}α(x, y) = α(ϕ^{−1}(x), ϕ^{−1}(y)) and by (ϕ, α) → ^{ϕ}α the induced action of Aut(L) on H^{2}(L, O^{×}). For q a power of p, denote by α^{(q)} the map L × L → O^{×} defined by α^{(q)}(x, y) = α(x, y)^{\frac{1}{q}} and by α^{(q)} the image of α under the induced isomorphism H^{2}(L, O^{×}) ≅ H^{2}(L, O^{×}).

Lemma 2.1. Let L be a finite abelian p′-group and let ϕ : L → L be the group automorphism defined by ϕ(x) = x^{p} for all x ∈ L. Then for all α ∈ H^{2}(L, O^{×}), we have ^{ϕ}α = α^{(p^{2})}.

Proof. It is well-known that since L is a finite p′-group, it follows that the canonical map O → k induces an isomorphism H^{2}(L, O^{×}) ≅ H^{2}(L, k^{×}). Thus we may assume that O = k.

Consider the universal coefficient sequence

0 → Ext^{1}(H_{1}(L, ℤ), k^{×}) → H^{2}(L, k^{×}) → Hom(H_{2}(L, ℤ), k^{×}) → 0.

Since k is algebraically closed, k^{×} is divisible, and therefore injective as an abelian group. So the first term in this sequence is zero. For the third term, we have H_{2}(L, ℤ) ≅ Λ^{2}(L), the exterior square in the category of abelian groups. Therefore we obtain an isomorphism

H^{2}(L, k^{×}) ≅ Hom(Λ^{2}(L), k^{×}).

which by naturality is Aut(L) equivariant and which commutes with the Frobenius morphism of k. More precisely, if α ∈ H^{2}(L, k^{×}) corresponds to τ ∈ Hom(Λ^{2}(L), k^{×}) under the above isomorphism, then for any ψ ∈ Aut(L), and any power q of p, ^{ψ}α corresponds to the homomorphism ^{ψ}τ defined by ^{ψ}τ(x ∧ y) = τ(ψ^{−1}(x) ∧ ψ^{−1}(y)) and α^{(q)} corresponds to the homomorphism τ^{(q)} defined by τ^{(q)}(x ∧ y) = τ(x ∧ y)^{\frac{1}{q}}. The result follows since for any τ ∈ Hom(Λ^{2}(L), k^{×}) we have τ(x^{q} ∧ y) = τ(x ∧ y)^{q} = τ(x ∧ y^{q}). □

The isomorphism H^{2}(L, k^{×}) ≅ Hom(Λ^{2}(L), k^{×}) in the above proof can be explicitly described as follows. If α ∈ Z^{2}(L, k^{×}), then the image of the class of α in Hom(Λ^{2}(L), k^{×}) is the group homomorphism x ∧ y ↦ α(x, y)α(y, x)^{−1}, where x, y ∈ L. One can either verify directly, using the 2-cocycle identity, that this assignment is a group homomorphism in each component, or one can observe that α(x, y)α(y, x)^{−1} is equal to the commutator of lifts of x, y in a central extension determined by α. More precisely, let

1 → k^{×} → L̃ → L → 1

be a central extension defined by α. For each x ∈ L, choose an element x̃ ∈ L̃ lifting x. An easy calculation shows that α(x, y)α(y, x)^{−1} = [x̃, ỹ]. This commutator does not depend on the choices of the lifts x̃ and since L̃ is a central extension of the abelian group L, this commutator is a group homomorphism in each component. In particular, [x̄^{p}, ȳ^{p}] = [x̄, ȳ]^{p^{2}}, which explains the statement of the above Lemma.

Lemma 2.2. Let P be a finite p-group and let Φ(P) be the Frattini subgroup of P. The kernel of the natural group homomorphism Aut(P) → Aut(P/Φ(P)) is a p-group. If P is homocyclic, then the map Aut(P) → Aut(P/Φ(P)) is surjective.

Proof. For the first assertion see [6, Chapter 5, Theorem 1.4]. Assume that P is homocyclic and let {x_{1}, x_{2}, ⋯, x_{r}} be a minimal generating set of P. Let ψ ∈ Aut(P/Φ(P)). For each i, 1 ≤ i ≤ r, pick an element u_{i} ∈ P such that ψ(x_{i}Φ(P)) = u_{i}Φ(P). Since P is homocyclic, there exists a homomorphism ψ̃ : P → P such that ψ̃(x_{i}) = u_{i}, 1 ≤ i ≤ r. Clearly, ψ̃ lifts ψ. Since ψ is an automorphism, Im(ψ̃)Φ(P) = P whence Im(ψ̃) = P and ψ̃ ∈ Aut(P). □

Lemma 2.3. Let P be a finite abelian p-group and let L be an abelian p′-group acting on P. There exists an automorphism ϕ of P ⋊ L such that ϕ(L) = L and ϕ(x) = x^{p} for all x ∈ L.

Proof. Denote by L′ the image of L in Aut(P). Proving the existence of ϕ is equivalent to showing that there exists τ ∈ Aut(P) of P such that τyτ^{−1} = y^{p} for all y ∈ L′. Suppose first that P = P_{1} × P_{2} for L′-invariant subgroups P_{1} and P_{2} of P. If for each i = 1, 2, there exists τ_{i} ∈ Aut(P_{i}) with τ_{i}(y↓_{P_{i}})τ_{i}^{−1} = (y↓_{P_{i}})^{p} for all y ∈ L′, then the map τ: P → P sending x_{1}x_{2} to τ_{1}(x_{1})τ_{2}(x_{2}) for x_{1} ∈ P_{1}, x_{2} ∈ P_{2} has the required properties. Hence we may assume that P is indecomposable for the action of L′ and consequently that P is homocyclic (see [6, Chapter 5, Theorem 2.2]).

We claim that it suffices to prove the result for the case that P is elementary abelian. Indeed, let U be the kernel of the map Aut(P) → Aut(P/Φ(P)). By Lemma 2.2, U is a p-group. Let L̄′ be the image of L′ in Aut(P/Φ(P)) and suppose that there exists τ̄ ∈ Aut(P/Φ(P)) such that τ̄ητ̄^{−1} = η^{p} for all η ∈ L̄. By Lemma 2.2, there exists τ ∈ Aut(P) lifting τ̄. Since L′U is the full inverse image of L̄′ in Aut(P), and L̄′ is τ̄-invariant, τL′Uτ^{−1} = L′U. Hence L′ and τL′τ^{−1} are both complements to the normal Sylow p-subgroup U of L′U. By the Schur–Zassenhaus theorem, there exists u ∈ U such that uL′u^{−1} = τL′τ^{−1}. Replacing τ by u^{−1}τ we may assume that τL′τ^{−1} = L′. Then for any y ∈ L′, τy′τ^{−1} and y′^{p} are elements of the p′-group L′ lifting the same element of L̄′. The claim follows by Lemma 2.2.

By the discussion above we may assume that P is elementary abelian and that P is an indecomposable, faithful 𝔽_{p}L′-module. Since L′ is an abelian p′-group, P is in fact an irreducible 𝔽_{p}L′-module and L′ is cyclic. Let L′ = ⟨y⟩ and let f(X) ∈ 𝔽_{p}[X] be the characteristic polynomial of y as an element of End_{𝔽_{p}}(P). Since f(y^{p}) = f(y)^{p} = 0, f(X) is also the characteristic polynomial of y^{p}. Thus, y and y^{p} are conjugate in GL(P) ≅ Aut(P). □

Proof of Theorem 1.1. Let ϕ be as in Lemma 2.3. Then ϕ induces a O-algebra isomorphism O_{α}(P ⋊ L) ≅ O_{^{ϕ}α}(P ⋊ L). The result follows by Lemma 2.1 since for any power q of p, O_{α}(P ⋊ L)^{(q)} ≅ O_{α^{(q)}}(P ⋊ L) as O-algebras. □

3. On characters of groups of class two

For a finite group H denote by Irr(H) the set of ordinary irreducible characters of H. If N is a normal subgroup of H and χ ∈ Irr(H), denote by Irr(H | χ) the subset of Irr(H) covering χ. Recall that if H/N is abelian, then the group of irreducible (i.e. linear) characters of H/N acts on Irr(H | χ) via multiplication and this action is transitive.

Proposition 3.1. Let H be a finite group which is nilpotent of class 2. Let χ be a faithful irreducible character of Z := [H, H]. Set m = \sqrt{|H : Z(H)|}.

(i) For any ϕ ∈ Irr(Z(H) | χ), ϕ↑^{H} = mτ_{ϕ} for some τ_{ϕ} ∈ Irr(H). In particular, m = τ_{ϕ}(1) is an integer.

(ii) The map ϕ → τ_{ϕ}, ϕ ∈ Irr(Z(H) | χ), is a bijection between Irr(Z(H) | χ) and

Irr(H | χ). (iii) The actions of Irr(H/Z) on Irr(H | χ) and of Irr(Z(H)/Z) on Irr(Z(H) | χ) are

compatible with the bijection in (ii). More precisely, let η ∈ Irr(H/Z), and let ϕ ∈ Irr(Z(H) | χ). Then τ_{η↓_{Z(H)}ϕ} = ητ_{ϕ}. Consequently, ητ_{ϕ} = τ_{ϕ} if and only if η restricts to the trivial character of Z(H).

Proof. Let τ be an irreducible character of H covering χ. We claim that τ(x) = 0 for all x ∈ H \Z(H). Indeed, since Z(H) is the intersection of all maximal abelian subgroups of H, it suffices to prove that if A is a maximal subgroup of H, then τ(x) = 0 if x ∉ A. So, let A be a maximal abelian subgroup of H. Then Z(H) ≤ A and since H is of class 2, A is normal in H. Let ψ be an irreducible constituent of the restriction of τ to A and suppose that g ∈ H is such that ^{g}ψ = ψ. Then for all a ∈ A, ψ(gag^{−1}) = ψ(a) and so ψ(gag^{−1}a^{−1}) = 1. Since the restriction of ψ to Z equals χ, we have that χ(gag^{−1}a^{−1}) = 1 for all a ∈ A. The faithfulness of χ and the maximality of A now imply that g ∈ C_{G}(A) = A. Consequently, τ = ψ↑^{H} and τ(x) = 0 for all x ∉ A, proving the claim.

Let ϕ be the unique linear character of Z(H) covering χ and which is covered by τ. Since ϕ is linear, the restriction of τ to Z(H) consists of τ(1) copies of ϕ. By the claim above,

1 = ⟨τ, τ⟩ = \frac{1}{|H|} ∑_{x∈Z(H)} τ(x)τ(x^{−1}) = \frac{τ(1)^{2}}{|H|} ∑_{x∈Z(H)} ϕ(x)ϕ(x^{−1}) = \frac{τ(1)^{2}}{m^{2}} .

Thus τ(1) = m and

τ(1)m = |H : Z(H)| = ϕ↑^{H}(1).

On the other hand, by Frobenius reciprocity the multiplicity of τ as a constituent of ϕ↑^{H} equals τ(1). So ϕ↑^{H} = mτ. Setting τ_{ϕ} = τ proves part (i) of the proposition. Part (ii) is immediate from (i) and the fact that every element of Irr(H | χ) covers a unique element of Irr(Z(H) | χ). By the induction formula, η(ϕ↑^{H}) = (η↓_{Z(H)}ϕ)↑^{H}, hence (i) gives that τ_{η↓_{Z(H)}ϕ} = ητ_{ϕ}. Now (ii) yields that ητ_{ϕ} = τ_{ϕ} if and only if η↓_{Z(H)}ϕ = ϕ if and only if η↓_{Z(H)} is trivial. □

4. The basic algebra.

Lemma 4.1. Let 1 → A → B → C → 1 be a short exact sequence of abelian groups and let π : D → A be a surjective homomorphism of abelian groups. For each α ∈ C, choose a pre-image u_{α} in B. Then there exists a 2-cocycle (α, β) → f_{α,β} from C × C to D such that π(f_{α,β}) = u_{α}^{−1}u_{β}^{−1}u_{αβ} and f_{α,β} = f_{β,α} for all α, β ∈ C.

Proof. It is well-known that Ext_{ℤ}^{n}(C, A) = {0} for n ≥ 2, and hence the connecting homomorphism Ext_{ℤ}^{1}(C, A) → Ext_{ℤ}^{2}(C, ker(π)) = {0} is zero. Thus the map Ext_{ℤ}^{1}(C, D) → Ext_{ℤ}^{1}(C, A) induced by π is surjective. In particular, the element in Ext_{ℤ}^{1}(C, A) represented by the given short exact sequence lifts to an element in Ext_{ℤ}^{1}(C, D). Rephrased in terms of extensions this means that there is a commutative diagram of abelian groups with exact rows of the form

[figure: 1; D; π; B̂; τ; C; 1; 1; A; B; C; 1]

Note that τ is surjective and restricts to the map π on D. For α ∈ C, choose a preimage v_{α} of u_{α} in B̂ and set f_{α,β} = v_{α}^{−1}v_{β}^{−1}v_{αβ} for all α, β ∈ C. Since B̂ is abelian, it follows that f_{α,β} = f_{β,α}. Thus (α, β) ↦ f_{α,β} is a 2-cocycle with the properties as stated. □

We recall the following result on the structure of twisted group algebras.

Lemma 4.2. Let G be a finite group and let α ∈ H^{2}(G, k^{×}). Then there exists a central extension

1 → Z → G̃ → G → 1

with Z a finite cyclic p′-group and a linear character χ: Z → k^{×} such that k_{α}G ≅ kG̃e where e = \frac{1}{|Z|} ∑_{z∈Z} χ(z^{−1})z is the idempotent of kZ corresponding to χ. Moreover, Z may be chosen to be contained in [G̃, G̃].

Proof. This is well known, but for completeness we provide a proof. Let m be the order of the cohomology class α. Since k is algebraically closed, k^{×} is a divisible group. So we have a short exact sequence

0 → μ_{m} → k^{×} → k^{×} → 0.

Consider the corresponding maps of universal coefficient sequences

\xymatrix{0 \ar[r] & Ext^{1}(H_{1}(G, ℤ), μ_{m}) \ar[r] \ar[d] & H^{2}(G, μ_{m}) \ar[r] \ar[d] & Hom(H_{2}(G, ℤ), μ_{m}) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Ext^{1}(H_{1}(G, ℤ), k^{×}) \ar[r] \ar[d] & H^{2}(G, k^{×}) \ar[r]^{≅} \ar[d]^{m} & Hom(H_{2}(G, ℤ), k^{×}) \ar[r] \ar[d]^{m} & 0 \\ 0 \ar[r] & Ext^{1}(H_{1}(G, ℤ), k^{×}) \ar[r] & H^{2}(G, k^{×}) \ar[r]^{≅} & Hom(H_{2}(G, ℤ), k^{×}) \ar[r] & 0}

We have Ext^{1}(H_{1}(G, ℤ), k^{×}) = 0 since k^{×} is divisible, so H^{2}(G, k^{×}) → Hom(H_{2}(G, ℤ), k^{×}) is an isomorphism. Since α has order m, its image in Hom(H_{2}(G, ℤ), k^{×}) lifts to a surjective element of order m in Hom(H_{2}(G, ℤ), μ_{m}). An inverse image α̃ ∈ H^{2}(G, μ_{m}) again has order m. Let

1 → Z → G̃ → G → 1

be a central extension corresponding to α̃, with Z ≅ μ_{m}.

Now choose a presentation of G by generators and relations

1 → R → F → G → 1.

By freeness, the identity map on G lifts to a map F → G̃. This map sends R into Z, and [F, F] into [G̃, G̃]. It has [F, R] in its kernel since μ_{m} is central. This gives us a map

H_{2}(G, ℤ) = (R ∩ [F, F])/[F, R] → Z

which is the image of α in Hom(H_{2}(G, ℤ), μ_{m}). This map is surjective, but it lands in Z ∩ [G̃, G̃] and hence Z ⊆ [G̃, G̃].

The formula for the idempotent e can be found in the statement and proof of Thévenaz [10, Chapter 2, Proposition 10.5]. The linear character χ sends each element z ∈ Z to its image under Z ≅ μ_{m} ↪ k^{×}. □

Let P be an abelian p-group, L an abelian p′-subgroup of Aut(P) and α ∈ H^{2}(L, k^{×}). Let Z, G̃ and χ be as in the conclusion of Lemma 4.2 applied to G = P ⋊ L and with α regarded as an element of H^{2}(G, k^{×}) via the pull back along G → G/P. Let H be the full inverse image of L in G̃. Then G̃ = P ⋊ H and e is a central idempotent of H.

We have a natural homomorphism

ρ: H → Hom(H, k^{×})

sending g to ρ(g): h → χ[g, h]. The kernel of this map is Z(H) and the image is Hom(H/Z(H), k^{×}). We denote by

ρ̄: H/Z(H) → Hom(H/Z, k^{×})

the induced isomorphism.

Now P/Φ(P) is naturally a faithful 𝔽_{p}L-module. The extension of scalars k ⊗_{𝔽_{p}} P/Φ(P) gives a kL-module isomorphic to J(kP)/J^{2}(kP). Let ψ be the character of kL on this module, and write

ψ = ⨁_{i=1}^{r} ψ_{i}

where r is the rank of P/Φ(P) and the ψ_{i} are one dimensional kL-modules (there may be repetitions). We choose an H-invariant complement W for J^{2}(kP) in J(kP), and a basis w_{i} of W so that for g ∈ H we have

(4.3)  gw_{i}g^{−1} = ψ_{i}(g)w_{i}.

Since a p′-group of automorphisms of an abelian p-group preserves some decomposition into homocyclic summands (see for example Chapter 5, Theorem 2.2 in Gorenstein [6]), we may assume that

kP = k[w_{1}, …, w_{r}]/(w_{1}^{p^{n_{1}}}, …, w_{r}^{p^{nr}})

with n_{1} ⩾ ⋯ ⩾ n_{r} and |P | = p^{n_{1}+···+n_{r}}. Thus we have relations

(4.4)  w_{i}^{p^{n_{i}}} = 0.

Applying the results of Section 3, the irreducible characters τ_{ϕ} of H lying over χ are in one to one correspondence with the one dimensional characters ϕ of Z(H) lying over χ. The corresponding central idempotents are

(4.5)  e_{ϕ} = \frac{1}{|Z(H)|} ∑_{h∈Z(H)} ϕ(h^{−1})h.

Choose one of these, say τ = τ_{ϕ_{0}}, and choose a matrix representation T_{ϕ_{0}} : H → Mat_{m}(k) affording τ_{ϕ_{0}}. Then for each ϕ choose a one dimensional representation ξ_{ϕ} of H whose restriction to Z(H) is ϕϕ_{0}^{−1} (and hence whose restriction to Z is trivial) and chosen so that ξ_{ϕ_{0}} = 1. We assume that these ξ_{ϕ} have been chosen, one for each ϕ, and we define T_{ϕ}: H → Mat_{m}(k) via T_{ϕ}(h) = ξ_{ϕ}(h).T_{ϕ_{0}}(h). Then T_{ϕ} is a matrix representation affording τ_{ϕ}. So the map

kHe → Mat_{m}(k) × ⋯ × Mat_{m}(k)  (|Z(H) : Z| copies)
he ↦ (T_{ϕ_{0}}(h), …, T_{ϕ}(h), …)

is an isomorphism. Elements of kHe of the form ∑_{ϕ} ξ_{ϕ}^{−1}(h)e_{ϕ}.h are sent to diagonal elements (T_{ϕ_{0}}(h), …, T_{ϕ_{0}}(h)), and therefore span a copy of Mat_{m}(k) in kHe containing e as its identity element. Let us write 𝔐 for this subalgebra of e.kH.

Now for each ψ_{i} and each ϕ, the character ϕ.(ψ_{i}|_{Z(H)}) is some ϕ′, which we denote ϕψ_{i} for convenience. So ξ_{ϕψ_{i}}ξ_{ϕ}^{−1}ψ_{i}^{−1} is trivial on Z(H). Thus there exists an element g_{i,ϕ} ∈ H such that

(4.6)  ρ(g_{i,ϕ}) = ξ_{ϕψ_{i}}ξ_{ϕ}^{−1}ψ_{i}^{−1},

where ρ(g_{i,ϕ})(h) = χ([g_{i,ϕ}, h]). We choose such elements g_{i,ϕ}, one for each ψ_{i} and ϕ. On the other hand, using (4.3) and (4.5), we have

∑_{h∈Z(H)} ϕ(h^{−1})w_{i}h = ∑_{h∈Z(H)} ϕ(h^{−1})ψ_{i}(h^{−1})hw_{i}

and so

(4.7)  w_{i}e_{ϕ} = e_{ϕψ_{i}}w_{i}.

Lemma 4.8. For h ∈ H we have

(g_{i,ϕ}w_{i})(ξ_{ϕ}(h)^{−1}e_{ϕ}.h) = (ξ_{ϕψ_{i}}(h)^{−1}e_{ϕψ_{i}}.h)(g_{i,ϕ}w_{i}).

Thus g_{i,ϕ}w_{i}e_{ϕ} = e_{ϕψ_{i}}g_{i,ϕ}w_{i} commutes with 𝔐.

Proof. Scalars commute with everything, and the e_{ϕ}, being in Z(kH), commute with all but the w_{i}. By (4.3) we have hw_{i} = ψ_{i}(h)w_{i}x, and by (4.7) we have w_{i}e_{ϕ} = e_{ϕψ_{i}}w_{i}. We are in kGe, and eg_{i,ϕ}h = eχ([g_{i,ϕ}, h])^{−1}hg_{i,ϕ}. Putting these together gives

ξ_{ϕ}(x)^{−1}(g_{i,ϕ}w_{i})(e_{ϕ}.h) = ξ_{ϕ}(h)^{−1}ψ_{i}(h)^{−1}χ([g_{i,ϕ}, h])^{−1}(e_{ϕψ_{i}}h)(g_{i,ϕ}w_{i}).

Finally, applying (4.6), the scalar on the right hand side is equal to ξ_{ϕψ_{i}}(h)^{−1}. For the final statement, we have

(g_{i,ϕ}w_{i}e_{ϕ}) (∑_{ϕ′} ξ_{ϕ′}^{−1}(h)e_{ϕ′}.h) = (g_{i,ϕ}w_{i})(ξ_{ϕ}^{−1}(h)e_{ϕ}.h)

= (ξ_{ϕψ_{i}}^{−1}(h)e_{ϕψ_{i}}.h)(g_{i,ϕ}w_{i})
= (∑_{ϕ′} ξ_{ϕ′}^{−1}(h)e_{ϕ′}.h) (e_{ϕψ_{i}}g_{i,ϕ}w_{i}).  □

Definition 4.9. Let 𝔄 be the subalgebra of kHe generated by the elements e_{ϕ} and g_{i,ϕ}w_{i}e_{ϕ}. Thus by the lemma, 𝔄 and 𝔐 commute.

We claim that 𝔄 is a basic algebra of dimension |P | · |H : Z(H)|, and that multiplication in kHe induces an isomorphism

𝔄 ⊗_{k} 𝔐 → kHe,

so that kHe ≅ Mat_{m}(𝔄). For this purpose, we shall use the following.

Lemma 4.10. Let A ⩽ B be k-algebras with A an Azumaya algebra (i.e., a finite dimensional central separable k-algebra). Then the map A ⊗_{k} C_{B}(A) → B is an isomorphism.

Proof. See for example Chapter 3, Corollary 4.3, in Bass [1]. □

Remark 4.11. Note that the hypotheses of the lemma include the statement that the identity element of A is equal to the identity element of B.

We display 𝔄 as kQ/I where Q is a quiver and I ⩽ J^{2}(kQ) is an ideal of relations. The quiver Q has |Z(H) : Z| vertices labelled [ϕ] corresponding to the idempotents e_{ϕ} ∈ kZ(H) lying over χ, and directed edges labelled with the w_{i} corresponding to

g_{i,ϕ}w_{i}e_{ϕ} = e_{ϕψ_{i}}g_{i,ϕ}w_{i} = e_{ϕψ_{i}}g_{i,ϕ}w_{i}e_{ϕ}.

going from [ϕ] to [ϕψ_{i}]. For brevity, we can illustrate these vertices and directed edges as

[ϕ] ⟶^{i} [ϕψ_{i}]

Lemma 4.12. (i) For suitable elements z_{i,j,ϕ} ∈ Z(H), we have

g_{j,ϕψ_{i}} g_{i,ϕ} = g_{i,ϕψ_{j}} g_{j,ϕ} z_{i,j,ϕ}.

(ii) The following relations hold in 𝔄:

(g_{j,ϕψ_{i}}w_{j}e_{ϕψ_{i}})(g_{i,ϕ}w_{i}e_{ϕ}) = q_{i,j,ϕ}(g_{i,ϕψ_{j}}w_{i}e_{ϕψ_{j}})(g_{j,ϕ}w_{i}e_{ϕ})

where q_{i,j,ϕ} = ϕ(z_{i,j,ϕ}) ∈ k^{×}. (iii) By changing the choices of g_{i,ϕ} by elements of Z(H), we may ensure that z_{i,j,ϕ} ∈ Z

and q_{i,j,ϕ} = χ(z_{i,j,ϕ}).

Proof. (i) By (4.6), we have

ρ(g_{j,ϕψ_{i}}g_{i,ϕ}) = ρ(g_{j,ϕϕ_{i}})ρ(g_{i,ϕ})
= (ξ_{ϕψ_{i}ψ_{j}}ξ_{ϕψ_{i}}^{−1}ψ_{j}^{−1})(ξ_{ϕψ_{i}}ξ_{ϕ}^{−1}ψ_{i}^{−1})
= ξ_{ϕψ_{i}ψ_{j}}ξ_{ϕ}^{−1}ψ_{j}^{−1}ψ_{i}^{−1}.

This is symmetric in i and j, and so

ρ(g_{j,ϕψ_{i}}g_{i,ϕ}) = ρ(g_{i,ϕψ_{j}}g_{j,ϕ}).

Since the kernel of ρ is Z(H) it follows that for some element z_{i,j,ϕ} ∈ Z(H) we have

g_{j,ϕψ_{i}} g_{i,ϕ} = g_{i,ϕψ_{j}} g_{j,ϕ} z_{i,j,ϕ}.

(ii) This follows from the fact that we have z_{i,j,ϕ} e_{ϕ} = ϕ(z_{i,j,ϕ})e_{ϕ}.

(iii) We apply Lemma 4.1 with A = Irr(H/Z(H)), B = Irr(H/Z), C = Irr(Z(H)/Z), the map from B to C the restriction map, D = H/Z, π the composition of the natural surjection H/Z → H/Z(H) with ρ̄ and u_{α} = ξ_{αϕ_{0}}, α ∈ Irr(Z(H)/Z). Let f_{α,β} ∈ H/Z be as in the conclusion of the lemma, and let f̃_{α,β} ∈ H be any lift of f_{α,β} to H. Denote also by ψ_{i} the restriction of ψ_{i} to Z(H). So ψ_{i}^{−1}u_{ψ_{i}} is an element of Irr(H/Z(H)). Choose an element g_{i} ∈ H such that ρ(g_{i}) = ψ_{i}^{−1}u_{ψ_{i}} and set

g_{i,ϕ} = g_{i}f̃_{ψ_{i},ϕϕ_{0}^{−1}}.

Then

ρ(g_{i,ϕ}) = ψ_{i}^{−1}u_{ψ_{i}}ρ(f̃_{ψ_{i},ϕϕ_{0}^{−1}})
= ψ_{i}^{−1}u_{ψ_{i}}u_{ψ_{i}}^{−1}u_{ϕϕ_{0}^{−1}}^{−1}u_{ψ_{i}ϕϕ_{0}^{−1}}
= ψ^{−1}ξ_{ϕ}^{−1}ξ_{ψ_{i}ϕ}

and

g_{j,ϕψ_{i}} g_{i,ϕ}Z = g_{j}f̃_{ψ_{j},ϕψ_{i}ϕ_{0}^{−1}} g_{i}f̃_{ψ_{i},ϕϕ_{0}^{−1}}Z
= g_{j}g_{i}f̃_{ψ_{j},ϕψ_{i}ϕ_{0}^{−1}} f̃_{ψ_{i},ϕϕ_{0}^{−1}}Z
= g_{j}g_{i}f̃_{ψ_{j},ϕψ_{i}ϕ_{0}^{−1}} f̃_{ϕϕ_{0}^{−1},ψ_{i}}Z
= g_{j}g_{i}f̃_{ψ_{j},ϕϕ_{0}^{−1}} f̃_{ψ_{j}ϕϕ_{0}^{−1},ψ_{i}}
= g_{j}g_{j}if̃_{ψ_{j},ϕϕ_{0}^{−1}} f̃_{ψ_{i},ψ_{j}ϕϕ_{0}^{−1}}Z
= g_{i,ϕψ_{j}} g_{j,ϕ}Z.  □

Lemma 4.13. For each i and each ϕ we have

(g_{i,ϕψ_{i}^{p^{n_{i}} −1}}w_{i}e_{ϕψ_{i}^{p^{n_{i}} −1}}) … (g_{i,ϕψ_{i}^{2}} w_{i}e_{ϕψ_{i}^{2}})(g_{i,ϕψ_{i}} w_{i}e_{ϕψ_{i}})(g_{i,ϕ} w_{i}e_{ϕ}) = 0.

Here, we have written down the only composable sequence of arrows in Q beginning with e_{ϕ} with each arrow involving w_{i}, and there are p^{n_{i}} terms in the product:

[ϕ] ⟶^{i} [ϕψ_{i}] ⟶^{i} [ϕψ_{i}^{2}] ⟶^{i} ⋯ ⟶^{i} [ϕψ_{i}^{p^{n_{i}}}].

Proof. Relations (4.3) and (4.7) allow us to push the w_{i} terms past the other terms so that they are directly multiplied together. Then we can then use relation (4.4) to conclude that we get zero. □

Theorem 4.14. The relations on the quiver algebra kQ given by

(g_{j,ϕψ_{i}}w_{j}e_{ϕψ_{i}})(g_{i,ϕ}w_{i}e_{ϕ}) = q_{i,j,ϕ}(g_{i,ϕψ_{j}}w_{i}e_{ϕψ_{j}})(g_{j,ϕ}w_{i}e_{ϕ})

and

(g_{i,ϕψ_{i}^{p^{n_{i}} −1}}w_{i}e_{ϕψ_{i}^{p^{n_{i}} −1}}) … (g_{i,ϕψ_{i}^{2}} w_{i}e_{ϕψ_{i}^{2}})(g_{i,ϕψ_{i}} w_{i}e_{ϕψ_{i}})(g_{i,ϕ} w_{i}e_{ϕ}) = 0

as in Lemmas 4.12 and 4.13 are a complete set of relations among the arrows g_{i,ϕ}w_{i}e_{ϕ} of Q to define the quotient 𝔄. Thus 𝔄 ≅ kQ/I where I ⩽ J^{2}(kQ) is the two-sided ideal generated by these relations.

Proof. Using Lemmas 4.12 and 4.13, we have an obvious homomorphism from the algebra kQ/I given by these generators and relations to 𝔄. Now kQ/I is a finite dimensional algebra whose socle elements are products which involve p^{n_{i}} − 1 arrows of type i for each i. Such an element maps to something of the form (element of H)(w_{1}^{p^{n_{1}}−1} … w_{r}^{p^{nr} −1}e_{ϕ}) in 𝔄, and such an element is non-zero in kG. □

Theorem 4.15. The multiplication in kGe induces an isomorphism 𝔄 ⊗_{k} 𝔐 → kGe.

Proof. Applying Lemma 4.10 with A = 𝔐 and B the subalgebra generated by 𝔄 and 𝔐, we see that the given map is injective. The dimensions are given by dim(𝔄) = |Z(H) : Z| · |P |, dim(𝔐) = |H : Z(H)| and dim(kGe) = |G : Z|, so dim(kGe) = dim(𝔄) · dim(𝔐) and the map is an isomorphism. □

Corollary 4.16. The algebra kGe is isomorphic to Mat_{m}(𝔄). In particular, 𝔄 is the basic algebra of kGe, and is Morita equivalent to it. □

References

1. H. Bass, Lectures on topics in algebraic K-theory, Tata Institute, Bombay, 1967.

2. D. J. Benson and E. Green, Nonprincipal blocks with one simple module, Quarterly J. Math (Oxford) 55 (2004), 1–11.

3. D. J. Benson and R. Kessar, Blocks inequivalent to their Frobenius twists, J. Algebra 315 (2007), 588–599.

4. C. W. Eaton and M. Livesey, Donovan’s conjecture and blocks with abelian defect groups, arXiv:1803.03539v1, 2018.

5. , Towards Donovan’s conjecture for abelian defect groups, arXiv:1711.05357v2, 2018.

6. D. Gorenstein, Finite groups, Harper & Row, 1968.

7. M. Holloway and R. Kessar, Quantum complete rings and blocks with one simple module, Quarterly J. Math (Oxford) 56 (2005), 209–221.

8. R. Kessar, A remark on Donovan’s conjecture, Arch. Math. (Basel) 82 (2004), 391–394.

9. B. Külshammer, Crossed products and blocks with normal defect groups, Commun. in Algebra 13 (1985), no. 1, 147–168.

10. J. Thévenaz, G-algebras and modular representation theory, Oxford University Press, 1995.

David Benson, Institute of Mathematics, Fraser Noble Building, University of Aberdeen, King’s College, Aberdeen AB24 3UE, United Kingdom

Radha Kessar and Markus Linckelmann, School of Mathematics, Computer Science & Engineering, Department of Mathematics, City, University of London, Northampton Square, London EC1V 0HB, United Kingdom