Some examples of Picard groups of blocks

We calculate the Picard groups for $2$-blocks with abelian defect groups of $2$-rank at most three with respect to a complete discrete valuation ring. In particular this shows directly that all such Picard groups are finite and the subgroup $\Piccent$ of Morita equivalences fixing the centre is trivial. These are amongst the first calculations of this kind. Further we prove some general results concerning Picard groups of blocks with normal defect groups as well as some other cases.


Introduction
. One is that, whilst the Picard group of a k-block is usually infinite, it is not clear whether the Picard group of an O-block of a finite group must be finite. A related question is whether every element of Pic(B) can be taken to be a bimodule with endopermutation source. There are also no known examples of blocks B where the subgroup Piccent(B) of Pic(B) inducing the identity map on Z(B) is nontrivial.
The purpose of this article is to find the Picard groups of some classes of examples, both to provide evidence for the main open questions, but also as tools for the classification of Morita equivalence classes. Previous examples where the Picard groups have been calculated are the blocks with cyclic or Klein four defect groups, and blocks of groups P ⋊ E where P is an abelian p-group with E abelian and [P, E] = P , all in [1]. Further it follows from [13] that Pic(B) is finite when B is the unique block of a finite group G with a self-centralizing normal p-subgroup.
In [10] and [23] the Morita equivalence classes of 2-blocks with abelian defect group of 2-rank at most three were classified. Our examples of Picard groups include those for representatives of each Morita equivalence class of such blocks. Further we give general results in the case that the defect group of a block is abelian and normal.
Our main results are as follows. The groups L(B) and T (B) are the subgroups of Pic(B) of bimodules with linear and trivial sources respectively. Throughout, G m denotes the group (C 2 m × C 2 m ) ⋊ C 3 for m ≥ 1, so that G 1 ∼ = A 4 .   It is observed in [9] that derived equivalence preserves finiteness of Picard groups. In particular as by [10] Broué's conjecture holds for 2-blocks with abelian defect groups of 2-rank at most three, finiteness also holds by Proposition 4.5.
The structure of the paper is as follows. In Section 2 we define Picard groups and certain distinguished subgroups, and give some background results from [1]. The groups of perfect self-isometries of blocks play a major role in much of this paper, and Section 3 contains the relevant definitions and calculations. In Section 4 we apply the results of the previous section together with Weiss' criterion to prove most of Theorem 1.1. In Section 5 we calculate the Picard groups in the final three cases of Theorem 1.1.

Picard groups of blocks
The following is based on [1]. For further detail we also recommend [16,17]. Let G be a finite group and B be a block of OG with defect group D. Let F be the fusion system for B on D, defined using a maximal B-subpair , the inertial quotient. Aut(D, F ) denotes the subgroup of Aut(D) of automorphisms stabilizing F . Write Out(D, F ) = Aut F (D). Now let A be a source algebra for B, so A is a D-algebra and we may consider the fixed points A D under the action of D. Write Aut D (A) for the group of algebra automorphisms of A fixing each element of the image of D in A, and Out D (A) for the quotient of Aut D (A) by the subgroup of automorphisms given by conjugation by elements of (A D ) × . As noted in [1], by [20, 14.9] Out D (A) is isomorphic to a subgroup of Hom(E, k × ). The where foc(D) is the focal subgroup of D with respect to F , generated by the elements ϕ(x)x −1 for x ∈ D and ϕ ∈ Hom F ( x , D).
Automorphisms α of B give rise to elements of the Picard group as follows. Define the B-B-bimodule α B by taking α B = B as sets and defining a 1 ·m·a 2 = α(a 1 )ma 2 for a 1 , a 2 , m ∈ B. Inner automorphisms give isomorphic bimodules and α → α B gives rise to an injection Out(B) → Pic(B).
Each element of Pic(B) induces an automorphism of Z(B). The subgroup consisting of those which induce the identity morphism is denoted Piccent(B). Note Piccent(B) is precisely the subgroup of bimodules in Pic(B) that fix every irreducible character.
The following is clear from [1] but we state it here for convenience as it will be used frequently.
Proof. Let ϕ be the inflation of an irreducible character of G/N . An element of Out D (A) is realised by taking the bimodule α B inducing the Morita equivalence given by the automorphism α of B given by α(x) = ϕ(x)x. It is clear from the permutation of Irr(G) given by α B that distinct characters of G/N give rise to distinct elements of Pic(B).
In order to find T (B) when G is a direct product of groups we need to show that Out D (A) factorises according to the factorisation of G.
Write i for the identity element of the source algebra A, so A = iOGi. As described in [1, Remark 1.2] elements of Out D (A) correspond to direct summands of OGi ⊗ OD iOG as B-B-bimodules inducing Morita equivalences. In the following we will have to pass temporarily to the source algebra defined with respect to k in order to apply [8,Lemma 10.37]. We use the notation kB (or KB) to denote B ⊗ O k (or B ⊗ O K), and use similar notation for related objects.
Lemma 2.2. Let G 1 and G 2 be finite groups, B j a block of OG j and B = and so we have shown that i is a source idempotent of B. Now every indecomposable B-B-summand of

Perfect isometries
Before we calculate some Picard groups of blocks it will be necessary to determine some perfect self-isometry groups. We first introduce some notation. Let G be a finite group and B a block of OG. We write Irr(B) for the set of irreducible characters in B (with respect to K). Write G p ′ for the set of p-regular elements of G, IBr(B) for the set of irreducible Brauer characters of B and prj(B) for the set characters of projective indecomposable B-modules. B 0 (OG) will denote the principal block of OG. Now in addition let H be a finite group and C a block of OH. A perfect isometry between B and C is an isometry such that

. (Note that by an isometry we mean an isometry with respect to the usual inner products on Z Irr(B) and
Z Irr(C), so for all χ ∈ Irr(B), I(χ) = ±ψ for some ψ ∈ Irr(C)).
If H = G and C = B then we describe I as a perfect self-isometry of B. We denote by Perf(B) the group of perfect self-isometries of B.

Remark 3.2. An alternative way of phrasing the condition that I K induces an isomorphism between CF
The following two well-known lemmas are both proved in [5].

Lemma 3.4. Any Morita equivalence of blocks induces a perfect isometry.
Before proceeding with some specific examples we need a lemma about Picard groups and perfect self-isometry groups of group algebras of p-groups.
Lemma 3.5. Let P be a finite p-group. Then we have the following isomorphisms of groups. Proof.
(a) This is well-known.
(b) Since there is only one indecomposable projective module for OP , every perfect self-isometry of OP must have all positive or all negative signs. Now by Lemma 3.3 the induced permutation of Irr(OP ) induces an automorphism of Aut(Z(OP )) = Aut(OP ). Hence the result.
We recall the character table of A 4 , where we also set up some labelling of characters.
For the rest of this section we assume p = 2.
Let P be a finite abelian 2-group.
where J is a perfect isometry of OP induced by an O-algebra automorphism, σ ∈ S 4 and ǫ ∈ {±1}.
Proof. We proceed as in the proof of [10, Theorem 2.11]. The projective indecomposable characters are . By counting constituents we see that for 1 ≤ l ≤ 3. Consider the set (1) we have shown that |X m | = 1 or 2 for every 1 ≤ m ≤ 4. If |X 1 | = 2, then by considering (1) for l = 1 we see that X 4 = X 1 . Similarly by considering I(χ P2 ), we get that X 2 = X 4 . This is now a contradiction as then has at most 2|P | constituents with non-zero multiplicity. Therefore |X 1 | = 1 and so by considering I(χ P1 ) we get that |X 4 | = 1 and then by considering I(χ P2 ) and I(χ P3 ) we get that |X 2 | = |X 3 | = 1. Moreover, X 1 , X 2 , X 3 , X 4 must all be disjoint. By composing I with the perfect isometry (Id, I σ,1 ), for some appropriately chosen σ ∈ S 4 , we may assume X m = {m} for all 1 ≤ m ≤ 4. Therefore I(χ P l ) = ±χ P l for 1 ≤ l ≤ 3 and by considering we see that in fact all these signs are the same and we may assume, possibly by composing I with (Id, I Id,−1 ), that for 1 ≤ m ≤ 4. Next we note that (see Proposition 3.6 for the definition of δ m ), and so for all θ ∈ Irr(P ). Now set θ m ⊗ χ m := I(θ ⊗ χ m ), for 1 ≤ m ≤ 4. Evaluating (2) at (x, 1), (x, (123)) and (x, (132)), for some x ∈ P , gives Proceeding as in the proof of [10, Theorem 2.11] we have We have shown that we may assume I is of the form for all θ ∈ Irr(P ), where σ is a permutation of Irr(P ). In particular the Oalgebra automorphism of Z(O(P × A 4 )) induced by I leaves OP invariant. Therefore the permutation σ of Irr(P ) must induce an automorphism of OP and the theorem is proved.
where the action of S 3 is given by permuting a, b and ab, where a and b are generators for the two cyclic factors. In addition set G n ≤ H n to be (C 2 n × C 2 n ) ⋊ C 3 , where the action of C 3 is given by cyclically permuting a, b and ab. For 1 ≤ i ≤ 4 set χ i ∈ Irr(G n ) to be the character of A 4 with the same label inflated to G n and IBr Proposition 3.8. Let P be a finite abelian 2-group and n ∈ N. Suppose I is a permutation of Irr(P × G n ) induced by a Morita self-equivalence of O(P × G n ). Then there exists σ ∈ S 3 and J ∈ Perf(OP ), with all signs positive, such that I satisfies I(θ ⊗ χ i ) = J(θ) ⊗ χ σ(i) for all θ ∈ Irr(P ) and 1 ≤ i ≤ 3.
Proof. For 1 ≤ i ≤ 3 set X i := {θ ⊗ χ i |θ ∈ Irr(P )}. I must permute the X i 's as each X i is exactly the subset of Irr(P × G n ) of characters that reduce to φ i . By composing with a Morita equivalence induced by tensoring with a linear character of G n and/or conjugation by some element of H n , we may assume that I leaves each X i invariant. Now for 1 ≤ i ≤ 3, we define J i : Irr(P ) → Irr(P ) by By Lemma 3.4, I must be a perfect isometry. As if and only if θ∈Irr(P ) α θ θ ∈ CF(P, OP, O), each J i must be a perfect isometry with all signs positive. Now for each θ ∈ Irr(P ) Proceeding exactly as in the proof of Theorem 3.7 proves that J 1 = J 2 = J 3 and hence the result is proved.
We recall the decomposition matrix of B 0 (OA 5 ), where we also set up some labelling of characters and Brauer characters.
Proof. We first note that I leaves {χ i ⊗ ψ j |1 ≤ i ≤ 3, 1 ≤ j ≤ 3} invariant as this is exactly the subset of Irr(B 0 (O(G n × A 5 ))) consisting of characters that reduce to 1 or 2 Brauer characters. Similarly is exactly the subset of Irr(B 0 (O(G n ×A 5 ))) consisting of characters that reduce to a sum of 3 distinct irreducible Brauer characters and so is also left invariant by I. Now for 1 ≤ i ≤ 3 In particular, if M is a left OG-module we adopt the above notation by viewing M as an OG-O{1}-bimodule. Before we apply Weiss' criterion we need a proposition. Let G be a finite group, P a normal p-subgroup and B a block of OG. We denote by B P the sum of blocks of O(G/P ) dominated by B, that is those blocks not annihilated by the image of e B under the natural O-algebra homomorphism p P : OG → O(G/P ), where e B ∈ OG is the block idempotent corresponding to B. With this notation we have  Proof.
(a) First note that for each χ ∈ Irr(G), χ ↓ P is a sum of trivial or a sum of non-trivial irreducible characters of P . Now we can identify O(G/P ) with OG( g∈P g) and view p P as multiplication by g∈P g. The claim now follows by applying p P ⊗ O K to KGe χ for each χ ∈ Irr(B), where e χ ∈ KB is the character idempotent corresponding to χ.
where V χ is the simple KB-module corresponding to χ and f is the permutation of Irr(B) induced by M . So The following corollary is a consequence of Weiss' criterion and Proposition 4.2, and will be the main tool used in proving Theorem 4.6.  Our first consequence is the boundedness of Picard groups for general blocks with normal defect groups. Applying these methods, we have for example: Theorem 4.6. Let p = 2, P be a finite abelian 2-group and n, n 1 , n 2 ∈ N.
Proof. Write D for a defect group of the block B of the group G under consideration and A for a source algebra. Let E be the inertial quotient of the block and F for the fusion system associated to B. In the arguments that follow we will make free use of the results of [1] as presented in Section 2. We also make repeated use of Lemma 3.4 without further reference to it.
(a) Let M ∈ Pic(O(P × G n )). Recall that we are writing χ 1 , χ 2 , χ 3 for the linear characters of G n . By Proposition 3.8 and Lemma 3.5 part (2) we may compose M with some element of Aut O (OP ) such that the induced permutation I of Irr(P × G n ) satisfies I(θ ⊗ χ i ) = θ ⊗ χ σ(i) , where σ ∈ S 3 , for all θ ∈ Irr(P ) and 1 ≤ i ≤ 3. Now assume, as we may, that M satisfies the conditions of Proposition 4.2 with respect to S ∈ Syl p (G n ). Then S M induces a Morita equivalence of O(P × C 3 ), the sum of three blocks each isomorphic to OP . Moreover, by construction, this Morita equivalence is given by identifying each of these three blocks with OP and permuting them. Any such Morita equivalence certainly has trivial source and so by  A 2 is a source algebra of kG n , B 0 (kA 5 ) respectively, D 1 = O 2 (G n ) and D 2 is a Sylow 2subgroup of A 5 so that D = D 1 ×D 2 . It is obtained by tensoring with either of the non-trivial modules of dimension one. We have Out(D, F ) ∼ = C 2 ≀ C 2 . By considering G n × A 5 ✁ H n × S 5 we have S 3 × C 2 ≤ T (B). Note that T (B) must be a quotient group of S 3 ≀ C 2 . However S 3 ≀ C 2 has no quotient group of order 24, so Pic(B) = T (B) ∼ = S 3 × C 2 as required. 5 Blocks with abelian defect groups of 2-rank at most three In [10], which uses also results of [23], the Morita equivalence classes of 2-blocks with abelian defect groups of 2-rank at most three were classified. In this section we complete the computation of the Picard groups of a representative from each Morita equivalence class, so that the isomorphism type of the Picard group is known for every such block.
In order to prove Corollary 1.2, following Lemma 3.5 and Theorem 4.6 it remains to calculate the Picard groups for SL 2 (8), Aut(SL 2 (8)) and J 1 .
The calculation of the Picard group of the principal 2-block of SL 2 (2 n ) is a generalisation of the arguments in [1] for A 5 ∼ = SL 2 (4).  mod (B) gives a stable self-equivalence of Morita type preserving the trivial b-module. By [6,Corollary 3.3] M H must then preserve every simple b-module, and so it is a Morita equivalence. The subgroup of Pic(b) consisting of bimodules fixing the trivial module is the subgroup of bimodules induced by group automorphisms of G as described above, and is isomorphic to C n . Since M H ∈ T (b) and we have a splendid stable equivalence between b and B, it follows that M ∈ T (B). By the description of T (B), it follows that the Morita equivalence given by M is also induced by an automorphism of G. Since Out(G) ∼ = C n , the result follows, noting that every such automorphism acts nontrivially on Irr(B) so Piccent(B) = 1.

The principal 2-block of Aut(SL 2 (8))
The calculation of Pic(B 0 (O Aut(SL 2 (8)))) is complicated by the fact that we may no longer use [6,Corollary 3.3], which requires the inertial quotient to be cyclic. Instead we must show directly that any Morita equivalence of B 0 (O Aut(SL 2 (8))) restricted to the principal block b of the normalizer of a Sylow 2-subgroup gives rise to a self-equivalence of b permuting the simple modules. Let M ∈ Pic(B). Since D is a trivial intersection subgroup of G, induction and restriction gives a splendid stable equivalence of Morita type F : mod (b) → mod (B). Hence M H := F * • M • F : mod (b) → mod (b) induces a stable self-equivalence of Morita type of b. We show that this stable equivalence sends simple modules to simple modules.
b has simple modules of dimensions 1, 1, 1, 3, 3, labelled 1 1 , 1 2 , 1 3 , 3 1 , 3 2 . We may choose our labelling so that I, 1, 1 * has Green correspondent 1 1 , 1 2 , 1 3 respectively. By [15] Res G H (6) = 3 1 3 2 and Writing P S for the projective cover of the simple module S, examination of the structure of the projective indecomposable modules given in [15] yields that Ind G H (3 1 ) ∼ = P 12 / rad 4 (P 12 ) = 12 We deduce that M fixes the Green correspondents of 3 1 and 3 2 , and permutes those of the linear modules, so M H permutes the simple modules. It follows by [16,Theorem 4.14.10] that M H induces a Morita self-equivalence of b. By Theorem 4.6 Pic(b) = T (b), so M H has trivial source as a bimodule. Since F is induced by a trivial source bimodule it follows that M also has trivial source, i.e., M ∈ T (B).
Florian Eisele has already shown that Pic(B 0 (OJ 1 )) = 1 using different methods in (at present) unpublished notes.
We use the computer algebra package MAGMA [2] to find B 0 (kG)-modules which could occur as images of simple modules under F * . We do so by calculating the submodules of projective indecomposable modules with the given head and socle.
Suppose that U is a kG-module satisfying U/ rad(U ) ∼ = soc(U ) ∼ = 76. Then U is isomorphic to either 76, P 76 or the unique submodule of P 76 with structure 76 I 76 56 1 56 2 I 76 .
By the discussion above F is a splendid stable equivalence of Morita type, hence F lifts to a stable equivalence with respect to O. Let M H := F * • M • F : mod (B 0 (OH)) → mod (B 0 (OH)), inducing a stable self-equivalence of Morita type of B 0 (OH). We show that this stable equivalence sends simple modules to simple modules.
Since M ⊗ O k permutes the images of the simple modules under F * , it follows that M H ⊗ O k must permute the simple B 0 (kH)-modules. Hence M H permutes the simple B 0 (OH)-modules and by [16,Theorem 4.14.10] M H induces a Morita equivalence.
We have by Proposition 5.4 that Pic(B 0 (OH)) = T (B 0 (OH)) ∼ = C 3 . Note that M H either induces the trivial equivalence or permutes cyclically the three simple modules of dimension one (fixing the other two). It is clear that M ⊗ O k cannot permute F * (I), F * (1) and F * (1 * ) transitively, so M H and hence M must induce the trivial equivalence. We have shown that Pic(B 0 (OG)) = 1.