Quivers of blocks with normal defect group

Mary Schaps  Dana Shapira*  Orly Shlomo^{∗}

Bar-Ilan University Department of Mathematics and Computer Science Ramat-Gan, 52900

Abstract. For a finite group H̃ and a character χ of H̃, the McKay graph is a directed graph D(H̃, χ) whose vertices are the characters of H̃ and for which the arrows leaving θ depend on the constituents of χθ. Let G be a finite group and K a sufficiently large field with prime field F →^{∼} ℤ/pℤ. We show that if B is a block of KG with normal defect group P, then there is a p′ group H^{∗}, and a character χ^{∗} with ker(χ^{∗}) ⊂ Z(H^{∗}), such that the quiver of B is a connected component of D(H^{∗}, χ^{∗}).

Conversely, given a p′-group H̃ and an F character χ with ker(χ) ⊂ Z(H̃), then for any connected component of D(H̃, χ) we can find a block with a normal defect group having this quiver. If (m_{1}, …, m_{s}) are the degrees of the constituents of χ, and (i_{1}, …, i_{s}) are arbitrary natural numbers, we can take P = (C_{p^{i1}} )^{m_{1}} ×⋯×(C_{p^{is}})^{m_{s}}. In particular, P = (C_{p})^{m} will always give the desired quiver. For P abelian, we determine the relations on the quiver giving the group block.

§1. Introduction

The purpose of this paper is to describe the quiver of a block with normal defect group in terms of its character ring, or, conversely, to obtain information about the character ring from the quiver. As such it is intended to bridge a gap between modular group representation theory, which emphasizes the character table, and algebra representation theory where, there being no character table, the quiver is the fundamental object of interest.

Blocks b with normal defect group play a central role in group representation theory as the Brauer correspondents of group blocks B. When the defect group P is abelian, b and B are conjectured to be derived equivalent. When this happens, the

*Work done toward a Master’s degree at Bar-Ilan University. 1991 Mathematics Subject Classification. 20C05, 20C20, 16G20.

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group block B will be derived equivalent to a block b whose quiver and relations can be calculated from character tables. We hope this will facilitate the application of results from algebra representations to group representations.

We have chosen to write this paper from the point of view of group representation theory, and therefore take for granted the definitions of character table, blocks, defect groups, Brauer’s main theorems and the representation group. The needed definitions can be found in standard texts or, more briefly, in [S3],[MS], which were written to be accessible to experts in representations of algebras.

As a matter of notation, an ordinary group algebra of a finite group G over a field K will be denoted KG. If A is an algebra and φ : G → Aut_{K-alg}(A) is a homomorphism, then we write A[G] for the A-algebra with basis G on which g ·a = φ(g)(a) · g, and we then denote φ(g)(a) by gag^{−1}. If γ ∈ H^{2}(G, K^{∗}) is a cocycle, then we denote the twisted group algebra by K^{γ}[G], with g_{1} ∗ g_{2} = γ(g_{1}, g_{2})(g_{1}g_{2}). Finally, the twisted skew group algebra will be denoted by A^{γ}[G], with

a_{1}g_{1} ∗ a_{2}g_{2} = γ(g_{1}, g_{2})a_{1}(g_{1}a_{2}g_{1}^{−1})(g_{1}g_{2}).

In §2 we define the McKay graph D(H̃, χ) [M]. In §3 we prove:

Theorem 2. Let G be a finite group, and let K be a field of characteristic p which is a splitting field for G. Let B be a block of KG with normal defect group P. Then there is a p′-subquotient H of G, an embedding H ↪ Aut(P), and a faithful ℤ/pℤcharacter χ of H, such that if χ^{∗} is the lifting of χ to the representation group H^{∗} of H, the quiver of B is a connected component of the McKay graph D(H^{∗}, χ^{∗}).

In §4 we derive information showing which quivers can be associated to a given defect group. In §5 we consider different abelian defect groups corresponding to a fixed quiver. The relations, on the other hand, do depend on the defect group, and we give them in the case of abelian defect group in §6.

§2. The McKay graph

Let H̃ be a finite group, and let F be a prime field of characteristic zero or of positive characteristic not dividing |H̃|. Let K be an extension of F which is a splitting field for H̃. Let the character table of H̃ be considered as having entries in K.

Let χ be a character of H̃, with kernel N. If θ_{1}, …, θ_{t} is the complete set of irreducible characters of H̃, and χθ_{j} is defined by pointwise multiplication on conjugacy classes, define the natural number n_{ij} as the multiplicity of the constituent θ_{i} in the product character χθ_{j}. Since the θ_{i} form an orthonormal basis, this is given by n_{ij} = (θ_{i}, χθ_{j}), where (−, −) is defined in the usual way. The following is a slight adaptation of the “representation graph” defined in [M].

Definition. If H̃ is a finite group, K is a sufficiently large field of characteristic prime to |H̃|, and χ is a K character, the \underline{McKay graph} D(H̃, χ) is the directed graph with

(1) t vertices labelled by the irreducible characters θ_{1}, …, θ_{t} of H̃.

(2) For each pair θ_{i}, θ_{j} of characters, n_{ij} arrows from θ_{j} to θ_{i}, where

n_{ij} = (θ_{i}, χθ_{j})

Remark. In all our applications, the kernel N of χ will be central in H̃.

Example 1. D_{6} = ⟨a, b | a^{6} = b^{2} = e, bab^{−1} = a^{−1}⟩, the dihedral group of order 12. D_{6} has four linear characters θ_{1}, θ_{2}, θ_{4}, θ_{5}, and two irreducible characters of degree 2, which are θ_{3} and θ_{6}. The numbering is chosen so that θ_{1}, θ_{2}, and θ_{3} restrict to multiples of the trivial character on the center ⟨a^{3}⟩, and θ_{4}, θ_{5}, θ_{6} to multiples of the nontrivial character. D(D_{6}, θ_{3}) has two connected components, shown in Fig. 1.

Fig. 1

Example 2. Q_{4} = ⟨i, j | i^{4} = e, i^{2} = j^{2}, jij^{−1} = i^{−1}⟩, the quaternian group of order 8. Let θ_{1} be the trivial character, θ_{2}, θ_{3} and θ_{4} the other linear characters, and θ_{5} the character of degree 2. Set χ = θ_{2} + θ_{3}. Then D(Q_{4}, χ) has two components, as in Fig. 2.

Fig. 2

Lemma 1. If N is central in H̃, then the connected components of D(H̃, χ) are in one-to-one correspondence with the non-trivial characters of N. Proof. By the Burnside-Brauer theorem, [I, p. 49] the characters whose kernel contains N = (ker χ) form a connected component, since the powers of the faithful character induced by χ in H = H̃/ ker(χ) contain as constituents all irreducible characters of H. By Clifford theory, if θ ∈ Irr(H̃), then θ |_{N} is a multiple k of the sum of all conjugates of an irreducible character ψ of N [I, p. 79]. Since N is central, ψ^{h} = ψ for all h ∈ H̃, so θ |_{N}= kψ. Consider the set S_{ψ} of characters for which all restrictions to N are multiples of a fixed character ψ of N. It is obvious that for θ ∈ S_{ψ}, the constituents of χθ lie in S_{ψ}, from the fact that if χ |_{N}= k1_{N}, and if θ |_{N}= ℓψ, then χθ |_{N}= kθ |_{N}= kℓψ. Thus, the set of vertices S_{ψ} generate a subgraph D^{ψ}(H̃, χ) of D(H̃, χ). The set S_{ψ} is non-empty since the constituents of the induced character ψ^{G} all lie in S_{ψ}.

We wish to show that the subgraph D^{ψ}(H̃, χ) is connected. Suppose that θ_{1}, θ_{2} ∈ S_{ψ}. Then, using the tensor identity [I], we have

(θ_{1}, (1_{N})H̃ · θ_{2})H̃ = (θ_{1}, θ_{2})_{N}

which is non-zero. So there is some η in 1H̃ ; N such that (θ_{1}, ηθ_{2}) ≠ 0. By the Burnside-Brauer theorem, there is some n ≥ 1 such that η occurs in χ^{n} and then (θ_{1}, χ^{n}θ_{2}) ≠ 0.

§3. The quiver of a local block

Let A be a K-algebra, where K is a field sufficiently large that A/ Rad(A) →^{∼} ⨁_{i=1}^{t} M_{n_{i}}(K). Let f_{1}, …, f_{t} be a set of primitive idempotents, one from each matrix block, and set

n′_{ij} = dim f_{i}(J/J^{2})f_{j},

for J = Rad(A).

Definition. The quiver of A is the directed graph with t vertices, and n′_{ij} arrows from j to i.

The Brauer Main Theorems allow us to study the representations of a block with defect group P in terms of the blocks b of defect group P in N_{G}(P). Külshammer, generalizing Reynolds [Re], has shown that these “local” blocks have the structure M_{n}(K^{γ}[P ⋊ H]), where H is a p′-subgroup of Aut(P) and γ ∈ H^{2}(H, K^{∗}). Since the quiver depends only on the Morita equivalence class of the algebra, we consider only blocks of form K^{γ}[P ⋊ H]. Since γ depends only on H, this is a twisted skew group algebra KP^{γ}[H] over KP. We want to construct the quiver of this block b, knowing that its radical is given by Jb, where J = Rad(KP).

To eliminate the cocycle γ, we pass to the representation group H^{∗}, a central extension of H with central kernel Z isomorphic to the character group of H^{2}(H, K^{∗}) [CR, p. 361]. Thus, for any z ∈ Z and any γ ∈ H^{2}(H, K^{∗}) we have z(γ) ∈ K^{∗}.

Let H = \{h_{1}, …, h_{n}\}. Then H^{∗} is a group whose elements are z\overline{h}_{i}, with z ∈ Z \overline{h}_{i}z = z\overline{h}_{i}, and \overline{h}_{i}\overline{h}_{j} = z_{ij}\overline{h_{i}h_{j}}, where z_{ij}(γ) = γ(h_{i}, h_{j}). The natural projection of H^{∗} on H induces a homomorphism H^{∗} → Aut(P), so G^{∗} = P ⋊ H^{∗} is defined.

The embedding of H in Aut(P) induces a homomorphism of H into Aut_{K-alg}(KP). However, since P is embedded in the units of KP, and we will show below that an automorphism which is trivial on KP is trivial on P, thus we actually have an embedding of H in Aut_{K-alg}(KP).

As the theorem below will show, the exact structure of KP, and in particular, the fact that KP is a group algebra, is entirely irrelevant to the construction of the quiver. All that we need is a local algebra A, with radical J, on which H acts without kernel as a group of automorphisms.

If A is a local algebra, and H is embedded in Aut_{K-alg}(A), then H acts on J and on J^{2}, so we have an induced action on J/J^{2}. Since the characteristic of K does not divide |H| and K is sufficiently large, KH is separable. Thus, J/J^{2} as a KH module is completely reducible, and therefore determines a character χ of H. The character χ can be extended to a character χ^{∗} of H^{∗} which is trivial on the kernel Z. Since KH is separable, there is a splitting of J/J^{2} into J with image V. A K-basis for V is a set of generators for A over K. Since an automorphism of A is determined by the images of a set of generators, if h ∈ H acted trivially on V, it would act trivially on A. Since no h ∈ H induces a trivial automorphism of A, we conclude that χ is faithful.

Theorem 1. Let K be a sufficiently large field of characteristic not dividing the order of a finite group H, and let γ be an element of H^{2}(H, K^{∗}). Let A be a local K-algebra for which H is embedded in Aut_{K-alg}(A). Let χ be the character of the corresponding representation of H on J/J^{2}, and let χ^{∗} be the extension of χ to H^{∗}, with central kernel Z which is isomorphic to the character group of H^{2}(H, K^{∗}). Then the quiver of the skew group algebra A[H^{∗}] is D(H^{∗}, χ^{∗}) and the quiver of the twisted skew group algebra A^{γ}[H] is D^{γ}(H^{∗}, χ^{∗}), that portion of D(H^{∗}, χ^{∗}) whose vertices are the characters whose restriction to Z is a multiple of the double dual φ_{γ} of γ.

Proof. We want to construct the quiver of B = A[H^{∗}]. If K is a field of positive characteristic p, H^{2}(H, K^{∗}) is a p′-group, so H^{∗} is a p′-group, and thus KH^{∗} is separable by Maschke’s theorem. Then B/JB →^{∼} KH^{∗} is separable, and (JB)^{i} ⊂ J^{i}B, so JB is a nilpotent ideal with separable quotient B/JB, whence we conclude

that JB = Rad(B). Set B/JB = ⨁_{i=1}^{t} M_{n_{i}}(K).

Let e_{1}, …, e_{t} be the centrally primitive idempotents of KH^{∗}, and θ_{1}, …, θ_{t} the corresponding characters of H^{∗}. For each e_{i} we chose a primitive idempotent f_{i} ∈ K[H^{∗}] such that f_{i} · e_{i} = f_{i}. The f_{i}, which label the vertices of the quiver Q of A[H^{∗}], are thus in natural one-to-one correspondence with the vertices of D(H^{∗}, χ^{∗}).

We now wish to show that the vertices of the quiver of A^{γ}[H] are in natural one-to-one correspondence with those characters of H^{∗} whose restriction to Z is a multiple of the double dual φ_{γ} of γ, the character of Z defined by φ_{γ}(z) = z(γ). Since Z is the character group of the abelian group H^{2}(H, K^{∗}), the character group of Z is naturally isomorphic to H^{2}(H, K^{∗}). Letting γ_{1}, …, γ_{s} be the elements of H^{2}(H, K^{∗}), we denote by e_{γ} the central idempotent of KZ whose corresponding character is φ_{γ}. For any z ∈ Z, e_{γ} ·z = e_{γ}φ_{γ}(z) = e_{γ}z(γ). Since Z is contained in the center of H^{∗}, Clifford theory says that for every irreducible character θ_{i} of H^{∗} there is an irreducible character φ_{γ} of Z such that θ_{i} |_{Z}= θ_{i}(1) · φ_{γ}, so (θ_{i}, φ_{γ}^{G}) = θ_{i}(1). Since Z is central, φ_{γ}^{G}(z) = [G: Z]φ_{γ}(z) for z ∈ Z and φ_{γ}^{G}(g) = 0 for g ∉ Z. Thus,

e_{γ} = \frac{1}{z} ∑_{z∈Z} φ_{γ}(z^{−1}) z

= \frac{1}{G} ∑_{g∈G} φ_{γ}^{G}(g^{−1}) g

= \frac{1}{G} ∑_{g∈G} ∑_{θ_{i}∈S_{φγ}} θ_{i}(1)θ_{i}(g^{−1}) g

= ∑_{θ_{i}∈S_{φγ}} e_{i}.

Thus e_{i} · e_{γ} = e_{i}.

Let \overline{h}_{1}, …, \overline{h}_{n} be a set of coset representatives of cosets of Z in H^{∗} with \overline{h}_{i} mapping to h_{i} in H,and define an A-algebra homomorphism α : e_{γ}A[H^{∗}] → A^{γ}[H] by setting α(e_{γ}z\overline{h}_{i}) = α(e_{γ}z(γ)\overline{h}_{i}) = z(γ)h_{i} and extending linearly over A. The homomorphism α is onto because α(e_{γ} \overline{h}_{i}) = h_{i}, it is one-to-one because both the domain and the range are free A-modules of the same rank |H|, and it is a homomorphism because the definition of the representation group H^{∗} requires that if z = \overline{h}_{i}\overline{h}_{j}\overline{h_{i} h_{j}}^{−1}, then z(γ) = γ(h_{i}, h_{j}). Thus the quiver of A^{γ}[H] is the quiver of e_{γ}A[H^{∗}].

Since Z = ker(χ), we get χ |_{Z}= χ(1) 1_{Z}. As described above, θ_{j} |_{Z}= θ_{j}(1)φ_{γ} for some γ, and thus

χ · θ_{j} |_{Z}= χ(1)1_{Z} · θ_{j}(1)φ_{γ} = χ(1)θ_{j}(1) φ_{γ},

which is only possible if every constituent θ_{i} of χθ_{j} satisfies θ_{i} |_{Z}= θ_{i}(1)φ_{γ}. To simplify the notation, we denote D^{φ_{γ}} (H^{∗}, χ^{∗}) by D^{γ}(H^{∗}, χ^{∗}). Thus D(H^{∗}, χ^{∗}) is a disjoint union of D^{γ}(H^{∗}, χ^{∗}), and the fact that D^{γ}(H^{∗}, χ^{∗}) is the quiver of e_{γ}A[H^{∗}] will follow, as soon as we prove that D(H^{∗}, χ^{∗}) is the quiver of A[H^{∗}].

Let f_{i} be the primitive idempotent corresponding to the matrix block with character θ_{i}, and f_{j} the primitive idempotent for the block of θ_{j}. We need to show that

n_{ij} = n′_{ij} for all i, j = 1, …, t,

where n_{ij} = (θ_{i}, χθ_{j}) and n′_{ij} = dim f_{i} (Rad(B)/ Rad^{2}(B)) f_{j}.

The KH^{∗}-module homomorphism J → J/J^{2} splits, with image V = ⟨x_{1}, …, x_{m}⟩ ⊂ J, which is an m dimensional KH^{∗}-submodule of J. If δ : H^{∗} → Aut_{K-alg}(A), then by the definition of multiplication in A[H^{∗}], h · x_{ℓ} = δ(h)(x_{ℓ})h, so we write δ(h)(x_{ℓ}) = hx_{ℓ}h^{−1} ∈ V. Since JA = J, Rad(B) = JB = JA[H^{∗}] = J · KH^{∗}, and Rad(B)/ Rad^{2}(B) is isomorphic as left KH^{∗}-module to V · KH^{∗} ⊂ B, with action h · x_{ℓ}g = hx_{ℓ}h^{−1}hg. We have assumed that the primitive idempotents f_{i}, f_{j} are all in KH^{∗}. The vector space U_{j} = KH^{∗}f_{j} is a simple KH^{∗}-submodule of KH^{∗}, with basis {u_{1}, …, u_{θ_{j}(1)}} and character θ_{j}. Since J ∩ KH^{∗} = 0, the set {x_{k}u_{ℓ}} is linearly independent, forming a basis of V U_{j}.

For any x_{k} and u_{ℓ}, h · x_{k}u_{ℓ} = (hx_{k}h^{−1})hu_{ℓ}, so the linear transformation

V ⊗ U_{j} → V U_{j}

is an isomorphism of KH^{∗}-modules. Let V ⊗ U_{j} = ⨁_{i=1}^{t} W_{i} be a decomposition of V ⊗ U_{j} into components corresponding to characters. Then by the definition of n_{ij}

n_{ij} = \frac{1}{θ_{i}(1)} · dim W_{i}.

However, by the properties of block idempotents, W_{i} = e_{i} (V ⊗ U_{j}) . We conclude that

n_{ij} = \frac{1}{θ_{i}(1)} dim e_{i}(V ⊗ U_{j})

= \frac{1}{θ_{i}(1)} dim e_{i}(V U_{j})

= \frac{1}{θ_{i}(1)} dim e_{i}V KH^{∗}f_{j}

= \frac{1}{θ_{i}(1)} dim e_{i}(Rad(B)/ Rad^{2}(B))f_{j}

Since e_{i} is the sum of θ_{i}(1) primitive idempotents which can be permuted by an automorphism, we get

n_{ij} = dim f_{i} (Rad(B)/ Rad^{2}(B)) f_{j}
= n′_{ij}

Corollary. If Q is the quiver of a skew group ring A[H], J = Rad(A), then the constituents of the character χ of the action of H on J/J^{2} are the characters of end points of the arrows starting at the trivial character 1_{H}, with the multiplicity of a constituent θ_{i} corresponding to the number of arrows from 1_{H} to θ_{i}. Proof. χ = χ · 1_{H}. Example 3. Let K = ℤ/7ℤ, and let A = K[x, y]/(x, y)^{2}. Let G be the extra special group of order 27, G = ⟨a, b, c | a^{3} = b^{3} = c^{3} = e, aba^{−1} = bc, bc = cb, ac = ca⟩. Let θ_{2}(a) = ω, θ_{2}(b) = θ_{2}(c) = 1 and θ_{3}(b) = ω, θ_{3}(a) = θ_{3}(c) = 1. Set χ = θ_{2} + θ_{3}. Let H = C_{3} × C_{3} generated by h_{1} and h_{2}. It is known that H^{2}(H, K^{∗}) →^{∼} C_{3}, and one can show that the representation group H^{∗} of H is isomorphic to G when π: H^{∗} → G sends \overline{h}_{1} to a and \overline{h}_{2} to b.

Let ba be the coset representative of the coset of Z = ⟨c⟩ containing ab. Then by our previous definition we get z_{12} = a · b(ba)^{−1} = c and z_{21} = b · a(ba)^{−1} = 1. Thus a character η_{ℓ} of Z sending c to ω^{ℓ} corresponds to a cocycle

γ_{ℓ}(a, b) = z_{12}(γ_{ℓ}) = η_{ℓ}(c) = ω^{ℓ},
γ_{ℓ}(b, a) = z_{21}(γ_{ℓ}) = η_{ℓ}(1) = 1.

and

γ_{ℓ}(a, a) = γ_{ℓ}(b, b) = 1

Thus D(G, χ) has three components, corresponding to the three characters η_{0}, η_{1}, η_{2} of Z.

Fig. 3

Theorem 2. Let G be a finite group, and let K be a field of characteristic p which is a splitting field for G. Let B be a block of KG with normal defect group P. Then there is a p′-subquotient H of G, an embedding H ↪ Aut(P), and a faithful ℤ/pℤcharacter χ of H, such that if χ^{∗} is the lifting of χ to the representation group H^{∗} of H, the quiver of B is a connected component of the McKay graph D(H^{∗}, χ^{∗}).

Proof. By the Brauer main theorems, there is a block b_{o} of C_{G}(P) with defect group P, such that B = b_{0}^{G}, and such that

H = N_{G}(P, b_{0})/P C_{G}(P)

is a p′-group. Since H is the quotient of N_{G}(P, b_{0})/C_{G}(P) by the normal psubgroup P C_{G}(P)/C_{G}(P), the Zassenhaus lemma implies that N_{G}(P, b_{0})/C_{G}(P) has a subgroup isomorphic to H. The natural embedding of N_{G}(P, b_{0})/C_{G}(P) into Aut(P) thus induces an embedding β: H ↪ Aut(P). This induces a homomorphism β̂: H ↪ Aut_{K−algebra}(KP). As we mentioned before Theorem 1, β̂ is also an embedding, since K-algebra P is embedded in KP as a subgroup of the units. Let χ be the induced representation of H in J/J^{2}, where J = Rad(KP).

By the Külshammer theorem quoted above, there is a cocycle γ ∈ H^{2}(H, K^{∗}) such that

B →^{∼} M_{n}(KP^{γ}[H])

n = [G: N_{G}(P, b_{0}] · deg(b_{0}/ Rad b_{0}). Since the quiver is invariant under Morita equivalence, it suffices to find the quiver of KP^{γ}[H]. Setting A = KP, we see from Theorem 1 that this quiver is given by D^{γ}(H^{∗}, χ^{∗}), which is a connected component of D(H^{∗}, χ^{∗}).

The assertions that χ is faithful and that it is the character of a representation defined over ℤ/pℤ follow from Lemma 1 below.

Lemma 1. Given an embedding H ↪ Aut(P) for a p′-group H, the induced character χ on J/J^{2} is faithful and depends only on the Frattini quotient P/ϕ(P) of P. There is a canonical embedding H ↪ GL(m, ℤ/pℤ) whose character is χ.

Proof. For any element a ∈ P, the element x_{a} = a − e lies in the radical J of KP. Let \overline{x}_{a} be its image in J/J^{2}. We have

x_{ab} = (ab − e)
= (a − e)(b − e) + (a − e) + (b − e)
= x_{a}x_{b} + x_{a} + x_{b}
So  \overline{x}_{ab} = \overline{x}_{a} + \overline{x}_{b},

since x_{a}x_{b} ∈ J^{2}. Thus we have a homomorphism of abelian groups β : P → J/J^{2}, whose kernel is the Frattini subgroup ϕ(P) of P. A set of generators for P/ϕ(P), which is isomorphic to (ℤ/pℤ)^{m}, maps onto a K-vector space basis for J/J^{2}, by Jennings Theorem [HB, p. 258]. Thus we have a homomorphism α: Aut(P) → Aut(P/ϕ(P)) →^{∼} GL(m, ℤ/pℤ). This homomorphism determines the representation of H on J/J^{2}, and thus χ, the character of that representation, is the character of a representation. It remains to show that it is faithful.

As mentioned in the proof of the theorem, there is a submodule V of J which is isomorphic to J/J^{2} as an H module. Suppose an element h of H induces the identity on J/J^{2}. Then it induces the identity on V. However, the elements of V generate KP, so h induces the identity on KP. However, since P is embedded in KP, it induces the identity on P, so h = e in H. When P is elementary abelian, Aut(P) = GL(m, F).

§4. Quivers associated to a fixed defect group

As we showed in the last section, for giving the quiver the only important contribution of the p-group is the representation of H on J/J^{2}, induced from an embedding of H in Aut(P). One can consider the connection between the quiver and the defect group from two points of view: (1) given a defect group, what are the possible quivers of blocks with normal defect group P, and (2) given a quiver, what are the different primes p and p-groups P for which this is the quiver of a block with normal defect group P? Approach (1) is more natural for group theorists, and approach (2) for representation of algebra researchers, but we will address both.

Let P be a fixed p group, with m generators. Let K be a field of characteristic p with prime field F. Set J = Rad(KP), an ideal with m generators. What are the McKay graph components D^{ψ}(H̃, χ̃) which can occur as quivers of local blocks with defect group P? By Theorem 2, H = H̃/ ker χ̃ must be a p′ subgroup of Aut(P), and χ̃ must be lifted from a faithful F −representation χ induced on J/J^{2} by the action of H on KP. Furthermore, ker χ̃ is contained in the center of H̃, with linear character ψ. As in theorem 2, the character ψ induces a cocycle by the formula γ (\overline{h_{i}}, \overline{h_{j}}) = ψ (\overline{h_{i}} \overline{h_{j}} \overline{h_{i}h_{j}}^{−1}) . If γ is a trivial cocycle, then D^{ψ}(H̃, χ̃) = D(H, χ). The block is Morita equivalent to KP^{γ}[H]. Example 4. If P is cyclic, the only p′ subgroups in Aut(P) are C_{e} for e|p − 1, and the only ℤ/pℤ faithful characters are characters θ which send a generator of C_{e} to a primitive e-th root of unity. The complete set of characters is 1, θ, θ^{2}, … θ^{e−1}, and θ^{i+1} = θ · θ^{i}, giving an arrow from θ^{i} to θ^{i+1}. The quiver is a cycle on e vertices. A cyclic group has no non-trivial cocycles.

Example 5. If P = Q_{4}, the only p′ subgroup of GL(2, ℤ/2ℤ) →^{∼} S_{3} is C_{3}. Thus there are three irreducible characters θ_{1} = 1, θ_{2}, θ_{3}. The group P is generated by two elements. There are five possible faithful characters θ_{1} + θ_{2}, θ_{1} + θ_{3}, 2θ_{2}, 2θ_{3} and θ_{2} + θ_{3}. The first two pairs are carried into each other by the automorphism of C_{3} exchanging θ_{2} and θ_{3}, giving three non isomorphic quivers

Fig. 4

Example 6. If P = C_{2} ×C_{2} ×C_{2}, Aut(P) = GL(3, ℤ/2ℤ), whose p′ part has order (p^{3} − 1)(p^{2} − 1)(p − 1), i.e., 7 · 3 = 21. We have the following three possibilities:

\underline{H = C_{3}} : The only faithful characters of degree 3 defined over ℤ/2ℤ is the permutation character, i.e., the sum of all three linear characters, giving quiver 4a.

\underline{H = C_{7}} : Let θ be a non trivial character. The only degree 3 characters defined over ℤ/2ℤ are two reducible characters of degree 3 given by θ+θ^{2}+θ^{4} or θ^{3}+θ^{5}+θ^{6}, giving the graph 4b or its dual.

\underline{H = C_{7} ⋊ C_{3}} : H has only five characters, three of degree 1 and two of degree 3. Then the faithful characters defined over ℤ/2ℤ are the two irreducible degree 3 characters. Therefore we get the graph in Fig. 5 or its dual.

For examples with nontrivial cocycles, see Example 2 and Example 3.

Fig. 5 §5. The abelian defect groups associated to a fixed quiver

To go in the opposite direction, finding all the p-groups which are normal defect groups of a block with a given quiver, we will restrict our attention to abelian defect groups, since for nonabelian defect groups, this involves the difficult problem of reconstructing a group from information about its automorphisms.

Lemma 2. Let H be a group and χ a faithful representation. Let H^{∗} be the representation group, and χ^{∗} the lifted character. Suppose that p is a prime for which there is a representation of H in GL(m, ℤ/pℤ) with character χ and that p ∤ |H|, and suppose that χ_{1}⊕⋯⊕χ_{v} are the direct summands of χ over ℤ/pℤ, then for any v-tuple of natural numbers (j_{1}, …, j_{v}), the group P = (C_{p^{j_{1}}} )^{χ_{1}(1)} ×…×(C_{p^{jv}})^{χ_{v}(1)} can be the defect group of a block with a quiver which is a connected component of D(H^{∗}, χ^{∗}), and in particular the connected component of characters whose vertices are H^{∗}-characters whose restriction to ker χ^{∗} is trivial will be D(H, χ).

Proof. We have a representation χ of a p′-group H in GL(m, ℤ/pℤ). Passing to a ℤ/pℤ irreducible component, we get a representation χ_{i} of H in GL(m_{i}, ℤ/pℤ). Letting j be any natural number, the natural projection ℤ/p^{j}ℤ → ℤ/pℤ induces a homomorphism GL(m_{i}, ℤ/p^{j}ℤ) into GL(m_{i}, ℤ/pℤ) whose kernel is a normal p-subgroup of GL(m_{i}, ℤ/p^{j}ℤ). The inverse image of H in GL(m_{i}, ℤ/p^{j}ℤ) is a group with normal p-Sylow subgroup and complement isomorphic to H, so by the Zassenhaus lemma, there is a subgroup isomorphic to H in GL(m_{i}, ℤ/p^{j}ℤ). However, GL(m_{i}, ℤ/p^{j}ℤ) is just the automorphism group of (C_{p^{j}})^{m_{i}}. So we get an embedding of H in Aut (C_{p^{j}})^{m_{i}} inducing the character χ_{i} on J_{i}/J_{i}^{2}, with J_{i} = Rad (K (C_{p^{j}})^{m_{i}}) . Setting j = j_{i} and taking the product of all these embeddings gives an embedding of H in Aut(C_{p^{j_{1}}}^{m_{1}} × ⋯ × C_{p^{jv}}^{m_{v}}), with character χ. Example 7. Let H = A_{5}. Let ζ be a primitive 5th root of 1. The character table is

60  15  12  12  20
(1)  (12)(34)  (12345)  (12354)  (123)
1  1  1  1  1
4  0  − 1  − 1  + 1
5  1  0  0  − 1
3  − 1  − ζ^{3} − ζ^{2}  − ζ − ζ^{4}  0
3  − 1  − ζ − ζ^{4}  − ζ^{2} − ζ^{3}  0

\underline{If p ≡ 1 mod 5,} then all the non trivial irreducible representations are faithful and defined over ℤ/pℤ, and the possible quivers for (C_{p^{i}}^{χ(1)}) with irreducible χ are 6a, 6b, 6c and its dual.

6a  6b  6c

Fig. 6

\underline{If p ≢ 1 mod 5,} then the characters of degree three are not defined in ℤ/pℤ, so we must take their sum, getting the quiver in 7a.

More complicated quivers can be formed by combining these and copies of the trivial character, as in Example 5 above or 7b, where χ = θ_{1} + θ_{2}.

7a 7b Fig. 7

§6. Relations on a block with normal abelian defect group The relations on a block of the form KP^{γ}[H] were determined in [MS], p.513.

Given a quiver Q and a set of primitive idempotents, we have

KP^{γ}[H] →^{∼} K[Q]/(e_{ℓ} · x_{i}^{p^{α_{i}}} , e_{ℓ} · (x_{i}x_{j} − x_{j}x_{i}))_{ℓ=1}^{t}

for all idempotents ℓ = 1, …, t of the quiver. At any idempotent, there is a linear combination of arrows giving x, since V K[H] is generated by the elements in the bases of each component of its decomposition. The twist by γ may produce some strange effects, as in the quiver of (C_{3} × C_{3}) ⋊ Q_{4} in Example 2, where the block corresponding to the non-trivial character of the center of Q_{4} gives M_{2}(K[X, Y]/(X^{3}, Y^{3}, XY + Y X)) as shown in [MS]. The twist converted the symmetry relation XY − Y X into the form XY + Y X.

Example 8. Let P = C_{2} × C_{2} × C_{2} with H = C_{7} ⋊ C_{3} and let χ be one of the characters of degree 3, as in Example 6. The relations are easier to derive if we replace the McKay graph with different directed graph in which each character θ is represented by θ(1) points, representing the primitive idempotents, so that each regular arrow represents one element of V KH, while the arrows with solid heads represent matrix units (this is just the weight ≥ 1 part of the basis graph defined in [S1]). We let a, b, c be generators for P, with an automorphism g of order 7 given by (a, b, c, ac, abc, ab, bc) and ℓ of order 3 given by (b c abc)(ab ac bc). This gives an embedding into GL(3, ℤ/2ℤ). However, we prefer to work over K, the field of eight elements. Let ζ be a 7th root of unity, a solution to the irreducible polynomial h(x) = x^{3} + x + 1, whose canonical matrix is given by the matrix of g. The three eigenvectors of g for the eigenvalues ζ^{i} are

x_{i} = ∑_{j=0}^{6} ζ^{−ij}a^{g^{j}}, i = 1, 2, 4

If we are looking only at the action of g, we would get the quiver 4(b). However, when we add the action of ℓ, the stable idempotent e_{0} = \frac{1}{7} (∑_{j=0}^{6} g^{j}) splits into three, q_{0}, q_{1}, q_{2}, and the others form two matrix blocks. The arrows between matrix block idempotents are the original arrows, but the arrows to q_{0}, q_{1}, q_{2} are linear combinations. We will show what happens between q_{0}, q_{1}, q_{2} and e_{1}, e_{2}, e_{4}. Let α_{j}^{i} go from q_{i} to e_{j}, β_{k}^{j} from e_{j} to e_{k}, j ≠ k, j, k ≠ 0 and γ_{k}^{t} from e_{k} to q_{t}

Fig. 8

α_{1}^{0} = q_{0}e_{0}x_{1}e_{1} = \frac{1}{3}(1 + ℓ + ℓ^{2})x_{1}e_{1}  = \frac{1}{3}x_{1}e_{1} + \frac{1}{3}x_{2}(ℓe_{1}) + \frac{1}{3}x_{4}(ℓ^{2}e_{1})
α_{1}^{1} = q_{1}e_{0}x_{1}e_{1} = \frac{1}{3}(1 + w^{2}ℓ + wℓ^{2})x_{1}e_{1} = \frac{1}{3}x_{1}e_{1} + \frac{1}{3}w^{2}x_{2}(ℓe_{1}) + \frac{1}{3}wx_{4}(ℓ^{2}e_{1})
α_{1}^{2} = q_{2}e_{0}x_{1}e_{1} = \frac{1}{3}(1 + wℓ + w^{2}ℓ^{2})x_{1}e_{1} = \frac{1}{3}x_{1}e_{1} + \frac{1}{3}wx_{2}(ℓe_{1}) + \frac{1}{3}w^{2}x_{4}(ℓ^{2}e_{1})

To multiply two such arrows, one expresses the product as an idempotent times a linear combination of monomials in the x_{i} times an element of KH. This is done by sending all the ℓ to the left, conjugating the idempotents and x_{i} as necessary. All x_{i}^{2} = 0. One can check, rather laboriously, that α_{j}^{i}β_{k}^{j}γ_{k}^{t} (or any cyclic permutation of) is non zero if i = t, j = 1, 2, 4, and k = 3, 5, 6, in which case there is a non zero monomial x_{1}x_{2}x_{4}.

Conclusion. The original motivation for this work was an attempt to solve the Donald-Flanigan problem [DF], the conjecture that every group algebra has a deformation to a separable algebra, see [ES], [S2],[S3], [S4]. For abelian defect group, the conjecture has been proven for local blocks [GS1] [MS], and it has been shown in [MS] that if the Broué conjecture holds, it is true for every block of abelian defect

group. The Broué conjecture asserts that a block with abelian defect group can be

obtained from its corresponding local block as the endomorphism ring of a tilting

complex [B1], [B2], [R].

The current project was undertaken in the hope that a better knowledge of the

quiver of the local block would provide information about the possible tilting com-

plexes, and thus about the possible structures of the global blocks. However, since

the work shows that the quivers of the local blocks can be determined from char-

acter table information alone, we hope these methods will have wider application

inside group theory.

We wish to thank the referee for pointing out that the character ring diagram

D(H, χ) already occurred in the literature as McKay’s representation graph [M].

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