BLOCKS WITH A QUATERNION DEFECT

GROUP OVER A 2-ADIC RING: THE CASE Ã_{4}

Thorsten Holm, Radha Kessar, Markus Linckelmann

Abstract. Except for blocks with a cyclic or Klein four defect group, it is not known in general whether the Morita equivalence class of a block algebra over a field of prime characteristic determines that of the corresponding block algebra over a p-adic ring. We prove this to be the case when the defect group is quaternion of order 8 and the block algebra over an algebraically closed field k of characteristic 2 is Morita equivalent to kÃ_{4}. The main ingredients are Erdmann’s classification of tame blocks [6] and work of Cabanes and Picaronny [4, 5] on perfect isometries between tame blocks.

Introduction

Throughout these notes, O is a complete discrete valuation ring with algebraically closed residue field k of characteristic 2 and with quotient field K of characteristic 0. According to Erdmann’s classification in [6], if G is a finite group and if b is a block of OG having the quaternion group Q_{8} of order 8 as defect group, then the block algebra kGb̄ is Morita equivalent to either kQ_{8} or kÃ_{4} or the principal block algebra of kÃ_{5}, where here b̄ is the canonical image of b in kG. In the first case the block is it nilpotent (cf. [3]), and it follows from Puig’s structure theorem of nilpotent blocks in [8] that OGb is Morita equivalent to OQ_{8}. In the remaining two cases one should expect that OGb is Morita equivalent to OÃ_{4} or the principal block algebra of OÃ_{5}, respectively. We show this to be true in one of these two cases under the assumption that K is large enough:

Theorem A. Let G be a finite group, and let b be a block of OG having a quaternion defect group of order 8. Denote by b̄ the image of b in kG. Assume that KGb is split. If kGb̄ is Morita equivalent to kÃ_{4} then OGb is Morita equivalent to OÃ_{4}.

By Cabanes-Picaronny [4, 5], in the situation of Theorem A there is a perfect isometry between the character groups of OGb and of OÃ_{4}. Thus Theorem A is a consequence of the following slightly more general Theorem which characterises OGb in terms of its center, its character group and kÃ_{4}; see the end of this section for more details regarding the notation.

Typeset by AMS-TEX

Theorem B. Let A be an O-free O-algebra such that K ⊗_{O} A is split semi-simple

and such that k ⊗_{O} A is Morita equivalent to kÃ_{4}. Assume that there is an isometry

Φ : ℤIrr_{K}(A) ≅ ℤIrr_{K}(OÃ_{4}) which maps Proj(A) to Proj(OÃ_{4}) such that the map sending e(χ) to e(Φ(χ)) for every χ ∈ Irr_{K}(A) induces an O-algebra isomorphism of the centers Z(A) ≅ Z(OÃ_{4}). Then A is Morita equivalent to OÃ_{4}.

Theorem B is in turn a consequence of the more precise Theorem C, describing A in terms of generators and relations:

Theorem C. Let A be a basic O-free O-algebra such that K ⊗_{O} A is split semi-simple

and such that k ⊗_{O} A is isomorphic to kÃ_{4}. Assume that there is an isometry Φ :

ℤIrr_{K}(A) ≅ ℤIrr_{K}(OÃ_{4}) which maps Proj(A) to Proj(OÃ_{4}) such that the map sending e(χ) to e(Φ(χ)) for every χ ∈ Irr_{K}(A) induces an O-algebra isomorphism of the centers Z(A) ≅ Z(OÃ_{4}). Then A is isomorphic to the unitary O-algebra with set of generators {e_{0}, e_{1}, e_{2}, β, γ, δ, η, λ, κ} of A, such that e_{0}, e_{1}, e_{2} are pairwise orthogonal idempotents whose sum is 1 and satisfying the following relations:

β = e_{0}β = βe_{1}, γ = e_{1}γ = γe_{0} ;
δ = e_{1}δ = δe_{2}, η = e_{2}η = ηe_{1} ;
λ = e_{2}λ = λe_{0}, κ = e_{0}κ = κe_{2} ;
βδ = −2κ + κλκ ; ηγ = −2λ + λκλ ; δλ = −2γ + γβγ ;
κη = −2β + βγβ ; λβ = −2η + ηδη ; γκ = −2δ + δηδ ;
γβδ = −4δ + 2δηδ ; δηγ = −4γ + 2γβγ ; λκη = −4η + 2ηδη ;
βγκ = −4κ + 2κλκ ; ηδλ = −4λ + 2λκλ ; κλβ = −4β + 2βγβ ;
ηγβ = −4η + 2ηδη ; βδη = −4β + 2βγβ ; δλκ = −4δ + 2δηδ ;
λβγ = −4λ + 2λκλ ; κηδ = −4κ + 2κλκ ; γκλ = −4γ + 2γβγ ;
βδλβ = −8β + 4βγβ ; δλβδ = −8δ + 4δηδ ; λβδλ = −8λ + 4λκλ ;

When reduced modulo 2, these relations seem to be more than those occuring in Erdmann’s work [6] over k (we recall these more precisely in §2); but they are not, since the extra relations over k can be deduced from those given by Erdmann. We need to add in extra relations over O in order to ensure that the algebra we construct is O-free of the right rank.

Since OÃ_{4} fulfills the hypotheses of Theorem C it follows that A ≅ OÃ_{4}, hence Theorem C indeed implies Theorem B. The proof of Theorem C is given at the end of Section 2.

Notation. If A is an O-algebra such that K ⊗_{O} A is split semi-simple, we denote by

Irr_{K}(A) the set of characters of the simple K ⊗_{O} A-modules, viewed as central functions

from A to O and we denote by Irr_{k}(k ⊗_{O} A) the set of isomorphism classes of simple

k ⊗_{O} A-modules. We denote by ℤIrr_{K}(A) the group of characters of A, and we denote

by Proj(A) the subgroup of ℤIrr_{K}(A) generated by the characters of the projective indecomposable A-modules. We denote by L^{0}(A) the subgroup of ℤIrr_{K}(A) of all elements which are orthogonal to Proj(A) with respect to the usual scalar product in ℤIrr_{K}(A). For any χ ∈ Irr_{K}(A), we denote by e(χ) the corresponding primitive idempotent in Z(K ⊗_{O} A). If A = OG for some finite group G we have the well-known

formula

e(χ) = \frac{χ(1)}{|G|} ∑_{x∈G} χ(x^{−1})x .

We refer to [1, 2] for the concept and basic properties of perfect isometries, and to [9] for general block theoretic background material.

1 Characters and perfect isometries of OÃ_{4}

We identify Ã_{4} = Q_{8} ⋊ C_{3}. Let t be a generator of C_{3} and let y be an element of order 4 in Q_{8}. Set z = y^{2}; that is, z is the unique central involution of Ã_{4}. Then the seven elements 1, z, y, t, t^{2}, tz, t^{2}z are a complete set of representatives of the conjugacy classes in Ã_{4}.

Let ω be a primitive third root of unity in O. The character table of Ã_{4} is as follows:

1 z y  t  t^{2} tz t^{2}z
η_{0}  1 1 1  1  1 1 1
η_{1}  1 1 1  ω  ω^{2} ω ω^{2}
η_{2}  1 1 1 ω^{2}  ω ω^{2} ω
η_{3}  3 3 −1 0  0 0 0
η_{4}  2 −2 0 −ω^{2} −ω ω^{2} ω
η_{5}  2 −2 0 −ω −ω^{2} ω ω^{2}
η_{6}  2 −2 0 −1 −1 1 1

The algebra OÃ_{4} has three simple modules T_{0}, T_{1}, T_{2}, up to isomorphism. Choosing for T_{0} the trivial module and after possibly exchanging the notation for T_{1}, T_{2}, the ordinary decomposition matrix of OÃ_{4} is as follows:

( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{matrix} )

The Cartan matrix of OÃ_{4} is the product of the decomposition matrix with its transpose, hence equal to

( \begin{matrix}4 & 2 & 2 \\ 2 & 4 & 2 \\ 2 & 2 & 4 \end{matrix} )

Let e_{0}, e_{1}, e_{2} be primitive idempotents in OÃ_{4} such that OÃ_{4}e_{i} is a projective cover of T_{i}, 0 ≤ i ≤ 2. By the above decomposition matrix, the characters of the projective indecomposable OÃ_{4}-modules OÃ_{4}e_{i} are

η_{0} + η_{3} + η_{4} + η_{5} ,
η_{1} + η_{3} + η_{4} + η_{6} ,
η_{2} + η_{3} + η_{5} + η_{6} ,

respectively. Their norm is 4, and the differences of any two different characters of projective indecomposable OÃ_{4}-modules yields the following further elements in Proj(OÃ_{4}) having also norm 4:

η_{0} − η_{1} + η_{5} − η_{6} ,
η_{0} − η_{2} + η_{4} − η_{6} ,
η_{1} − η_{2} + η_{4} − η_{5} .

It is easy to check, that up to signs, these are all elements in Proj(OÃ_{4}) having norm 4.

A self-isometry Φ of ℤIrr_{K}(OÃ_{4}) maps every η_{i} to ϵ_{i}η_{π(i)} for some signs ϵ_{i} ∈ {1, −1} and a permutation π of {0, 1, …, 6}. In other words, Φ is determined by the permutation τ of the set {1, −1} × {0, 1, …, 6} satisfying τ(1, i) = (ϵ_{i}, π(i)) and τ(−1, i) = (−ϵ_{i}, π(i)) for all i, 0 ≤ i ≤ 6. If we write i, −i instead of (1, i), (−1, i), respectively, this becomes τ(i) = ϵ_{i}π(i) and τ(−i) = −ϵ_{i}π(i), with the usual cancellation rules for signs. In this way, every self-isometry Φ of ℤIrr_{K}(OÃ_{4}) gets identified to a permutation of the set of symbols {i, −i|0 ≤ i ≤ 6}.

A perfect self-isometry of ℤIrr_{K}(OÃ_{4}) preserves necessarily Proj(OÃ_{4}). The next Proposition implies that the converse is true, too:

Proposition 1.1. The group of all perfect self-isometries of ℤIrr_{K}(OÃ_{4}) is equal to the group of all self-isometries of ℤIrr_{K}(OÃ_{4}) which preserve Proj(OÃ_{4}). This group is generated by −Id together with the set of permutations

(0, 1, 2)(4, 6, 5) ,
(1, 2)(4, 5) ,
(2, −3)(5, −6) .

Every algebra automorphism of OÃ_{4} induces a permutation on Irr_{K}(OÃ_{4}) which is in fact a perfect isometry on ℤIrr_{K}(OÃ_{4}). Since η_{1} has degree 1, it is an algebra homomorphism from OÃ_{4} to O, and hence the map sending x ∈ OÃ_{4} to η_{1}(x)x is an algebra automorphism of OÃ_{4} whose inverse sends x ∈ OÃ_{4} to η_{2}(x)x. The following statement is an immediate consequence from the character table of OÃ_{4}:

Lemma 1.2. Let γ be the algebra automorphism of OÃ_{4} defined by γ(x) = η_{1}(x)x for all x ∈ OÃ_{4}. The permutation π of {0, 1, …, 6} defined by η_{i} ∘ γ = η_{π(i)} is equal to π = (0, 1, 2)(4, 6, 5).

The anti-automorphism of OÃ_{4} sending x ∈ Ã_{4} to x^{−1} induces also a permutation of the set Irr_{K}(OÃ_{4}), and this is also a perfect isometry (this holds for any finite group). This permutation can also be read off the character table:

Lemma 1.3. Let ι be the algebra anti-automorphism of OÃ_{4} mapping x ∈ Ã_{4} to x^{−1}. The permutation π of {0, 1, …, 6} defined by η_{i} ∘ ι = η_{π(i)} is equal to π = (1, 2)(4, 5).

Proof of 1.1. The first two permutations are perfect isometries by 2.2 and 2.3, respectively. An easy but painfully long verification shows that the bicharacter sending (g, h) ∈ Ã_{4} × Ã_{4} to

η_{0}(g)η_{0}(h)+η_{1}(g)η_{1}(h)−η_{2}(g)η_{3}(h)−η_{3}(g)η_{2}(h)+η_{4}(g)η_{4}(h)−η_{5}(g)η_{6}(h)−η_{6}(g)η_{5}(h)

is perfect; that is, its value at any (g, h) is divisible in O by the orders of CÃ_{4}(g) and CÃ_{4}(h) and it vanishes if exactly one of g, h has odd order. Thus the isometry given by the permutation (2, −3)(5, −6) is perfect. It remains to show that these permutations, together with −Id, generate the group of all self-isometries which preserve Proj(OÃ_{4}).

We described above a complete list of all elements in Proj(OÃ_{4}) having norm 4. Since the characters of the projective indecomposable modules are in that list, a selfisometry Φ of ℤIrr_{K}(OÃ_{4}) preserves Proj(OÃ_{4}) if and only if it permutes this set of norm 4 elements.

Let Φ be a self-isometry of ℤIrr_{K}(OÃ_{4}) which preserves Proj(OÃ_{4}). Then Φ preserves also the group L^{0}(OÃ_{4}) of generalised characters which are orthogonal to all characters in Proj(OÃ_{4}). Up to signs, the complete list of elements in L^{0}(OÃ_{4}) having norm 3 is

η_{0} + η_{1} − η_{4} , η_{0} + η_{2} − η_{5} , η_{0} − η_{3} + η_{6} ,
η_{1} + η_{2} − η_{6} , η_{1} − η_{3} + η_{5} , η_{2} − η_{3} + η_{4} .

Up to signs again, the complete list of elements in L^{0}(OÃ_{4}) having norm 4 is

η_{0} + η_{1} + η_{2} − η_{3} ,
η_{0} − η_{1} − η_{5} + η_{6} , η_{0} − η_{2} − η_{4} + η_{6} , η_{0} + η_{3} − η_{4} − η_{5} ,
η_{1} − η_{2} − η_{4} + η_{5} , η_{1} + η_{3} − η_{4} − η_{6} , η_{2} + η_{3} − η_{5} − η_{6} .

The first norm 4 element in this list, η_{0} + η_{1} + η_{2} − η_{3}, is the only norm 4 element which is orthogonal to all other norm 4 elements in L^{0}(OÃ_{4}). Thus Φ has to permute the characters η_{0}, η_{1}, η_{2}, η_{3} amongst each other.

Suppose first that Φ fixes η_{3}. Then, by composing Φ with a suitable product of powers of the first two permutations in the statement, we may assume that Φ fixes η_{0}, η_{1}, η_{2} up to signs. By considering the first of the above norm 4 elements in L^{0}(OÃ_{4}) we get that Φ fixes η_{0}, η_{1}, η_{2} all with positive signs. By considering the norm 3 elements in L^{0}(OÃ_{4}), it follows that Φ fixes also η_{4}, η_{5} and η_{6} with positive signs. Thus a self-isometry of ℤIrr_{K}(OÃ_{4}) which preserves Proj(OÃ_{4}) and which fixes η_{3} is in the group generated by the set of two permutations (0, 1, 2)(4, 6, 5) and (1, 2)(4, 5).

Suppose next that Φ does not fix η_{3}. By precomposing Φ with a suitable of power of (0, 1, 2)(4, 6, 5) we may assume that Φ sends η_{2} to −η_{3}. By composing Φ with a suitable power of (0, 1, 2)(4, 5, 6) we may assume that Φ fixes η_{0}, up to a sign. Since Φ preserves the norm 4 element η_{0} + η_{1} + η_{2} − η_{3}, we necessarily have Φ(η_{0}) = η_{0}. Then Φ maps η_{1} either to η_{1} or η_{2} (with positive signs, again because of that same norm 4 element). In the first case, Φ fixes both η_{0}, η_{1}, and by checking the norm 3 elements in L^{0}(OÃ_{4}) one gets Φ = (2, −3)(5, −6). In the second case, again checking on norm 3 elements, one gets Φ = (1, 2, −3)(4, 5, −6), but this is already the product of (1, 2)(4, 5) and (2, −3)(5, −6). □

2 The algebra A

Let A be a basic O-algebra fulfilling the hypotheses of Theorem B; that is, K ⊗_{O} A

is split semi-simple, k ⊗_{O} A is isomorphic to kÃ_{4}, and there is an isometry ℤIrr_{K}(A) ≅

ℤIrr_{K}(OÃ_{4}) mapping Proj(A) to Proj(OÃ_{4}) and inducing an isomorphism Z(A) ≅ Z(OÃ_{4}). There is a “compatible choice” for these isomorphisms:

Proposition 2.1. There is an algebra isomorphism α : k ⊗_{O} A ≅ kÃ_{4} and an isom-

etry Φ : ℤIrr_{K}(A) ≅ ℤIrr_{K}(OÃ_{4}) mapping Proj(A) to Proj(OÃ_{4}) with the following properties:

(i) Φ maps Irr_{K}(A) onto Irr_{K}(OÃ_{4}); that is, all signs are +1.

(ii) The map sending e(χ) to e(Φ(χ)) for every χ ∈ Irr_{K}(A) induces an isomorphism Z(A) ≅ Z(OÃ_{4}).

(iii) For any primitive idempotents e ∈ A and f ∈ OÃ_{4} and every χ ∈ Irr_{K}(A) such that α(ē) = f̄ we have χ(e) = Φ(χ)(f); that is, A and OÃ_{4} have the same decomposition matrices through α and Φ.

Proof. The O-rank of A is 24 and also the sum of the squares of the seven irreducible K-linear characters of A; thus every irreducible character of A has degree smaller than 5. Also, there is no character of degree 4 because 24 − 4^{2} = 8 cannot be written as a sum of six squares of the six remaining characters. But there must be a character of degree 3; if not, 24 would be the sum of seven squares all either 1 or 4, which is not possible. Thus the squares of the six remaining characters add up to 24 − 3^{2} = 15, and the only way to do this is with three characters of degree 1 and three characters of degree 2.

This proves that the character degrees of the irreducible characters of A and of OÃ_{4} coincide for some bijection Irr_{K}(A) ≅ Irr_{K}(OÃ_{4}). Since the decomposition matrix of A multiplied with its transpose yields the Cartan matrix of A - which is equal to that of kÃ_{4} - the algebra A has in fact the same decomposition matrix as OÃ_{4} for a suitable bijection Φ : Irr_{K}(A) ≅ Irr_{K}(OÃ_{4}) and the bijection Irr_{k}(k ⊗_{O} A) ≅ Irr_{k}(kÃ_{4}) induced

by α. Extend Φ to a ℤ-linear isomorphism ℤIrr_{K}(A) ≅ ℤIrr_{K}(OÃ_{4}), still denoted by Φ. By construction, Φ sends the characters of the projective indecomposable A-modules to the characters of the projective indecomposable OÃ_{4}-modules; in particular, Φ maps Proj(A) to Proj(OÃ_{4}). It remains to see that the map sending e(χ) to e(Φ(χ)) for every χ ∈ Irr_{K}(A) induces an isomorphism Z(A) ≅ Z(OÃ_{4}). For any i, 0 ≤ i ≤ 6, denote by χ_{i} the irreducible character of A such that Φ(χ_{i}) = η_{i}. As in the proof of 1.1, we have a distinguished norm 4 element in L^{0}(A) which is orthogonal to all other norm 4 elements in L^{0}(A), namely χ_{0} +χ_{1} +χ_{2} −χ_{3}. Thus, if Ψ : ℤIrr_{K}(A) ≅ ℤIrr_{K}(OÃ_{4}) is some isometry mapping Proj(A) to Proj(OÃ_{4}) and inducing an isomorphism Z(A) ≅ Z(OÃ_{4}), then Ψ(χ_{0} + χ_{1} + χ_{2} − χ_{3}) = ±(η_{0} + η_{1} + η_{2} − η_{3}). By Proposition 1.1, there is a perfect self-isometry μ of ℤIrr_{K}(OÃ_{4}) such that Φ = μ ∘ Ψ. □

Remark 2.2. If we assume that A is Morita equivalent to some block algebra with Q_{8} as defect group, then Proposition 2.1 follows also from the work of Cabanes and Picaronny in [4, 5].

Since k ⊗_{O} A ≅ kÃ_{4}, the quiver of A is the same as that of kÃ_{4}, thus of the following

form:

\xymatrix{0 \ar[rr]^{β} \ar[dr]^{κ} &  & 1 \ar[ll]^{γ} \ar[dl]^{δ} \\  & 2 \ar[ul]^{λ} \ar[ur]^{η}}

Write ā for the image of a ∈ A in Ā = k ⊗_{O} A ≅ kÃ_{4}. The generators β, γ, δ, κ, λ, η

can be chosen such that their images in Ā fulfill the following relations:

β̄δ̄ = κ̄λ̄κ̄ ,
η̄γ̄ = λ̄κ̄λ̄ ,
δ̄λ̄ = γ̄β̄γ̄ ,
κ̄η̄ = β̄γ̄β̄ ,
λ̄β̄ = η̄δ̄η̄ ,
γ̄κ̄ = δ̄η̄δ̄

and

γ̄β̄δ̄ = δ̄η̄γ̄ = λ̄κ̄η̄ = 0 .

In order to determine the algebra structure of A, we have to “lift” these relations over O.

We fix an algebra isomorphism α : k ⊗_{O} A ≅ kÃ_{4} and an isometry Φ : ℤIrr_{K}(A) ≅

ℤIrr_{K}(OÃ_{4}) satisfying the conclusions of Proposition 2.1. We denote by χ_{i} the unique irreducible K-linear character of A such that Φ(χ_{i}) = η_{i} for all i, 0 ≤ i ≤ 6.

The characters η_{0}, η_{1}, η_{2}, η_{3} of OÃ_{4} have height zero, the characters η_{4}, η_{5}, η_{6} have height one. Thus, via the isomorphism of the centers induced by Φ, it follows that for 0 ≤ i ≤ 3 we have 8e(χ_{i}) ∈ A, and for 4 ≤ j ≤ 6 we have 4e(χ_{j}) ∈ A. We can in fact describe an O-basis of Z(A) in terms of the centrally primitive idempotents e(χ_{i}). The strategy is now to play off the descriptions of Z(k ⊗_{O} A) in terms of the generators in

the quiver and of Z(A) in terms of the centrally primitive idempotents e(χ_{i}).

Lemma 2.3. The following elements of Z(K ⊗_{O} A) are all contained in the radical

J(Z(A)):

s = 2e(χ_{4}) + 2e(χ_{5}) + 2e(χ_{6}) ,
z_{0} = 4e(χ_{2}) + 4e(χ_{3}) + 2e(χ_{4}) ,
z_{1} = 4e(χ_{1}) + 4e(χ_{3}) + 2e(χ_{5}) ,
z_{2} = 4e(χ_{0}) + 4e(χ_{3}) + 2e(χ_{6}) ,
y_{0} = 4e(χ_{1}) + 4e(χ_{2}) + 2e(χ_{4}) + 2e(χ_{5}) ,
y_{1} = 4e(χ_{0}) + 4e(χ_{2}) + 2e(χ_{4}) + 2e(χ_{6}) ,
y_{2} = 4e(χ_{0}) + 4e(χ_{1}) + 2e(χ_{5}) + 2e(χ_{6}) .

Moreover, for any two different i, j in {0, 1, 2} the set

{1, z_{i}, z_{j}, s, 8e(χ_{3}), 4e(χ_{i+4}), 4e(χ_{j+4})}

is an O-basis of Z(A).

Proof. In view of Proposition 2.1 we may assume that A = OÃ_{4}. This is just an explicit verification, using the character table of Ã_{4}. One verifies first that z_{0} ∈ A. By symmetry, this implies that z_{1}, z_{2} are also in A. Then y_{0} = z_{0} + z_{1} − 8e(χ_{3}) is in A, similarly for the y_{1}, y_{2}. An equally easy computation shows that s ∈ A. Thus all the given elements belong to Z(A). None of these elements is invertible, so they all belong to J(Z(A)) because Z(A) is local.

In order to see the last statement on the basis of Z(A), we may assume that i = 0 and j = 1. For any x ∈ Ã_{4} denote by \underline{x} the conjugacy class sum of x in OÃ_{4}. The orthogonality relations imply the well-known formula

\underline{x} = ∑_{0≤m≤6} \frac{χ_{m}(\underline{x}^{−1})}{χ_{m}(1)}e(χ_{m}) .

Thus, for the seven conjugacy classes in Ã_{4}, we have

\underline{1} = e(χ_{0}) + e(χ_{1}) + e(χ_{2}) + e(χ_{3}) + e(χ_{4}) + e(χ_{5}) + e(χ_{6}) ;
\underline{z} = e(χ_{0}) + e(χ_{1}) + e(χ_{2}) + e(χ_{3}) − e(χ_{4}) − e(χ_{5}) − e(χ_{6}) ;
\underline{y} = 6e(χ_{0}) + 6e(χ_{1}) + 6e(χ_{2}) − 2e(χ_{3}) ;
\underline{t} = 4e(χ_{0}) + 4ω^{2}e(χ_{1}) + 4ωe(χ_{2}) − 2ωe(χ_{4}) − 2ω^{2}e(χ_{5}) − 2e(χ_{6}) ;
\underline{t}^{2} = 4e(χ_{0}) + 4ωe(χ_{1}) + 4ω^{2}e(χ_{2}) − 2ω^{2}e(χ_{4}) − 2ωe(χ_{5}) − 2e(χ_{6}) ;
\underline{tz} = 4e(χ_{0}) + 4ω^{2}e(χ_{1}) + 4ωe(χ_{2}) + 2ωe(χ_{4}) + 2ω^{2}e(χ_{5}) + 2e(χ_{6}) ;

\underline{t^{2}z} = 4e(χ_{0}) + 4ωe(χ_{1}) + 4ω^{2}e(χ_{2}) + 2ω^{2}e(χ_{4}) + 2ωe(χ_{5}) + 2e(χ_{6}) .

We show that they are all in the O-linear span of the elements in the set

{1, z_{0}, z_{1}, s, 8e(χ_{3}), 4e(χ_{4}), 4e(χ_{5})} .

Note first that

z_{2} = 4 · 1 − z_{0} − z_{1} − s + 8e(χ_{3}),
4e(χ_{6}) = 2s − 4e(χ_{4}) − 4e(χ_{5})

are in the O-linear span of this set. One easily verifies now that

\underline{z} = 1 − s ,
\underline{y} = 6 · 1 − 3s − 8e(χ_{3}) ,
\underline{t} = ωz_{0} + ω^{2}z_{1} + z_{2} − 4ωe(χ_{4}) − 4ω^{2}e(χ_{5}) − 4e(χ_{6}) ,
\underline{t}^{2} = ω^{2}z_{0} + ωz_{1} + z_{2} − 4ω^{2}e(χ_{4}) − 4ωe(χ_{5}) − 4e(χ_{6}) ,
\underline{tz} = ωz_{0} + ω^{2}z_{1} + z_{2} ,
\underline{t^{2}z} = ω^{2}z_{0} + ωz_{1} + z_{2} .

This concludes the proof of 2.3 □

The center of Ā = k ⊗_{O} A can easily be described in terms of the generators in the

quiver of A:

Lemma 2.4. The following set is a k-basis of Z(Ā).

{1, β̄γ̄ + γ̄β̄, κ̄λ̄ + λ̄κ̄, η̄δ̄ + δ̄η̄, β̄δ̄λ̄, δ̄λ̄β̄, λ̄β̄δ̄} .

Proof. Straightforward verification, using (β̄γ̄)^{2} = β̄δ̄λ̄ and the similar relations for the other elements in the given set. □

Proposition 2.5. For any primitive idempotent e in A we have Z(A)e = eAe. Moreover,

(i) the set {e_{0}, z_{0}e_{0}, z_{1}e_{0}, 4e(χ_{4})e_{0}} is an O-basis of e_{0}Ae_{0}.

(ii) the set {e_{1}, z_{0}e_{1}, z_{2}e_{1}, 4e(χ_{4})e_{1}} is an O-basis of e_{1}Ae_{1};

(iii) the set {e_{2}, z_{1}e_{2}, z_{2}e_{2}, 4e(χ_{5})e_{2}} is an O-basis of e_{2}Ae_{2}.

Proof. Since Z(A) ≅ Z(OÃ_{4}) and Z(Ā) ≅ Z(kÃ_{4}), the canonical map A → Ā maps Z(A) onto Z(Ā) and hence Z(A)e onto Z(Ā)ē. By Nakayama’s Lemma, it suffices to show that Z(Ā)ē = ēĀē. Now dim_{k}(ēĀē) = 4 by the Cartan matrix, and so we have only to show that dim_{k}(Z(Ā)ē) = 4. By the symmetry of the quiver of A, we may assume that e corresponds to the vertex labelled 0. Then the set {ē, β̄γ̄, κ̄λ̄, β̄δ̄λ̄} is a k-basis of Z(Ā)ē by 2.4; in particular, dim_{k}(Z(Ā)ē) = 4 as required. This shows that eAe = Z(A)e.

In order to prove (i), note that the set

{e_{0}, z_{0}e_{0}z_{1}e_{0}, se_{0}, 8e(χ_{3})e_{0}, 4e(χ_{4})e_{0}, 4e(χ_{5})e_{0}}

generates e_{0}Ae_{0} as O-module, by the first statement and by the O-basis of Z(A) described in 2.3. Now we have

8e(χ_{3})e_{0} = 2z_{0}e_{0} − 4e(χ_{4})e_{0} ,
4e(χ_{5})e_{0} = 2z_{0}e_{0} − 2z_{1}e_{0} + 4e(χ_{4})e_{0} ,
se_{0} = (z_{1} − z_{0} + 4e(χ_{4}))e_{0} .

Thus the set given in (i) generates e_{0}Ae_{0} as O-module, and hence is a basis since the O-rank of e_{0}Ae_{0} is 4. The same arguments show (ii), (iii). □

Proposition 2.6. We can choose the generators β, γ, δ, η, λ, κ in such a way that (i) Aγ is the unique O-pure submodule of Ae_{0} with character χ_{3} + χ_{4}; (ii) Aλ is the unique O-pure submodule of Ae_{0} with character χ_{3} + χ_{5}; (iii) Aη is the unique O-pure submodule of Ae_{1} with character χ_{3} + χ_{6}; (iv) Aβ is the unique O-pure submodule of Ae_{1} with character χ_{3} + χ_{4};

(v) Aκ is the unique O-pure submodule of Ae_{2} with character χ_{3} + χ_{5}; (vi) Aδ is the unique O-pure submodule of Ae_{2} with character χ_{3} + χ_{6}.

Proof. We are going to prove (i); by the symmetry of the quiver of A one gets all other statements. Observe first that Āγ̄ is the unique 5-dimensional submodule of Ae_{0} with composition factors 2[S_{0}], 2[S_{1}], [S_{2}]. Indeed, the set {γ̄, β̄γ̄, η̄γ̄, γ̄β̄γ̄, β̄γ̄β̄γ̄} is a k-basis of Āγ̄, and we have γ̄, γ̄β̄γ̄ ∈ ē_{0}Āē_{0}, yielding the two composition factors isomorphic to S_{0}, we have β̄γ̄, β̄γ̄β̄γ̄ ∈ ē_{1}Āē_{0}, yielding the two composition factors isomorphic to S_{1}, and finally η̄γ̄ ∈ ē_{2}Āē_{0}, yielding the remaining composition factor isomorphic to S_{2}. One checks that there is no other submodule with exactly these composition factors. Now there is exactly one O-pure submodule U of Ae_{0} whose reduction modulo J(O) has composition series 2[S_{0}] + 2[S_{1}] + [S_{2}], namely the unique O-pure submodule of Ae_{0} with character χ_{3} + χ_{4}; this is a direct consequence of the decomposition matrix. One constructs U as follows: write K ⊗_{O} Ae_{0} = X_{0} ⊕ X_{3} ⊕ X_{4} ⊕ X_{5}, where X_{j} is

the unique submodule of K ⊗_{O} Ae_{0} with character χ_{j} for j ∈ {0, 3, 4, 5}, and then

U = Ae_{0} ∩ (X_{3} ⊕ X_{4}). Take now for γ any inverse image in U of γ̄. Then Aγ ⊆ U and U ⊆ Aγ + J(O)U. Thus Aγ = U by Nakayama’s Lemma. □

Corollary 2.7. If the generators β, γ, δ η, λ, κ are chosen such that they fulfill the conclusions of 2.6 then, with the notation of 2.3, the following hold.

(i) y_{0}δ = y_{0}η = 0.

(ii) y_{1}λ = y_{1}κ = 0.

(iii) y_{2}γ = y_{2}β = 0.

Proposition 2.8. We can choose the generators β, γ, δ, η, λ, κ such that the following holds:

βγ = z_{0}e_{0} = 4e(χ_{3})e_{0} + 2e(χ_{4})e_{1};
γβ = z_{0}e_{1} = 4e(χ_{3})e_{1} + 2e(χ_{4})e_{1};
δη = z_{2}e_{1} = 4e(χ_{3})e_{1} + 2e(χ_{6})e_{1};
ηδ = z_{2}e_{2} = 4e(χ_{3})e_{2} + 2e(χ_{6})e_{2};
λκ = z_{1}e_{2} = 4e(χ_{3})e_{2} + 2e(χ_{5})e_{2};
κλ = z_{1}e_{0} = 4e(χ_{3})e_{0} + 2e(χ_{5})e_{0};
βδλ = κηγ = 8e(χ_{3})e_{0};
δλβ = γκη = 8e(χ_{3})e_{1};
λβδ = ηγκ = 8e(χ_{3})e_{2}.

Proof. In view of the decomposition matrix of A we have e_{0} = e(χ_{0})e_{0} + e(χ_{3})e_{0} + e(χ_{4})e_{0} + e(χ_{5})e_{0}. Moreover, the elements e(χ_{0})e_{0}, e(χ_{3})e_{0}, e(χ_{4})e_{0}, e(χ_{5})e_{0} are Klinearly independent because they are pairwise orthogonal idempotents in K ⊗_{O}A. Sim-

ilar statements hold for e_{1}, e_{2}.

We assume a choice of generators fulfilling 2.6. We have Aβγ ⊆ Aγ, and the submodule Aγ of Ae_{0} has character χ_{3} +χ_{4} by 2.6. Thus βγ is a K-linear combination of e(χ_{3})e_{1} and e(χ_{4})e_{1}. But also βγ is an O-linear combination of the basis elements e_{1}, z_{0}e_{1} z_{1}e_{1}, 4e(χ_{4})e_{1} given in 2.5 in which none of χ_{1}, χ_{5} shows up. Therefore βγ is in fact an O-linear combination of the elements z_{0}e_{0}, 4e(χ_{4})e_{0}; say

βγ = (μ_{0}z_{0}e_{0} + 4ν_{0}e(χ_{4}))e_{0} = (4μ_{0}e(χ_{3}) + 2(μ_{0} + 2ν_{0})e(χ_{4}))e_{0}

for some coefficients μ_{0}, ν_{0} ∈ O. Hence

(βγ)^{2} = (16μ_{0}^{2}e(χ_{3}) + 4(μ_{0} + 2ν_{0})^{2}e(χ_{4}))e_{0} .

Now (β̄γ̄)^{2} ≠ 0, and therefore μ_{0} ∈ O^{×}. Set now

a_{0} = 1 + ν_{0}μ_{0}^{−1}y_{0} .

Since y_{0} ∈ J(Z(A)) by 2.3 we have a_{0} ∈ Z(A)^{×}. A trivial verification, comparing coefficients, shows that we have

βγ = μ_{0}z_{0}a_{0}e_{0} .

Since γ = e_{1}γ = γe_{0}, multiplying this with γ on the left yields

γβγ = μ_{0}z_{0}a_{0}e_{1}γ .

Now both γβ and μ_{0}z_{0}a_{0}e_{1} are contained in the pure submodule Aβ of Ae_{1} with character χ_{3}+χ_{4}, by 2.6 and the nature of the element z_{0}. Right multiplication by γ on this submodule is therefore injective (the annihilator of γ in Ae_{1} is the pure submodule with character χ_{1} + χ_{6}). Hence the previous equality implies also the equality

γβ = μ_{0}z_{0}a_{0}e_{1} .

In an entirely analogous way one finds scalars μ_{1}, μ_{2} ∈ O^{×} such that, setting a_{1} = 1 + ν_{1}μ_{1}^{−1}y_{1} and a_{2} = 1 + ν_{2}μ_{2}^{−1}y_{2}, one gets the equalities

δη = μ_{2}z_{2}a_{2}e_{1} , ηδ = μ_{2}z_{2}a_{2}e_{2} ,
λκ = μ_{1}z_{1}a_{1}e_{2} , κλ = μ_{1}z_{1}a_{1}e_{0} .

Moreover, the equalities in 2.7 imply the following equalities:

a_{0}δ = δ , a_{0}η = η ,
a_{1}λ = λ , a_{1}κ = κ ,
a_{2}γ = γ , a_{2}β = β .

If we replace now β by a_{0}β, this is not going to change the properties stated in 2.6 and also this is not changing the relations over k of the quiver. Similarly, we can replace δ by a_{2}δ and λ by a_{1}λ. Then the generators β, γ, δ, η, λ, κ still fulfill 2.6, and in addition, we have now the following equalities:

βγ = μ_{0}z_{0}e_{0} , γβ = μ_{0}z_{0}e_{1} ,
δη = μ_{2}z_{2}e_{1} , ηδ = μ_{2}z_{2}e_{2} ,
λκ = μ_{1}z_{1}e_{2} , κλ = μ_{1}z_{1}e_{0} .

We have to get rid of the scalars μ_{0}, μ_{1}, μ_{2}. Since χ_{3} is the only character appearing in the characters of all projective indecomposable A-modules we have

βδλ = 8μe(χ_{3})e_{0}

for some μ ∈ O. Then actually μ ∈ O^{×} because β̄δ̄λ̄ ≠ 0. Moreover, βδλβ = 8μe(χ_{3})β, and hence also

δλβ = 8μe(χ_{3})e_{1} .

The same argument applied again yields

λβδ = 8μe(χ_{3})e_{2} .

Applying this argument to the arrows in the quiver in the opposite direction implies that there is μ′ ∈ O^{×} such that

κηγ = 8μ′e(χ_{3})e_{0} ,
ηγκ = 8μ′e(χ_{3})e_{2} ,
γκη = 8μ′e(χ_{3})e_{1} .

Now β̄δ̄λ̄ = κ̄λ̄κ̄λ̄ = κ̄η̄γ̄, and hence μ′ = μ(1 + ν) for some ν ∈ J(O). Note that we can always multiply any of the generators by any scalar in 1 + J(O) without modifying the relations over k. Thus, if we replace κ by (1 + ν)κ, we may assume that μ′ = μ.

Since the set {κ, κλκ} is an O-basis of e_{0}Ae_{2}, we can write

βδ = aκ + bκλκ

for some unique scalars a, b ∈ O. Multiplying this by λ yields

8μe(χ_{3})e_{0} = βδλ = aκλ + b(κλ)^{2} = (aμ_{1}z_{1} + bμ_{1}^{2}z_{1}^{2})e_{0} .

By comparing the coefficients at e(χ_{3})e_{0} and e(χ_{5})e_{0} of the left and right expression in this equality, we get the equations

8μ = 4aμ_{1} + 16bμ_{1}^{2} ,
0 = 2aμ_{1} + 4bμ_{1}^{2} .

An easy computation shows that b = \frac{μ}{μ_{1}^{2}} . Moreover, since β̄δ̄λ̄ = (κ̄λ̄)^{2} we have ā = 0 and b̄ = 1_{k}, hence b = \frac{μ}{μ_{1}^{2}} ∈ 1 + J(O). By repeating the same argument we find also that the coefficients \frac{μ}{μ_{0}^{2}} , \frac{μ}{μ_{2}^{2}} are in 1 + J(O).

Next, we compute βδλκηγ in two different ways: on one hand we have

(βδλ)(κηγ) = 64μ^{2}e(χ_{3})e_{0} ,

and on the other hand we have

β(δ(λκ)η)γ = μ_{0}μ_{1}μ_{2}z_{0}z_{1}z_{2}e(χ_{3})e_{0} = 64μ_{0}μ_{1}μ_{2}e(χ_{3})e_{0} .

Together we get

μ^{2} = μ_{0}μ_{1}μ_{2} .

Thus \frac{μ}{μ_{0}^{2}} \frac{μ}{μ_{1}^{2}} = \frac{μ_{2}}{μ_{0}μ_{1}} ∈ 1 + J(O). Similarly, \frac{μ_{1}}{μ_{0}μ_{2}} , \frac{μ_{0}}{μ_{1}μ_{2}} ∈ 1 + J(O). But then also

\frac{μ_{1}μ_{2}}{μ_{0}} \frac{μ_{1}}{μ_{0}μ_{2}} = \frac{μ_{1}^{2}}{μ_{0}^{2}} ∈ 1 + J(O). Since 2 ∈ J(O) this implies that \frac{μ_{1}}{μ_{0}} ∈ 1 + J(O). But then actually μ_{2} = \frac{μ_{1}μ_{2}}{μ_{0}} \frac{μ_{0}}{μ_{1}} ∈ 1 + J(O). Similarly, μ_{0}, μ_{1} ∈ 1 + J(O). So we can replace β by μ_{0}^{−1}β, or equivalently, we can assume that μ_{0} = 1. Similarly, we can assume that μ_{1} = μ_{2} = 1. Then μ^{2} = 1. If μ = −1 we multiply all generators by −1; since 2 ∈ J(O), this does not change the relations over k, but it does change the sign of any of the above expressions βδλ etc. involving three generators. Therefore, we can also assume that μ = 1.

□

We can now prove Theorem C from the introduction.

Proof of Theorem C. We assume a choice of generators of A fulfilling Proposition 2.8. We show that A satisfies the relations given in Theorem C. Those in the first three lines are obvious. Since the set {κ, κλκ} is an O-basis of e_{0}Ae_{2}, we can write

βδ = aκ + bκλκ

for some unique scalars a, b ∈ O. Multiplying this by λ yields

8e(χ_{3})e_{0} = βδλ = aκλ + b(κλ)^{2} = (4a + 16b)e(χ_{3})e_{0} + (2a + 4b)e(χ_{5})e_{0} .

By comparing the coefficients at e(χ_{3})e_{0} and e(χ_{5})e_{0} of the left and right expression in this equality, we get the equations

8 = 4a + 16b ,
0 = 2a + 4b .

Thus the coefficients a, b have values

a = −2 , b = 1 ,

and from this we get the following relation in the statement of Theorem C:

βδ = −2κ + κλκ .

In exactly the same way we get the following five relations in the Theorem:

ηγ = −2λ + λκλ ,
δλ = −2γ + γβγ ,
κη = −2β + βγβ ,

λβ = −2η + ηδη ,
γκ = −2δ + δηδ .

A similar technique is going to yield the remaining relations: write γβδ = cδ +dδηδ for some unique c, d ∈ O; as before, this is possible since {δ, δηδ} is an O-basis of e_{1}Ae_{2}. Multiplying by η yields

γβδη = cδη + d(δη)^{2} = cz_{2}e_{1} + dz_{2}^{2}e_{1} .

The left side is equal to (γβ)(δη) = z_{0}z_{2}e_{1}, so comparing coefficients yields now

16 = 4c + 16d ,
0 = 2c + 4d ,

and this implies c = −4 and d = 2. Thus we get indeed

γβδ = −4δ + 2δηδ

as claimed. The remaining relations of this type follow in exactly the same way.

Now consider the last three relations. Write βδλβ = rβ + sβγβ, for r, s ∈ O. Then βδλβγ = rβγ + sβγβγ. So

32e(χ_{3})e_{0} = (4r + 16s)e(χ_{3})e_{0} + (2r + 4s)e(χ_{4})e_{0}

which yields s = 4 and r = −8. The remaining two relations follow in exactly the same way. Thus A satisfies all relations given in Theorem C.

Let Ã be the O-algebra described by the generators and relations given in Theorem C. There is a surjective algebra morphism from Ã to A. In order to show that Ã and A are isomorphic it suffices therefore to show that the cardinality of a minimal generating set for A as an O-module is at most 24. Thus it suffices to check that the set

S :={e_{0}, e_{1}, e_{2}, β, γ, δ, η, λ, κ,
βγ, γβ, δη, ηδ, λκ, κλ,
βγβ, γβγ, δηδ, ηδη, λκλ, κλκ,
βδλ, δλβ, λβδ}

spans Ã as O-module. This is an easy consequence of the given relations; we give some details for the convenience of the reader: Let

G = {e_{0}, e_{1}, e_{2}, β, γ, δ, η, λ, κ}

From the given relations it is immediate that for any two elements x, y of G, xy is in the O-span of S. Thus it suffices to show that for any two elements x, y of G − {e_{0}, e_{1}, e_{2}} and any element u of S − {e_{0}, e_{1}, e_{2}, β, γ, δ, η, λ, κ}, xu and uy are in the O-span of S. From the given relations we may also assume that u is one of βγβ, γβγ, δηδ, ηδη, λκλ, κλκ or one of βδλ, δλβ, λβδ.

First, note that the relations κη = −2β + βγβ and δλ = −2γ + γβγ give that κηγ = βδλ. Similarly, we get ηγκ = λβδ and γκη = δλβ.

Now suppose u = βγβ. Then we may assume that x is one of γ or λ and that y is one of γ or δ. The relation κη = −2β + βγβ gives γκη = −2γβ + γβγβ, hence γβγβ is in the O-span of S. The relation κη = −2β + βγβ also gives λκη = −2λβ + λβγβ. It follows from the relation λκη = −4η + 2ηδη that λβγβ is in the O-span of S. We show similarly that βγβγ and βγβδ are in the O-span of S.

The cases u = γβγ, δηδ, ηδη, λκλ, κλκ are handled analogously.

Now suppose u = βδλ. Then we may assume that x is one of λ or γ and y is one of β or κ. The relation λβδλ = −8λ + 4λκλ shows that λβδλ is in the O-span of S. From the relation γβδ = −4δ + 2δηδ we get γβδλ = −4δλ + 2δηδλ. From γκ = −2δ + δηδ, we get δηδλ = γκλ + 2δλ. Hence δηδλ is in the O-span of S, and so is γβδλ. We argue similarly to show that βδλβ and βδλκ are in the O-span of S.

The cases u = δλβ and u = λβδ are handled in the same fashion. □

Remark 2.9. An interesting consequence of 2.5 is the structure of eAe for any primitive idempotent e in A. We have an O-algebra isomorphism

eAe ≅ O[X, Y]/ < X^{2} − Y^{2} − 2(X − Y) , XY − 2X^{2} + 4X > ;

indeed, we may assume that e = e_{0}, and then the assignment X ↦ z_{0}e_{0}, Y ↦ z_{1}e_{0} induces the required isomorphism. In particular, we have an isomorphism of k-algebras

ēĀē ≅ k[X, Y]/ < X^{2} − Y^{2} , XY > .

This is, by Erdmann [6, III.1, III.3], up to isomorphism the unique 4-dimensional symmetric k-algebra which is not isomorphic to the group algebra of the Klein four group. One might be tempted to ask whether any symmetric O-algebra is the endomorphism algebra of some projective module of some block algebra.

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3. M. Broué, L. Puig, A Frobenius theorem for blocks, Invent. Math. 56 (1980), 117–128.

4. M. Cabanes, C. Picaronny, Types of blocks with dihedral or quaternion defect groups, J. Fac. Sci. Univ. Tokyo 39 (1992), 141–161.

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6. K. Erdmann, Blocks of Tame Representation Type and Related Algebras, Springer Lecture Notes in Math. 1428 (1990).

7. W. Feit, The representation theory of finite groups, North-Holland, Amsterdam, 1982.

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Thorsten Holm Department of Pure Mathematics University of Leeds Leeds, LS2 9JT

U.K.

Radha Kessar Department of Mathematical Sciences Meston Building Aberdeen, AB24 3UE

U.K.

Markus Linckelmann Department of Mathematical Sciences Meston Building Aberdeen, AB24 3UE

U.K.