LIFTING MCKAY GRAPHS AND RELATIONS TO PRIME EXTENSIONS

Mohammed Hasan Ali and Mary Schaps

Abstract. We give an algorithmic procedure for constructing the quivers and homogeneous relations of the Brauer correspondents of blocks of abelian noncyclic defect group in the almost simple groups in the ATLAS. This is one step in a program to compute the structure of the indecomposable projectives for these blocks. As an illustration, we determine an explicit tilting complex for the non-principal 3-block of the central extension of PSL(3,4) by C_{2}.

§0. Introduction.

One of the important outstanding problems in the theory of modular group representations is the determination of the structure of the projective indecomposables of blocks B in simple or almost simple groups. Recent work on the Broué conjecture (the conjecture that blocks of abelian defect group are derived equivalent to the Brauer correspondent b, cf. [Br], [O],[H]) leads in many cases to explicit tilting complexes P^{∗}, cf. [BB]. In these cases, the desired block is Morita equivalent to the endomorphism ring of P^{∗} in the homotopy category of complexes of projective modules over the Brauer correspondent b of the block B. Both the verification that the complex P^{∗} is a tilting complex producing the desired block, and the calculation of the endomorphism ring of the complex in the derived category can be done more efficiently if the structure of b is well understood. This can be done in a particularly compact fashion by giving the quiver and relations of the block b.

Typeset by AMS-TEX 2 MOHAMMED HASAN ALI AND MARY SCHAPS

1991 Mathematics Subject Classification. 20C15, 20C05, 20C20.

Research partially supported by a grant from the Bar-Ilan Research Authority and the Emmy Noether Research Institute for Mathematics of the Minerva Foundation of Germany.

There is, furthermore, a natural grading on b, compatible with the filtration by the power of the radical in the group algebra of the defect group. It is induced by the image V of a splitting of Rad(b)/ Rad^{2}(b) into Rad b, a homogeneous basis for the subspace of weight d being given by monomials of degree d in basis vectors of V. If we could find a homogeneous basis for the endomorphism ring of the tilting complex, then we could try to transfer the grading to the blocks B of the larger group, as was done for the cyclic case in [SZ4]. This would also mean that the Donald-Flanigan deformation guaranteed in [MS] could be made homogeneous, with parameter-ring K[t].

In this paper we concentrate on the noncyclic case where the defect group Q of b is C_{q} × C_{q} or C_{q} × C_{q} × C_{q}. This covers most of the noncyclic blocks of abelian defect group in the ATLAS groups and their subgroups [C]. We first note a general result about the construction of the quiver of b in §2, and then give an algorithm for constructing the relations in §3, followed by some examples. In §4 we show that the differentials on the tilting complex can be chosen to be homogeneous. Finally, in §5 we give an explicit tilting complex for the non-principal 3-block of the central extension of PSL(3,4) by C_{2}.

§1. Quivers and relations for blocks with normal defect group.

Before discussing the case of abelian defect group in particular, we first review what is known in general about the quivers and relations of blocks of normal defect group. By a fundamental result of Külshammer [K], all such blocks are isomorphic to M_{n}(K^{γ}[Q ⋊ G′]), where K is a sufficiently large field of characteristic q, Q is the defect group, G′ is an q′ group acting faithfully on Q, and γ is a Hochschild cohomology two-cycle for G′ into K^{∗}. The matrix algebra does not affect the quivers and relations, and we incorporate γ into the group theory by passing to a central extension G of G′ with kernel N, as in [Re], where N is cyclic of order equal to the order of γ in H^{2}(G′, K^{∗}). The original block is thus Morita equivalent to one of the

LIFTING MCKAY GRAPHS AND RELATIONS TO PRIME EXTENSIONS 3

blocks of Q ⋊ G; these blocks all have the same K-dimension and are in one-to-one correspondence with the irreducible characters of the abelian group N.

We recall the definition of the McKay graph, which we will use to obtain the quiver of the desired block of Q ⋊ G. Definition. Let G be a finite group, let K be a sufficiently large field of characteristic q not dividing |G|, and let X be any character of G (not necessarily irreducible). The McKay graph D(G, X) is the directed graph with

(1) Vertices X_{i} labeled by elements of Irr(G), i = 1, …, r.

(2) For each pair of vertices X_{i}, X_{j}, a number n_{ij} = (X_{i}, X · X_{j})_{G} arrows from X_{j} to X_{i}, where X · X_{j} is the class function formed by pointwise multiplication, that is, (X · X_{j})(g) = X(g) · X_{j}(g) and n_{ij} is the number of

constituents of X_{i} in X · X_{j}, i.e., X · X_{j} = ∑_{i=1}^{r} n_{ij}X_{i}.

We recall that the quiver of a block B is a directed graph whose vertices correspond to the isomorphism classes of projective indecomposables [e_{i}B], where the number of arrows from [e_{j}B] to [e_{i}B] is dim_{k} e_{i}((Rad B)/ Rad(B)^{2})e_{j}. This is sometimes called the Ext-quiver of the block.

The main result of [SSS] states that if χ is the character of the action of G on J/J^{2}, J = Rad(KQ) and char K ∤ |G|, then the quiver of K[Q ⋊ G] is the McKay graph D(G, χ). The connected components of D(G, χ) are in one-to-one correspondence with the blocks of K[Q ⋊ G]. Let y_{1}, …, y_{s} be a complete set of primitive idempotents for KG, and let y_{1}, …, y_{m} be a completion to a basis of KG using matrix units. Then from [MS] we know that there exists a subvector space V = ⟨x_{1}, …, x_{r}⟩ of K[Q] which is a G-module and is isomorphic as a G-module to J/J^{2}. If r_{1}, …, r_{t} is a set of relations for K[Q] as a quotient of the free tensor algebra on V, then a complete set of relations for the algebra K[Q⋊G] as a quotient of the path algebra of the quiver is given by the relations y_{i}r_{j}y_{k}, where 1 ≤ i ≤ s, and 1 ≤ k ≤ m for j = 1, …, t.

4 MOHAMMED HASAN ALI AND MARY SCHAPS

In practice, when Q is not abelian, writing the relations of K[Q] in terms of the basis of V is not trivial because the basis elements of V are not simple linear compositions of group elements like h−e, but rather are obtained by some Maschke averaging. Thus it is not easy to convert group relations to algebra relations. However, when Q is abelian, the relations of K[Q] are of the simple form x_{i}^{p^{n}} = 0 and x_{i}x_{j} − x_{j}x_{i} = 0. The algebra K[Q] has a homogeneous basis, given by the monomials in the generators x_{1}, …, x_{r}. In the sequel, we will abbreviate K[Q] by KQ.

§2. Lifting McKay graphs.

We consider the case of a McKay graph D(G, X) where X is an irreducible character of G induced from an irreducible character W of a normal subgroup H, where [G : H] = p. Letting a be an element of G − H, and letting W_{i} = W^{a^{i}}, i = 0, …, p−1, we have X = ∑_{i=0}^{p−1} W_{i}. We note that the permutation α of the irreducibles induced by conjugation by a induces graph isomorphism D(H, W) →^{∼} D(H, W_{i}), i = 0, …, p − 1. The set of functions from the conjugacy classes of H which are ℤ-linear combinations of irreducible characters will be denoted by ℤ Irr(H). Lemma 2.1. Let X ∈ Irr(G) be a character induced from an irreducible character W of H, i.e., X = W^{G}. Then the McKay graph D(G, X) is completely determined by the McKay graph D(H, W) and the mapping of Irr(G) to ℤ Irr(H) induced by restriction. If R′, R′′ ∈ Irr(G), then the number of arrows from R′ to R′′ in D(G, X) is the total number of arrows from summands of R′_{H} to summands of R′′_{H} in D(H, W). Proof. By a result of Frobenius X · R′ = (W · R′_{H})^{G}. We now apply Frobenius reciprocity:

(R′′, X · R′)_{G} = (R′′, (W · R′_{H})^{G})_{G}
= (R′′_{H}, W · R′_{H})_{H}
□

LIFTING MCKAY GRAPHS AND RELATIONS TO PRIME EXTENSIONS 5

In the sequel, in our situation of H ⊴ G and [G : H] = p a prime, we let Irr(H) = S_{H} ∪ C_{H}, where S_{H} are the characters fixed under conjugation by G. Let Irr(G) = S_{G} ∪ C_{G} be the corresponding partition into characters which have split and characters which have collapsed. Example 1. G = C_{5} ⋊ C_{8}, H = C_{5} ⋊ C_{4}, the generalized quaternion group.

G = ⟨a, b | a^{8} = b^{5} = e, a^{−1}ba = b^{2}⟩, H = ⟨c, b | c^{4} = b^{5} = e, c^{−1}bc = b^{−1}⟩,

X = X_{9} ∈ C_{G}, W = X_{5} ∈ C_{H}, deg X = 4, deg W = 2.

Here c = a^{2}.

See Fig. 1 for the diagram of the McKay graphs.

Fig. 1

6 MOHAMMED HASAN ALI AND MARY SCHAPS

§3. Lifting relations.

The second author set up a database [S2], using GAP [G], of blocks of noncyclic abelian defect group with defect at least 2 in the ATLAS groups and their subgroups. There are none for primes q > 13, and only 10 for q = 11, 13. For q = 3, 5, and 7, the blocks are further subdivided. The main database, SortAt_{−}q, located at [S2], consists of blocks with elementary abelian defect group. There are 874 blocks in this database for q = 3, 360 for q = 5, and 81 for q = 7. There is a separate list, CyclicSortAt_{−}q, of blocks of abelian defect group containing C_{q^{2}}, most of which have cyclic defect, with the remainder being products of cyclic blocks.

In the interesting case where the defect group Q is elementary abelian, it usually has prime rank d(= 2, 3), and d = 2 is much more common than d = 3. Notable exceptions are the O’Nan group, whose principal 3-block is elementary of rank 4, and extensions of PSL(2, 81). Since |Q| = q^{d}, the integer d is the defect of the block.

A study of the Brauer correspondents b of the blocks with elementary abelian undertaken by R. Leabovich [L] and T. Berrebi [B] has determined that in almost all cases either H is itself abelian, or the Brauer correspondent is isomorphic to the principal block of the normalizer, and thus we are reduced to the abelian case, or b →^{∼} M_{t}(b′), where b′ →^{∼} K[Q ⋊ G], with [G : H] = 2 and H abelian. For example, the normalizer for blocks of the symmetric groups S_{n} are of the form (Q ⋊ G) × S_{n−2p}. Then b →^{∼} M_{t}(b′), where b′ is a block of K[Q ⋊ G] and t is the degree of a block of S_{n−2p} of defect zero. In general, as was shown by Reynolds [Re], if b is a block isomorphic to M_{t}(K^{γ}[Q ⋊ G′]), we can find a central extension G of G′ with cyclic kernel N such that b is Morita equivalent to a block of Q ⋊ G. The group Q ⋊ G, with N minimal, will be called the Reynolds group. These are listed in Tables 1, 2 and 3. When q = 3 and d = 3, the Reynolds group is generally a direct product. For nilpotent blocks, which are Morita equivalent to KQ, we take the Reynolds group to be Q. Those blocks for which the decomposition matrix was

LIFTING MCKAY GRAPHS AND RELATIONS TO PRIME EXTENSIONS 7

not known and no other information was available are not included.

Let the Reynolds group of b be Q ⋊ G, where G is a q′-group and the action of G on Q = C_{q} × ⋯ × C_{q} induces an action on Rad(KQ)/ Rad^{2}(KQ) with character χ. In Table 1-Table 3, we give Q ⋊ G, sorted as in the database by the invariants k(B) and ℓ(B) of the block B. If the block which is Morita equivalent to b is not the principal block of K[Q⋊G], then we indicate the block by writing b_{j}. Note that in all cases given here K[Q ⋊ G] has at most three blocks.

\underline{Table 1} Reynolds groups for 3-blocks in the database with elementary abelian defect group, k(B) ≤ 24

d  k(B)  ℓ(B)  # of blocks  Reynolds group
2  6  2  15  (C_{3} × C_{3}) : (C_{8} : C_{2}), b_{2}
2  6  2  8  (C4 × (C_{3} × C_{3}) : C_{2}) : C2), b_{3}
2  6  2  16  (C_{3} × C_{3}) : C_{2}
2  6  4  74  (C_{3} × C_{3}) : C_{4}
2  6  5  74  (C_{3} × C_{3}) : Q
2  9  1  35  C_{3} × C_{3}
2  9  2  48  C_{3} × S_{3}
2  9  4  141  S_{3} × S_{3}
2  9  5  182  (C_{3} × C_{3}) : D_{4}
2  9  7  89  (C_{3} × C_{3}) : (C_{8} : C_{2})
2  9  8  4  (C_{3} × C_{3}) : C_{8}
3  15  2  2  (C_{3})^{3} : C_{2}
3  15  5  1  (C_{3} × (C_{3} × C_{3}) : C_{4}) : C_{2}
3  15  7  1  (C_{3} × (C_{3} × C_{3}) : Q) : C_{2}
3  18  4  4  C_{3} × (C_{3} × C_{3}) : C_{4}
3  18  8  4  S_{3} × (C_{3} × C_{3}) : C_{4}
3  18  10  3  S_{3} × (C_{3} × C_{3}) : Q
4  18  14  2  (C_{3})^{4} : (C_{2}.(C_{2})^{4}) : D_{5}
4  24  14  1  (((C_{3} × C_{3}) : C_{4}) × ((C_{3} × C_{3}) : C_{4})).C_{2}
4  24  16  1  ((C_{3})^{4} : ((C_{2}.((C_{2})^{4}) : D_{5}).C_{2}
4  24  20  1  (C_{3})^{4} : (C_{80} : C_{2}), b_{2}
4  24  22  1  (C_{3})^{4} : (C_{40} : C_{4})

8 MOHAMMED HASAN ALI AND MARY SCHAPS

\underline{Table 2} Reynolds groups for 5-blocks

d  k(B)  ℓ(B)  # of blocks  Reynolds group
2  10  6  1  (C_{5} × C_{5}) : (C_{3} : (C_{8} : C_{2})), b_{2}
2  11  5  5  (C_{5} × C_{5}) : C_{2}.(C_{4} × C_{4}).C_{2}, b_{2}
2  13  10  8  (C_{5} × C_{5}) : (C_{8} : C_{2})
2  13  12  1  (C_{5} × C_{5}) : (C_{3} : C_{8})
2  14  12  6  (C_{5} × C_{5}) : C_{12}
2  16  10  8  (C_{5} × C_{5}) : (D_{4}Y C_{4})
2  16  12  9  (C_{5} × C_{5}) : (S_{3} × C_{4})
2  16  14  1  (C_{5} × C_{5}) : (QY C_{4})
2  20  14  19  (C_{5} × C_{5}) : (C_{4} × C_{4}) : C_{2}
2  26  24  3  (C_{5} × C_{5}) : C_{24}
\underline{Table 3}

Reynolds groups for 7-blocks

d  k(B)  ℓ(B)  # of blocks  Reynolds group
2  16  12  1  (C_{7} × C_{7}) : (C_{3} × D_{16})
2  22  18  3  (C_{7} × C_{7}) : (C_{3} × (C_{3} : C_{4}))
2  22  21  1  (C_{7} × C_{7}) : (C_{3} × Q_{8})
2  25  18  3  (C_{7} × C_{7}) : (S_{3} × C_{6})
2  25  21  2  (C_{7} × C_{7}) : (C_{3} × D_{8})
2  26  18  1  (C_{7} × C_{7}) : (C_{3} × C_{6})
2  26  24  3  (C_{7} × C_{7}) : C_{24}
2  27  21  8  (C_{7} × C_{7}) : (C_{3} × SL(2, 3))
2  27  24  1  (C_{7} × C_{7}) : (C_{3} × GL(2, 3))
2  35  27  7  (C_{7} × C_{7}) : (C_{3} × (C_{3} : D_{4}))
2  49  48  3  (C_{7} × C_{7}) : C_{48}

In this paper, we treat the common case when one of the following holds: Case 1: The character χ is a sum of linear characters. Case 2: The group G has an abelian normal subgroup H of index p such that χ_{H} = ϕ_{1} + ⋯ + ϕ_{p} is a sum of linear characters permuted by G and KH has a set of primitive idempotents permuted by a, where a ∈ G is an element of p power order whose residue in G/H generates G/H. Let N = ker(χ).

In Case 1, the relations are all of the form e_{i}x_{j}^{q} or e_{i}(x_{j}x_{k} − x_{k}x_{j}) for j, k = 1, …, p, i = 1, …, [G : N], as described in [MS].

LIFTING MCKAY GRAPHS AND RELATIONS TO PRIME EXTENSIONS 9

Case 2 is the case treated in detail by this paper. Clearly,

N = ⋂_{i=1}^{n} ker(ϕ_{i}).

(Even if H had not not been assumed abelian, since the derived group H′ of H is in each ker(ϕ_{i}), we could have gotten H′ ⊂ N, so that H/N would have been abelian.) Since the ϕ_{i} are conjugate via a, ϕ_{1}(a^{p}) = ϕ_{j}(a^{p}) for j = 2, …, p.

Let e_{1}, …, e_{r} be the block idempotents for H. Since conjugation by a is an automorphism, a permutes these idempotents of H. Any ϕ_{j}, for j = 1, …, p, induces a permutation τ_{j} on the linear characters θ_{i} of H by setting τ_{j}(i) = k iff ϕ_{j} ·θ_{i} = θ_{k}. As was proven in [MS], this precisely corresponds to the condition

e_{i}x_{j} = x_{j}e_{k}.

Since τ_{j} corresponds to multiplication by ϕ_{j}, the τ_{j} commute because character multiplication commutes. In particular, in the relation e_{i}(x_{j}x_{k}−x_{k}x_{j}), both monomials have the same idempotent e_{ℓ} acting on the left, where ℓ = τ_{j} ∘ τ_{k}(i) = τ_{k} ∘ τ_{j}(i).

The problem of finding quivers and relations for the case of H abelian and G = H ⋊ C_{p} was first treated by §2 of [RR]. We consider the nonsplit case and also construct the relations in a mechanical, algorithmic fashion which has been machine implemented for the case of H abelian and p = 2. We recall that C_{H} is the set of characters of H on which a acts nontrivially and refer to the set \overline{C}_{H} of the corresponding idempotents as “collapsing idempotents”. Similarly, S_{H} is the set of fixed characters, and the set \overline{S}_{H} containing the corresponding idempotents will be called “splitting idempotents”.

To give the quiver and relations of K[Q ⋊ G], we proceed as follows:

1) Choose one primitive idempotent eî of each nontrivial conjugacy class [e_{i}] of idempotents of H under the action of a. For each idempotent e_{k} in H, choose a natural number ℓ(k), 1 − p < ℓ(k) ≤ p − 1 such that ek̂ = a^{−ℓ(k)}e_{k}a^{ℓ(k)}, and set E_{kk̂} = a^{ℓ(k)}, E_{k̂k} = a^{−ℓ(k)}.

10 MOHAMMED HASAN ALI AND MARY SCHAPS

2) Construct the quiver D(G, χ) from D(H, ϕ_{1}) by collapsing and splitting as in Lemma 2.1, labeling each collapsed vertex by the idempotent ek̂ representing its conjugacy class of idempotents.

3) For each splitting idempotent e_{i}, let e_{i1}, e_{i2} …, e_{ip} be the idempotents into which it splits. (If p = 2, we abbreviate e_{i1} by e_{i}^{+} and e_{i2} by e_{i}^{−}.) Let θ be a nontrivial lifting of 1_{H}. By standard results in Clifford theory, if θ′_{i} is the character associated to e_{i1} then the character θ′_{ij} associated to e_{ij} is a multiple of θ′_{i} by a power of θ, and we may choose the numbering so that θ_{ij} = θ′_{i}θ^{j−1} is associated to e_{ij}.

Lemma 3.2. Let Q be an elementary abelian q-group of prime rank p. Let G be a q′-group acting irreducibly on Q and assume that G is a p-extension of an abelian group H. Let R = {r_{j}} be the set of all relations of KQ, i.e., x_{ℓ}^{q} = 0, ℓ = 1, …, p, and x_{t}x_{s} − x_{s}x_{t} = 0 for 1 ≤ t < s ≤ p. Each relation r_{j} determines a unique permutation σ_{j} such that e_{i}r_{j} = r_{j}e_{σ_{j}(i)}. Set k = σ_{j}(i). Each relation e_{i}r_{j}e_{k} of K[Q ⋊ H] determines the following relations of the quiver of K[Q ⋊ G] :

(a) If e_{i} does not split and is a representative of its conjugacy class and e_{k} does not split, then we get a relation e_{i}r_{j}E_{kk̂} = e_{i}r_{j}a^{ℓ}, where ℓ = ℓ(k) is chosen so that a^{−ℓ}e_{k}a^{ℓ} is the conjugacy class representative ek̂.

(b) If e_{i} does not split and is a representative of its conjugacy class and e_{k} does

split, we get a relation

e_{i}r_{j}e_{k},

which splits into p relations since e_{k} is splitting.

(c) If e_{i} splits and e_{k} does not, then we get relations e_{iℓ}r_{j}e_{k}a^{ℓ}, where ℓ = ℓ(k) is chosen so that a^{−ℓ}e_{k}a^{ℓ} is the conjugacy class representative ek̂.

(d) If e_{i}, e_{k} are both splitting, we get the p^{2} relations e_{it}r_{j}e_{kt′}.

LIFTING MCKAY GRAPHS AND RELATIONS TO PRIME EXTENSIONS 11

Proof. Let {r_{j}} be the complete set of relations such that K[Q] →^{∼} K⟨x_{1}, …, x_{p}⟩/{r_{j}}, as given in the statement of the lemma. Since the quiver of K[Q] is just p loops, then by the Lemma in Section 5 of [MS], substituting 0 for the deformation parameter, we know that the relations of K[Q ⋊ H] and K[Q ⋊ G], respectively, are obtained from {r_{j}} by multiplying on the left by a primitive idempotent corresponding to a vertex and, in the case of K[Q ⋊ G], on the right by a matrix unit.

Fix a relation r_{j} of K[Q]. If it is monomial in x_{1}, …, x_{p}, then we get σ_{j} as a composition of τ_{i}. If it is of the form x_{s}x_{t} − x_{t}x_{s}, then σ_{j} = τ_{t}τ_{s} = τ_{s}τ_{t}. We previously defined a partition Irr(H) = S_{H}∪C_{H}, according to whether the character was fixed under conjugation by G and would thus split, or whether it had nontrivial conjugates, and a similar partition of the idempotents \{e_{1}, …, e_{r}\} = \overline{S}_{H} ∪ \overline{C}_{H}. For any primitive idempotent e_{i} for which i = î, we have the following logical possibilities:

(a) If e_{i}, e_{k} ∈ \overline{C}_{H}, k = σ_{j}(i), then there is a relation e_{i}r_{j} of the quiver of K[Q ⋊ H]. Let ℓ = ℓ(k) be the integer such that ek̂ = a^{−ℓ}e_{k}a^{ℓ}. Then the corresponding relation of K[Q ⋊ G] will be

0 = eîr_{j}E_{kk̂} = eîr_{j}a^{ℓ}.

(b) If e_{i} ∈ \overline{C}_{H} and e_{k} ∈ \overline{S}_{H}, we multiply by the various idempotents which split it. These relations are distinct since they end at different points.

(c) If e_{i} ∈ \overline{S}_{H} and e_{k} ∈ \overline{C}_{H}, then we proceed as in (a) to show that the matrix unit given in the lemma of [MS] is a power of a.

(d) If e_{i}, e_{k} ∈ \overline{S}_{H}, then there are p^{2} possible relations. Corollary 3.3. If a splitting idempotent e_{s} ∈ \overline{S}_{H} can be inserted in e_{i}r_{j}, then the a^{ℓ} at the end in cases (a) and (c) can be conjugated forward until it reaches e_{s} when it can be converted into a coefficient. Since H is abelian, case (d) occurs only when p = 2, and there are only p non-zero relations in (d).

12 MOHAMMED HASAN ALI AND MARY SCHAPS

Proof. Conjugation by a introduces coefficients in two ways. First of all, ax_{p}a^{−1} = ϕ_{1}(a^{p}) · x_{1}. Secondly, when the power of a reaches an element of \overline{S}_{H}, it is converted into a coefficient by the formula a · e_{st} = e_{st} · a = θ′_{st}(a)e_{st}.

We must show that when H is abelian, case (d) occurs only for p = 2, and that then there are only p nonzero relations.

If r_{j} is a relation for which case (d) occurs, and m(x_{1}, …, x_{p}) is a monomial occurring in r_{j}, then we have σ_{j}(i) = m(τ_{1}, …, τ_{p})(i) = k, which on characters becomes m(ϕ_{1}, …, ϕ_{p}) · θ_{i} = θ_{k}. Since θ_{i}, θ_{k} ∈ S_{H}, we get that m(ϕ_{1}, …, ϕ_{p}) = θ_{i}^{−1}θ_{k} ∈ S_{H}, showing that m(ϕ_{1} …, ϕ_{p}) is invariant under the action of G.

This cannot occur for the monomial relation x_{t}^{q} = 0 because the cycles in the McKay graph D(H, ϕ_{1}) are of length r dividing |H|, so ϕ_{t}^{r} = ϕ_{t′}^{r} for all t, t′. If we had ϕ_{t}^{q} invariant under the action of G so that ϕ_{t}^{q} = ϕ_{t′}^{q}, then since |H| is a q′ integer, we would have ϕ_{t} = ϕ_{t′} for all t′, a contradiction since the irreducibility of the action of G on Q implies that the restriction of this action to H is a sum of distinct conjugates.

If r_{j} is the relation x_{t}x_{s} − x_{s}x_{t}, then ϕ_{t}ϕ_{s} is invariant. Then, acting by a^{s−t}, we get ϕ_{t}ϕ_{s} = ϕ_{s}ϕ_{2s−t}, so, multiplying by ϕ_{s}^{−1}, t ≡ 2s − t (mod p). Since p is a prime, t ≠ s, and 1 ≤ s, t ≤ p, we have 2t ≡ 2s (mod p) only when p = 2.

The monomial x_{1}x_{2} is an eigenvector for G, since x_{1} and x_{2} are eigenvectors for H, with characters ϕ_{1} and ϕ_{2}, while a^{−1}x_{1}x_{2}a = x_{2}(ϕ_{1}(a^{2})x_{1}) = ϕ_{1}(a^{2})x_{1}x_{2}. Thus if η is the linear character of G with η |_{H}= ϕ_{1}ϕ_{2} and η(a) = ϕ_{1}(a^{2}), then for any idempotent e_{is} with character ϕ_{is}, we have e_{is}r_{j} = r_{j}e_{ks′}, where s′ is so chosen that η · θ_{is} = θ_{ks′}. Thus there is only one value s′′ of s′ for which the relation e_{is}r_{j}e_{ks′′} is nonzero.

Example 2. Consider the Brauer correspondent of the principal block of the Janko group J_{2} for q = 5. It is a semidirect product of C_{5}×C_{5} by D_{6}, the dihedral group of order 12, and the action of D_{6} on J/J^{2} is determined by the unique two-dimensional representation without a kernel. The McKay graph is

Fig. 2

Let α_{01}^{±}, α_{12}, α_{23}^{±} be the arrows with increasing exponent and β_{10}^{±}, β_{21}, β_{32}^{±} the arrows in the opposite direction. The relations are derived as follows from the following cycles in H, e_{i}x_{1} = x_{1}e_{i+1} and e_{i}x_{2} = x_{2}e_{i−1}.

e_{1}x_{1}^{5}, e_{1}x_{2}^{5} = 0 :  α_{01}^{+}α_{12}(a_{23}^{+}β_{32}^{+} − α_{23}^{−}β_{32}^{−})β_{21} = 0
e_{2}x_{1}^{5}, e_{2}x_{2}^{5} = 0 :  β_{32}^{±}β_{21}(β_{10}^{+}α_{01}^{+} − β_{10}^{−}α_{01}^{−})α_{12} = 0
e_{1}x_{1}^{5} = 0 :  α_{12}(α_{23}^{+}β_{32}^{+} − α_{23}^{−}β_{32}^{−})β_{21}β_{10}^{±} = 0
e_{2}x_{2}^{5}x_{2}^{5} = 0 :  β_{21}(β_{10}^{+}α_{01}^{+} − β_{10}^{−}α_{01}^{−})α_{12}α_{23}^{±} = 0
e_{2}x_{1}^{5} = 0 :  (α_{23}^{+}β_{32}^{+} − α_{23}^{−}β_{32}^{−})β_{21}(β_{10}^{+}α_{01}^{+} − β_{10}^{−}α_{01}^{−}) = 0
e_{1}x_{2}^{5} = 0 :  (β_{10}^{+}α_{01}^{+} − β_{10}^{−}α_{01}^{−})α_{12}(α_{23}^{+}β_{32}^{+} − α_{23}^{−}β_{32}^{−}) = 0
e_{0}^{±}(x_{1}x_{2} − x_{2}x_{1}) = 0 : α_{01}^{+}β_{10}^{−} = 0, α_{01}^{−}β_{10}^{+} = 0
e_{3}^{±}(x_{1}x_{2} − x_{2}x_{1}) = 0 : β_{32}^{+}α_{23}^{−} = 0, β_{32}^{−}α_{23}^{+} = 0
e_{1}(x_{1}x_{2} − x_{2}x_{1}) = 0 : α_{12}β_{21} = β_{10}^{+}α_{01}^{+} + β_{10}^{−}α_{01}^{−}
e_{2}(x_{1}x_{2} − x_{2}x_{1}) = 0 : β_{21}α_{12} = α_{23}^{+}β_{32}^{+} + α_{23}^{−}β_{32}^{−}.

The projective modules needed to build the tilting complexes and the resolution of the Green correspondents in Okuyama’s method are then constructed by laying out the x_{1}^{i}, x_{2}^{j} in a grid. We give only P_{0^{+}} and P_{1}, the other projective modules being similar. The Cartan matrix, whose columns give the number of copies of every simple in each projective, is

C = [ \begin{matrix}5 & 0 & 2 & 2 & 4 & 4 \\ 0 & 5 & 2 & 2 & 4 & 4 \\ 2 & 2 & 5 & 0 & 4 & 4 \\ 2 & 2 & 0 & 5 & 4 & 4 \\ 4 & 4 & 4 & 4 & 9 & 8 \\ 4 & 4 & 4 & 4 & 8 & 9 \end{matrix} ]

Fig. 3

Note that our algorithm makes it quite natural to write the projectives in the rectangular form which was used with such good effect by Holloway [Hol] in his study of SL(2, q^{n}). §4. Homogeneous maps in tilting complexes.

Let us now consider the indecomposable projectives in the case where p = 2 and the normalizer of the defect group Q = C_{q} × C_{q} is of the form Q ⋊ G, where G

LIFTING MCKAY GRAPHS AND RELATIONS TO PRIME EXTENSIONS 15

has a normal abelian subgroup H of index 2. (A sample case with p = 3 was given in [SSS].) An examination of the Reynolds groups for blocks in our database [S2] given in the tables in §3 shows that either G is abelian or this condition is fulfilled with p = 2 for all but three blocks of defect 4 with q = 3, two blocks with q = 5 and two blocks for q = 7. If Rad(KQ) = J, the action of G on J/J^{2} is given by χ = ϕ_{1} +ϕ_{2}. We assume that a is an element of order 2^{c} of G not lying in H, which permutes ϕ_{1} and ϕ_{2}. We let x_{1} and x_{2} be eigenvectors for the actions of ϕ_{1} and ϕ_{2} on J/J^{2}. Set ζ = ϕ_{1}(a^{2}) = ϕ_{2}(a^{2}), where ζ is a primitive 2^{c′} root of unity for some integer 0 ≤ c′ ≤ c. We may choose the x_{i} so that ax_{1} = ζx_{2}a, x_{1}a = ax_{2}.

Whereas in calculating the relations in §3 we found it convenient to conjugate a to a splitting idempotent and convert it into a coefficient, in giving a homogeneous basis of the indecomposable projectives it will be more efficient to conjugate the element a to a collapsing idempotent.

Let A = K[Q ⋊ H], Ã = K[Q ⋊ G]. If, as before, we let e_{1}, …, e_{r} be the block idempotents of H, which are primitive because H is abelian, then the indecomposable projective e_{i}A has a homogeneous basis {e_{i}x_{1}^{j}x_{2}^{k}}_{j,k=0}^{p−1}.

Lemma 4.1. If m_{jk} = x_{1}^{j}x_{2}^{k}, then am_{jk}a^{−1} = ζ^{j}m_{kj}.

Proof.

ax_{1}^{j}x_{2}^{k}a^{−1} = (ax_{1}a^{−1})^{j}(ax_{2}a^{−1})^{k}
= (ζx_{2})^{j}x_{1}^{k}
= ζ^{j}x_{1}^{k}x_{2}^{j}

Definition. Let ρ_{jk} = τ_{1}^{j}τ_{2}^{k}, so that e_{i}m_{jk} = m_{jk}e_{ρ_{jk}(i)}.

Lemma 4.2. Let e_{i}^{±} be an idempotent corresponding to a splitting character θ_{i}. Then a homogeneous basis of the indecomposable projective e_{i}^{±}Ã is a union of the

16 MOHAMMED HASAN ALI AND MARY SCHAPS

following sets:

B_{i}^{C} = \{\frac{1}{2}e_{i}^{±} (m_{jk} ± ζ^{j}m_{kj}a) | e_{ρ_{jk}(i)} ∈ \overline{C}_{H}\}

B_{i}^{S} = {e_{i}^{±}m_{jk}e_{ℓ}^{±} | ρ_{jk}(i) = ℓ, e_{ℓ} ∈ S_{H}} ∪ {e_{i}^{±}m_{jj}}.

Each element in these bases lies in a Peirce component of Ã w.r.t. the primitive idempotents. Proof. For any idempotent e_{i} which splits into e_{i}^{+} and e_{i}^{−}, we have

e_{i}^{+} = \frac{1}{2}(e + a)e_{i} = \frac{1}{2}e_{i}(e + a)

and

e_{i}^{−} = \frac{1}{2}(e − a)e_{i} = \frac{1}{2}e_{i}(e − a).

Thus

e_{i}^{±}m_{jk} = \frac{1}{2}e_{i}(e ± a)m_{jk}

= \frac{1}{2}e_{i}(m_{jk} ± am_{jk})

= \frac{1}{2}e_{i} (m_{jk} ± ζ^{j}m_{kj}a) .

Case 1: \underline{ℓ = ρ_{jk}(i) is the index of a collapsing idempotent}: Let ℓ′ be the index of the conjugate idempotent so that e_{ℓ′} = ae_{ℓ}a^{−1}, then

ae_{i}m_{jk}e_{ℓ}a^{−1} = ζ^{j}e_{i}m_{kj} · (ae_{ℓ}a^{−1})
= ζ^{j}e_{i}m_{kj}e_{ℓ′},

so we see that ρ_{kj}(i) = ℓ′. Thus

\frac{1}{2}e_{i}^{±}(m_{jk} ± ζ^{j}m_{kj}a) = \frac{1}{2}(m_{jk} ± ζ^{j}m_{kj}a)e_{ℓ}.

Case 2: \underline{ℓ = ρ_{jk}(i) is the index of a splitting idempotent}: If j = k, with ℓ = ρ_{jk}(i), then

0 ≠ ae_{i}m_{jj}e_{ℓ}a^{−1} = ζ^{j}e_{i}m_{jj}(ae_{ℓ}a^{−1}),

so we conclude that e_{ℓ} ∈ \overline{S}_{H}, since otherwise we would have e_{i}m_{jj}e_{ℓ}(ae_{ℓ}a^{−1}) = 0. Finally, if j ≠ k, and ℓ = ρ_{jk}(i), then

e_{i}^{±}m_{jk}e_{ℓ}^{±} = \frac{1}{4}e_{i}^{±}m_{jk}(e ± a)e_{ℓ}

= \frac{1}{4}e_{i}^{±}(m_{jk} ± ζ^{j}am_{kj})e_{ℓ}.

The two different ± signs are independent, so these represent four different maps. Lemma 4.3. Let e_{i} be an idempotent corresponding to a collapsing character θ_{i} with i = î. A homogeneous basis of the projective indecomposable e_{i}Ã is a union of the following sets:

B_{i}^{S} = \{\frac{1}{2}(e_{i}m_{jk} ± m_{kj}a)e_{ℓ}^{±} | ρ_{jk}(i) = ℓ, e_{ℓ} ∈ \overline{S}_{H}\}

and

B_{i}^{C} = {e_{i}m_{jk} | ρ_{jk}(i) = ℓ = ℓ̂, e_{ℓ} ∈ C_{H}} ∪ {e_{i}m_{jk}a | ρ_{jk}(i) = ℓ ≠ ℓ̂, e_{ℓ} ∈ C_{H}}.

Proof. The proof for B_{i}^{S} is similar to the previous lemma. As for B_{i}^{C}, if ρ_{jk}(i) = ℓ = ℓ̂, the chosen representative of its conjugacy class, then e_{i}m_{jk} = eîm_{jk} ends with the idempotent e_{ℓ}, whereas if ρ_{jk}(i) = ℓ ≠ ℓ̂, then we must multiply by a matrix unit, which is equivalent to a times a constant since ζ = ϕ_{1}(a^{2}) is a constant. □

Now suppose that we have a tilting complex in which all the differentials are homogeneous and the irreducible components have length 2, as in the cyclic defect case or the elementary tilting complexes used by Okuyama [O]. Then we can choose representatives of the endomorphisms which are homogeneous in each degree. Furthermore, the set of homotopies also has a basis homogeneous in each degree.

18 MOHAMMED HASAN ALI AND MARY SCHAPS

§5. The Broué conjecture for the faithful 3-block of the covering group of P SL(3, 4).

We now give, as an application, a verification of the Broué conjecture for a block with defect group C_{3} × C_{3}. To the best of our knowledge, guided by the list of proven cases maintained at Bristol, this particular block has not been treated as of the time of this writing. Let L be the central extension with kernel C_{2} of P SL(3, 4). Then Q is a 3-Sylow subgroup, isomorphic to C_{3} × C_{3}. The centralizer of Q is Q × C_{2}. The centralizer has two blocks, each stable under the action of the normalizer. These correspond to the two blocks of the normalizer. In this case, since the defect group is abelian and thus contained in the centralizer, we consider N_{L}(Q)/C_{L}(Q), and calculation shows that it is isomorphic to Q_{8}, the quaternion group of order 8. Both blocks of the normalizer are isomorphic as algebras, and thus both are isomorphic to the group algebra of the unique semidirect product (C_{3} × C_{3}) ⋊ Q_{8} in which the center of Q_{8} acts non-trivially.

We are considering the non-principal block of the normalizer. This is the Brauer correspondent of the faithful 3-block B of L, i.e., the block consisting of ordinary characters which are nontrivial on the center Z(L) of order 2. A calculation of the decomposition matrix of this block at L shows that it is identical, up to permutation of rows and columns, with the decomposition matrix of the principal block of the Mathieu group M_{22}. The centralizers of the cyclic subgroups of the defect group Q are identical to the centralizer of Q itself, so the restriction map induces a stable equivalence. Calculating the Green correspondent of the simples of the block B with the C-MeatAxe developed by M. Ringe produced modules for the Brauer correspondent b of dimensions 1,1,4,4,6, with the same structure as the Green correspondents of M_{22}. Thus the same sequence of two elementary tilting complexes used by Okuyama [O] to settle the Broué conjecture for M_{22} will work for our block B.

LIFTING MCKAY GRAPHS AND RELATIONS TO PRIME EXTENSIONS 19

We want now to give the tilting complex explicitly, as a complex of b modules with homogeneous differentials. In the notation of section 3, we have G = C_{4}⋊C_{4} = C_{2}.Q_{8}, H = C_{4} × C_{2}, and the character χ is an irreducible character of G whose restriction to H is the sum of two linear characters ϕ_{1} and ϕ_{2} with common kernel N = Z(G) →^{∼} C_{2}. The McKay graph D(H, ϕ_{1}) is the disjoint union of two cycles of length four. The element a has order 8, and it acts on each of the two cycles by leaving two antipodal points fixed and exchanging the other two points. The element a^{2} is in H, and we have

ϕ_{1}(a^{2}) = ϕ_{2}(a^{2}) = −1.

Suppose 1 and 3 are the indices of the idempotents of the non-principal block of H left fixed by a, and let 2 and 4 be the indices of the idempotents interchanged by a. Choose 2 to be the representative of its orbit under the action of a, so that ˆ2 = ˆ4 = 2. Then the idempotents of G will be e_{1}^{±}, e_{3}^{±}, and e_{2}. These primitive idempotents then determine four indecomposable projective modules of dimension 9, which we will denote by P_{1}^{±} and P_{3}^{±}, and a single indecomposable projective module of dimension 18, which we will designate by P_{2}. The two Green correspondents G_{1}^{+} and G_{1}^{−} of dimension 1 are a pair of split simples, which we will identify with e_{1}^{±}. The two Green correspondents of dimension 4, G_{3}^{+} and G_{3}^{−}, have as tops a matched pair of split simples, which we will identify with e_{3}^{±}.

Using the rectangular representation of the projectives as in §3, we see that in fact G_{3}^{+}, which has composition factors 3^{+}, 2, 3^{−}, is the quotient of P_{3}^{+} by a submodule whose projective cover is P_{1}^{+} ⊕ P_{1}^{−}, and similarly G_{3}^{−} is a quotient of P_{3}^{−}. The Green correspondent G_{2}, with composition factors 2, 1^{+}, 1^{−}, 2, is the quotient of P_{2} by a submodule with projective cover P_{3}^{+} ⊕ P_{3}^{−}. We can represent these as

3^{+}  3^{−}  2
G_{3}^{+} =  2  G_{3}^{−} =  2  G_{2} =  1^{+} ⊕ 1^{−}
3^{−}  3^{+}  2

We now apply Okuyama’s method of elementary tilting complexes from [O]. For those who have difficulty obtaining this extremely important paper, the method is outlined, with an extension, in [S1]. We choose the index set I_{0} = {3^{+}, 3^{−}}. Denote the right regular representation of b by M. Letting g be the endomorphism corresponding to left multiplication by x_{1}, letting h be the endomorphism of M corresponding to left multiplication by x_{2}, and letting α be the automorphism of modules given by left multiplication by a, we get a tilting complex T_{1} = (P_{1}^{+})^{(1)} ⊕ (P_{1}^{−})^{(1)} ⊕ P_{2}^{(1)} ⊕ (P_{3}^{+})^{(1)} ⊕ P_{3}^{(1)}.

(P_{1}^{+})^{(1)} : P_{3}^{+} ⊕ P_{3}^{−} \underrightarrow{[ \begin{matrix}g^{2} + h^{2} \\ g^{2} − h^{2} \end{matrix} ]} P_{1}^{+}
(P_{1}^{−})^{(1)} : P_{3}^{+} ⊕ P_{3}^{−} \underrightarrow{[ \begin{matrix}g^{2} − h^{2} \\ g^{2} + h^{2} \end{matrix} ]} P_{1}^{−}
(P_{2})^{(1)} : P_{3}^{+} ⊕ P_{3}^{−} \underrightarrow{[ \begin{matrix}h − αg \\ h + αg \end{matrix} ]} P_{2}
(P_{3}^{+})^{(1)} :  P_{3}^{+}
(P_{3}^{−})^{(1)} :  P_{3}^{−}

The indecomposable projectives with indices in I_{0} occur in degree (−1) and the indecomposable projectives with indices in {1^{+}, 1^{−}, 2} occur in degree 0.

Each elementary tilting complex determines a linear combination of columns of the decomposition matrix D_{b} of b, with those in odd degrees multiplied by (−1). Multiplying negative rows by (−1) then produces the decomposition matrix of B_{1} = End_{D^{b}(b)}(T), from which the Cartan matrix can be calculated.

D_{b} = [ \begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 & 2 \end{matrix} ] → [ \begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ −1 & −1 & −1 & 0 & −1 \\ −1 & −1 & 0 & −1 & −1 \\ 0 & 0 & 0 & 0 & 1 \\ −1 & −1 & −1 & −1 & 0 \end{matrix} ] → [ \begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 \\ 1 & 1 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 & 0 \end{matrix} ] = D_{B_{1}}.

By Lemma 1.3(i) [O], the images G_{1}^{+(1)} and G_{1}^{−(1)} of G_{1}^{+} and G_{1}^{−} remain simple in B_{1}. Furthermore, by Lemma 1.3(ii) [O], G_{3}^{+(1)} and G_{3}^{−(1)} are now simple. This leaves only to calculate the Green correspondent G_{2}^{(1)}, using the resolution method in §4 of [R]. A projective resolution over b of G_{2} begins with the map in P_{2}^{(1)}, and thus a projective cover of this resolution in B_{1} has P_{2}^{(1)} as its top. The projective cover of the Heller translate over B_{1} has (P_{3}^{+})^{(1)}⊕(P_{3}^{−})^{(1)} as its top. Calculating the Cartan matrix from D_{B_{1}} and noting that there are unique maps in each direction between (P_{3}^{±})^{(1)} and P_{2}, we discover that G_{2}^{(1)} has, as before, composition factors 2^{(1)}, 1^{+(1)}, 1^{−(1)}, 2^{−(1)}. The Heller translate of G_{2}^{(1)} has socle 2^{(1)} and all the other simples as composition factors (with multiplicity 1), so it satisfies the conditions of Lemma 1.3(ii) [O]. Therefore, a second elementary tilting complex with index set I_{1} = {2} will complete the solution. Taking mapping cones, we can represent the new tilting complex as a complex of projectives from b, with homogeneous maps:

P_{2} ⟶^{h+gα} P_{1}^{+}

P_{2} ⟶^{h−gα} P_{1}^{−}

P_{3}^{+} ⊕ P_{3}^{−} \underrightarrow{[ \begin{matrix}h − αg \\ h + αg \end{matrix} ]}P_{2}
P_{3}^{−} ⟶^{h+αg} P_{2}
P_{3}^{+} ⟶^{h−αg} P_{2}

Conclusion.

We have concentrated here on an algorithmic construction of the indecomposable

projectives of the Brauer correspondent. We hope that by using the decomposition

matrix to find combinatorial tilting complexes, we will get a method for obtaining

quivers and relations for the group blocks themselves in the abelian defect case. The

method we are pursuing is outlined in [S1]. However, it requires a solid knowledge

of the projective modules of the correspondent to which this paper is intended to

contribute. It also involves questions of “folding” tilting complexes, as described

in [SZ1], [SZ2], [SZ3], [RS1], [SZ4]. We hope that, as in the cyclic case [RS2], the

Green correspondent will aid in choosing the proper “folding”.

References

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[Br] M. Broué, Equivalences of blocks of group algebras, Finite Dimensional Algebras and Related Topics, Kluwer, 1992, pp. 1–26.

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[M] J. McKay, Graphs, singularities and finite groups, Proc. Symp. Pure Math. 37 (1980), 183-1866.

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[Re] W.F. Reynolds, Blocks and normal subgroups of finite groups, Nagoya Math. J. 22 (1963), 15-32.

[R] J. Rickard, Derived categories and stable equivalence, J. of Pure and Applied Algebra 61 (1989), 303-317.

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[S1] M. Schaps, Deformations, tiltings, and decomposition matrices, Field Institute Publications, vol. 41, 2003.

[S2] M. Schaps et al., Databases of blocks, http://www.cs.biu.ac.il/∼mschaps/math.html.

[SSS] M. Schaps, D. Shapira and Orly Shlomo, Quivers of blocks with normal defect group, Proc. Symposia in Pure Math. 63 (1998), 497-510.

[SZ1] M. Schaps and E. Zakay-Illouz, Combinatorial partial tilting complexes for the Brauer star algebras, Representations of Algebra, Kluwer, 2001, pp. 187-207.

[SZ2] M. Schaps and E. Zakay-Illouz, Pointed Brauer trees, J. of Algebra 246 (2001), 647-672.

[SZ3] M. Schaps and E. Zakay-Illouz, Braid group action on the refolded tilting complexes of the Brauer algebra, Proceedings ICRA IX, Beijing, vol. II (2002), 434-449.

[SZ4] M. Schaps and E. Zakay-Illouz, Homogeneous deformations of Brauer tree algebras, Comm. Algebra 31 (2003), no. 11, 5243-5353.

Department of Mathematics, Al Quds University, Abu Dies

Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel