Splendid Morita equivalences for principal blocks with semidihedral defect groups
aa r X i v : . [ m a t h . R T ] O c t SPLENDID MORITA EQUIVALENCES FOR PRINCIPAL BLOCKSWITH SEMIDIHEDRAL DEFECT GROUPS
SHIGEO KOSHITANI, CAROLINE LASSUEUR AND BENJAMIN SAMBALE
Dedicated to Gunter Malle on his 60th Birthday.
Abstract.
We classify principal blocks of finite groups with semidihedral defect groupsup to splendid Morita equivalence. This completes the classification of all principal 2-blocks of tame representation type up to splendid Morita equivalence and shows thatPuig’s Finiteness Conjecture holds for such blocks. Introduction
The present article is motivated by Puig’s Finiteness Conjecture (see [Th´e95, (38.6)Conjecture]), strengthening Donovan’s Conjecture and predicting that for a given primenumber ℓ and a finite ℓ -group P there are only finitely many isomorphism classes ofinterior P -algebras arising as source algebras of ℓ -blocks of finite groups with defect groupsisomorphic to P , or equivalently that there are only a finite number of splendid Moritaequivalence classes of blocks of finite groups with defect groups isomorphic to P . Thecases where P is either cyclic [Lin96] or a Klein-four group [CEKL11] are the only casesfor which this conjecture has been proved to hold in full generality. Else, under additionalassumptions, Puig’s Finiteness Conjecture has also been proved for several classes of finitegroups, as for instance for ℓ -soluble groups [Pui94], for the symmetric groups [Pui94], forthe alternating groups and the double covers thereof, for Weyl groups, or for classicalgroups, see [HK00, HK05] and the references therein.Our principal aim in this article is to classify principal 2-blocks of finite groups withsemidihedral defect groups up to splendid Morita equivalence and deduce that Puig’sFiniteness Conjecture holds when letting the blocks vary through the class of all principal2-blocks of tame representation type. We show that the knowledge of the equivalenceclasses of principal blocks with dihedral defect groups up to splendid Morita equivalencesis enough to describe the splendid Morita equivalence classes of principal blocks withsemidihedral defect groups, as well as the bimodules realizing these equivalences. Werecall that Erdmann [Erd90] classified blocks of tame representation type up to Moritaequivalence by describing their basic algebras by generators and relations making in-tense use of the Auslander-Reiten quiver. However, given a splitting 2-modular system( K, O , k ), her results are not liftable to O in general and do not imply that the resultingMorita equivalences are necessarily splendid Morita equivalences. By contrast, if Puig’s Date : October 19, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Splendid Morita equivalence, semidihedral 2-group, Scott module.The first author was partially supported by the Japan Society for Promotion of Science (JSPS),Grant-in-Aid for Scientific Research (C)19K03416, 2019–2021. The second author acknowledges financialsupport by DFG SFB/TRR 195. The third author is supported by the DFG grants SA 2864/1-2 andSA 2864/3-1. initeness Conjecture holds over k , then it automatically holds over O , since the bimod-ules inducing splendid Morita equivalences are liftable from k to O .To state our main results, we introduce the following notation. For m ≥ q = p f let SL ± m ( q ) := { A ∈ GL m ( q ) | det( A ) = ± } , SU ± m ( q ) := { A ∈ GU m ( q ) | det( A ) = ± } . Now let q = p f where p is an odd prime. Then there are exactly three groups H withPSL ( q ) < H < PΓL ( q ) and | H : PSL ( q ) | = 2 . One is PGL ( q ), one is contained in PSL ( q ) ⋊ h F i where F is the Frobenius automorphismon F q , and the third one is denoted by PGL ∗ ( q ) (see [Go69]). Our main result is as follows: Theorem 1.1.
Let G be a finite group with a semidihedral Sylow -subgroup P of order n with n ≥ fixed and let k be an algebraically closed field of characteristic . Then thefollowing assertions hold. (a) The principal block B ( kG ) of kG is splendidly Morita equivalent to the principalblock of precisely one of the following groups: (bb) P ; (ba1) SL ± ( p f ) where p f + 1) = 2 n ; (ba2) SU ± ( p f ) where p f − = 2 n ; (ab) PGL ∗ ( p f ) where p f − = 2 n ; (aa1) PSL ( p f ) where p f + 1) = 2 n ; or (aa2) PSU ( p f ) where p f − = 2 n ;where p is an odd prime number and f ≥ . Moreover, the splendid Moritaequivalence is realized by the Scott module Sc( G × G ′ , ∆ P ) , where G ′ is the grouplisted in the corresponding case. (b) In particular, if G and G ′ are two groups such that | G | = | G ′ | and which areboth of type (ba1) , both of type (ba2) , both of type (ab) , both of type (aa1) , or bothof type (aa2) , then B ( kG ) and B ( kG ′ ) are splendidly Morita equivalent. In part (a) the labeling of the fusion patterns originates from [Ols75, p.231] (see also[CG12, Theorem 5.3]) and we emphasize that G ′ is not the derived subgroup [ G, G ].Finally, as Craven-Eaton-Kessar-Linckelmann proved in [CEKL11] that Puig’s Finite-ness Conjecture holds for 2-blocks with Klein-four defect groups and the first and thesecond authors proved in [KL20a, KL20b] that it holds as well for principal 2-blockswith dihedral and generalized quaternion defect groups, Theorem 1.1 yields the followingcorollary:
Corollary 1.2.
Puig’s Finiteness Conjecture holds for principal -blocks of tame repre-sentation type. Notation
Throughout this paper, unless otherwise stated we adopt the following notation andconventions. All groups considered are finite and all modules are finitely generated rightmodules. In particular G always denotes a finite group. We denote the dihedral group oforder 2 m with m ≥ D m , the generalized quaternion group of order 2 m with m ≥ Q m , and the cyclic group of order m ≥ C m . We denote by SD n the semidihedralgroup of order 2 n where n ≥ F P ( G ) the usion system of G on a Sylow p -subgroup P of G and by Syl p ( G ) the set of all Sylow p -subgroups of G . We write ∆ G := { ( g, g ) | g ∈ G } ≤ G × G . Given two subgroups N ⊳ G and L ≤ G with G = N L and N ∩ L = 1, N ⋊ L denotes the semi-direct productof N by L .We let k be an algebraically closed field of characteristic 2. We write B ( kG ) for theprincipal block of the group algebra kG . For a block B of kG , we write 1 B for the blockidempotent of B , C B for the Cartan matrix of B , and k ( B ) and ℓ ( B ), respectively, are thenumbers of irreducible ordinary and Brauer characters of G belonging to B . We denote bymod- B the category of finitely generated right B -modules and by mod- B the associatedstable module category.We denote by k G the trivial kG -module. Given a kG -module M and a 2-subgroup Q ≤ G we denote by M ( Q ) the Brauer construction of M with respect to Q . For H ≤ G we denote by Sc( G, H ) the Scott kG -module with respect to H . By definition Sc( G, H )is, up to isomorphism, the unique indecomposable direct summand of the induced module k H ↑ G which contains k G in its head (or equivalently in its socle) and is a 2-permutationmodule by definition. See [NT88, Chap.4 § G, H ) is the relative H -projective cover of k G (see [Th´e85, Proposition 3.1]).If G and H are finite groups, A and B are blocks of kG and kH respectively and M is an ( A, B )-bimodule inducing a Morita equivalence between A and B , then we view M as a right k [ G × H ]-module via the right G × H -action defined by m · ( g, h ) := g − mh for every m ∈ M, g ∈ G, h ∈ H . Furthermore, the algebras A and B are called splendidlyMorita equivalent (or source-algebra equivalent , or Puig equivalent ), if there is a Moritaequivalence between A and B induced by an ( A, B )-bimodule M such that M , viewed asa right k [ G × H ]-module, is a 2-permutation module. In this case, due to a result of Puig(see [Pui99, Corollary 7.4] and [Lin18, Proposition 9.7.1]), the defect groups P and Q of A and B respectively are isomorphic. Hence from now on we identify P and Q . Obviously M is indecomposable as a k ( G × H )-module and since M induces a Morita equivalence, A M and M B are both projective and therefore ∆ P ≤ G × H is a vertex of M . By aresult of Puig and Scott, this definition is equivalent to the condition that A and B havesource algebras which are isomorphic as interior P -algebras (see [Lin01, Theorem 4.1] and[Pui99, Remark 7.5]).In this paper, in order to produce splendid Morita equivalences between principalblocks of two finite groups G and G ′ with a common defect group P , we will use 2-permutation modules given by Scott modules of the form Sc( G × G ′ , ∆ P ), which are ob-viously ( B ( kG ) , B ( kG ′ ))-bimodules. Furthermore, we shall rely on the classification ofprincipal 2-blocks of finite groups with dihedral Sylow 2-subgroups, up to splendid Moritaequivalence, obtained in [KL20a], where the result for Klein-four groups is in [CEKL11].We will use the results of [CEKL11, KL20a, KL20b] without further introduction in thistext and refer the reader directly to the relevant material in these articles.3. Finite groups with semidihedral Sylow 2-subgroups
One of the starting points of this project is the following very useful observation dueto the third author:
Theorem 3.1 ([ABG70]) . Let G be a finite group with a semidihedral Sylow -subgroup P of order n with n ≥ , and assume that O ′ ( G ) = 1 . Then one of the following holds: (bb) G = P . (ba1) G = SL ± ( p f ) ⋊ C d where p f + 1) = 2 n and d is an odd divisor of f . ba2) G = SU ± ( p f ) ⋊ C d where p f − = 2 n and d is an odd divisor of f . (ab) G = PGL ∗ ( p f ) ⋊ C d where p f − = 2 n and d is an odd divisor of f . (aa1) G = PSL ( p f ) .H where p f + 1) = 2 n and H ≤ C (3 ,p f − × C d for an odd divisor d of f . (aa2) G = PSU ( p f ) .H where p f − = 2 n and H ≤ C (3 ,p f +1) × C d for an odd divisor d of f . (aa3) G = M .Proof. If G is 2-nilpotent, then Case (bb) holds since O ′ ( G ) = 1. In all other cases, G is a D -group, a Q -group or a QD -group with the notation of [ABG70, Definition 2.1].If G is a D -group, then G has a normal subgroup K of index 2 with a dihedral Sylow2-subgroup. Hence, the structure of K (and G ) follows essentially from the classificationof Gorenstein–Walter. The precise information can be extracted from Proposition 3.4 of[ABG70] and its proof. We see that Case (ab) holds. If G is a Q -group, then Case (ba1)or (ba2) occurs by Propositions 3.2 and 3.3 (and its proof) of [ABG70]. Finally, let G bea QD -group. Then by [ABG70, Proposition 2.2], N := O ′ ( G ) is simple and the possibleisomorphism types of N are given in the third main theorem of [ABG70], namely M ,PSL ( p f ) and PSU ( p f ). Since C G ( N ) ∩ N = Z ( N ) = 1 we have C G ( N ) ≤ O ′ ( G ) = 1.The possibilities for G/N ≤ Out( N ) can be deduced from the Atlas [Atlas]. In particular,Case (aa1) holds if N ∼ = M . Now let N be PSL ( p f ) or PSU ( p f ). Since, G/N has oddorder, it does not induce graph automorphisms on N . Hence, G/N ≤ C (3 ,p f − ⋊ C f or G/N ≤ C (3 ,p f +1) ⋊ C f . Again, since G/N has odd order,
G/N is abelian. (cid:3) The principal -blocks of M and PSL (3)Benson and Carlson [BC87, (14.1)] observed that the principal 2-blocks of the groupsPSL (3) and M are Morita equivalent by comparing their basic algebras. In this section,we prove that their result can be refined to a splendid Morita equivalence. More precisely,we prove that this Morita equivalence is induced by a Scott module using the gluingmethod developed by the first and the second author in [KL20a, Section 3 and Section 4]. Lemma 4.1.
Set G := PSL (3) , G ′ := M and let P ∈ Syl ( G ) ∩ Syl ( G ′ ) , so that P ∼ = SD . Then Sc( G × G ′ , ∆ P ) induces a stable equivalence of Morita type between B ( kG ) and B ( kG ′ ) .Proof. Set P := h s, t | s = t = 1 , tst = s i ∼ = SD , z := s and Z := h z i = Z ( P ) ∼ = C and observe that F P ( G ) = F P ( G ′ ) by [CG12, Theorem 5.3]. We readfrom the Atlas [Atlas, p.13 and p.18] that C G ( z ) ∼ = GL (3) ∼ = C G ′ ( z ). Note that k GL (3) has only one 2-block, namely the principal block since O ′ (GL (3)) = 1. Thus,Sc( C G ( z ) × C G ′ ( z ) , ∆ P ) realizes a (splendid) Morita equivalence between B ( k C G ( z ))and B ( k C G ′ ( z )) because Sc( C G ( z ) × C G ′ ( z ) , ∆ P ) = kC G ( z ) seen as ( kC G ( z ) , kC G ′ ( z ))-bimodule. On the other hand, Sc( C G ( z ) × C G ′ ( z ) , ∆ P ) | M (∆ Z ) by [KL20a, Lemma 3.2].Hence, as Sc( C G ( z ) × C G ′ ( z ) , ∆ P ) is Brauer indecomposable by [KT19, Theorem 1.2],we have in fact Sc( C G ( z ) × C G ′ ( z ) , ∆ P ) ∼ = M (∆ Z ). Thus, M (∆ Z ) induces a Moritaequivalence between B ( k C G ( z )) and B ( k C G ′ ( z )). Therefore, as all involutions in G are G -conjugate and F P ( G ) = F P ( G ′ ), [KL20a, proof of Case 1 of Proposition 4.6] yields thatfor every involution t ∈ P Sc( C G ( t ) × C G ′ ( t ) , ∆ P ) = M (∆ h t i )and induces a Morita equivalence between B ( kC G ( t )) and B ( kC G ′ ( t )). Therefore theassertion follows from [KL20a, Lemma 4.1]. (cid:3) roposition 4.2. With the notation of Lemma 4.1,
Sc( G × G ′ , ∆ P ) induces a splendidMorita equivalence between B ( kG ) and B ( kG ′ ) .Proof. Set M := Sc( G × G ′ , ∆ P ), B := B ( kG ) and B ′ := B ( kG ′ ). The block B has threesimple kG -modules: k G and two modules and of k -dimension 12 and 26 respectively.Similarly the block B ′ := B ( kG ′ ) has three simple kG ′ -modules: k G ′ and two modules and of k -dimension 12 and 26 respectively. (See [ModAtl]).To start with, we claim that these six simple modules are all trivial source modules.First, the trivial modules k G and k G ′ are obviously trivial source modules with vertex P ,and for G ′ := M , the module is a trivial source module with vertex Q by [Sch83,Lemma 2.1(a) and (d)], whereas the module is a trivial source module with vertex C × C by [Sch83, Lemma 2.2(a) and (c)]. Next, consider G := PSL (3) and its maximalsubgroup M := 3 ⋊ S = 3 ⋊ GL (3) where S is the symmetric group of degree 4(see [Atlas, p.13]). Using the Atlas [Atlas, p.13] and the 2-decomposition matrix of B given in [ModAtl] we easily compute that k M ↑ G = k G + as composition factors. Then,as k G and are self-dual, we must have k M ↑ G = k G ⊕ . Hence is a trivial sourcemodule. Moreover, the module is liftable and affords the ordinary character χ (in theAtlas notation [Atlas, p.13]). Therefore, it follows from [Lan83, II Lemma 12.6(ii)] andthe character values of χ at 2-elements that has vertex C × C . To prove that is a trivial source module, we consider SD = P < GL (3) =: ˜ G < M < G . We easilycompute that 1 P ↑ ˜ G = 1 ˜ G + ˜ χ a where ˜ χ a is the unique 2-rational irreducible ordinarycharacter of ˜ G of degree 2. Hence, as above by self-duality, k P ↑ ˜ G = k ˜ G ⊕ ˜2 where ˜2 isthe unique simple k ˜ G -module in B ( k ˜ G ), so that the simple module ˜2 is a trivial source k ˜ G -module. Again, we read from the the character table of ˜ G and [Lan83, II Lemma12.6(ii)] that ˜2 has vertex Q . Moreover, by the character tables of G and ˜ G , we have˜ χ a ↑ G = χ , so that ˜2 ↑ G = . Hence is also a trivial source kG -module with vertex Q .Next, we recall that there is a bijection between the set of isomorphism classes ofindecomposable trivial source kG ′ -modules (resp. kG -modules) with vertex X ≤ P andthe set of isomorphism classes of indecomposable projective k [ N G ′ ( X ) /X ]-modules (resp. k [ N G ( X ) /X ]-modules). (See [NT88, Chap.4, Problem 10]). Now consider Q ≤ P with Q ∼ = Q and K ≤ P with K ∼ = C × C . It is easy to compute (e.g. using GAP ) that N G ′ ( P ) /P = 1 and N G ′ ( Q ) /Q ∼ = N G ′ ( K ) /K ∼ = S and it is well-known that k S hastwo PIMs. Hence there are precisely two non-isomorphic indecomposable trivial source kG ′ -modules with vertex Q . One of them is by the above, and the other one has tobe Sc( G ′ , Q ), since Sc( G ′ , Q ) ≇ as it must contain a copy of the trivial module in itssocle. Namely,(1) { iso. classes of indec. trivial source B ′ -modules with vertex Q } = { Sc( G ′ , Q ) , } and similarly(2) { iso. classes of indec. trivial source B ′ -modules with vertex K } = { Sc( G ′ , K ) , } . For G we also have N G ( P ) /P = 1 and N G ( Q ) /Q ∼ = N G ( K ) /K ∼ = S (e.g. using GAP ).Thus, the same arguments as above yield:(3) { iso. classes of indec. trivial source B -modules with vertex Q } = { Sc(
G, Q ) , } and(4) { iso. classes of indec. trivial source B -modules with vertex K } = { Sc(
G, K ) , } . ow, let us consider the functor F : mod- B → mod- B ′ , X B ( X ⊗ B M ) B ′ . By Lemma 4.1, F is a functor realizing a stable equivalence of Morita type, hence an ad-ditive category equivalence between mod- B and mod- B ′ . Therefore, as F P ( G ) = F P ( G ′ )(see [CG12, Theorem 5.3]), first by [KL20a, Lemma 3.4(a)] we have F ( k G ) = k G ′ , and by [KL20a, Theorem 2.1(a)], F ( ) and F ( ) are both indecomposable kG ′ -modulesin B ′ . Next, we prove that F ( ) = . It follows from [KL20a, Lemma 3.4(b)] that F ( ) ∈ { Sc( G ′ , Q ) , } . If F ( ) = Sc( G ′ , Q ), then0 = Hom kG ′ ( F ( ) , k G ′ ) = Hom kG ′ ( ⊗ kG M, k G ′ )= Hom kG ( , k G ′ ⊗ kG ′ M ∗ ) by adjointness= Hom kG ( , k G ) by [KL20a, Lemma 3.4(a)]= 0 , a contradiction, so that we have F ( ) = . A similar argument using (2) and (4) yields F ( ) = . Therefore, by [Lin18, Theorem 4.14.10], F , namely M , induces a Moritaequivalence between B and B ′ because all simple B -modules are mapped to simple B ′ -modules. (cid:3) Proof of Theorem 1.1 (b)First of all, we give a lemma which is a direct consequence of a well-known resultdue to Alperin and Dade, see [Alp76] and [Dad77], restated in terms of splendid Moritaequivalences in [KL20a, Theorem 2.2].
Lemma 5.1.
Assume k is an algebraically closed field of arbitrary prime characteristic ℓ .Let G and G ′ be finite groups with a common Sylow ℓ -subgroup P ∈ Syl ℓ ( G ) ∩ Syl ℓ ( G ′ ) .Assume further that there are finite groups e G and ˜ G ′ such that e G ⊲ G and e G ′ ⊲ G ′ , e G/G and e G ′ /G ′ are ℓ ′ -groups, and e G = C G ( P ) G , e G ′ = C G ′ ( P ) G ′ . If Sc( G × G ′ , ∆ P ) realizes a(splendid) Morita equivalence between B ( kG ) and B ( kG ′ ) , then Sc( e G × e G ′ , ∆ P ) realizesa (splendid) Morita equivalence between B ( k e G ) and e B ( k e G ′ ) .Proof. Set B := B ( kG ), e B := B ( k e G ), B ′ := B ( kG ′ ) and e B ′ := B ( k e G ′ ). By [KL20a,Theorem 2.2], e B and B are splendidly Morita equivalent via 1 e B k e G B = Sc( e G × G, ∆ P ),and B ′ and e B ′ are splendidly Morita equivalent via 1 B ′ k e G ′ e B ′ = Sc( G ′ × e G ′ , ∆ P ). Fur-thermore Sc( G × G ′ , ∆ P ) induces a splendid Morita equivalence between B and B ′ byassumption. Hence composing these three splendid Morita equivalences, we have thatSc( e G × G, ∆ P ) ⊗ B Sc( G × G ′ , ∆ P ) ⊗ B ′ Sc( G ′ × e G ′ , ∆ P )=: M induces a splendid Morita equivalence between e B and e B ′ . It remains to see that M =Sc( e G × e G ′ , ∆ P ). Indeed, by the above M = 1 e B k e G B ⊗ B Sc( G × G ′ , ∆ P ) ⊗ B ′ B ′ k e G ′ e B ′ (cid:12)(cid:12)(cid:12) k e G ⊗ kG ( kG ⊗ kP kG ′ ) ⊗ kG ′ k f G ′ = k e G ⊗ kP k f G ′ = k ∆ P ↑ e G × f G ′ nd hence the definition of the Scott module yields Sc( e G × e G ′ , ∆ P ) (cid:12)(cid:12) M . However, as M induces a Morita equivalence between e B and e B ′ , which are both indecomposable asbimodules, e B M e B ′ must be indecomposable and it follows that M = Sc( e G × e G ′ , ∆ P ). (cid:3) We now prove Theorem 1.1(b) through a case-by-case analysis. For the remainder ofthis section, we let p, p ′ be prime numbers, f, f ′ ≥ n ≥ Proposition 5.2.
Let G := SL ± ( p f ) and G ′ := SL ± ( p ′ f ′ ) with p f + 1) = 4( p ′ f ′ + 1) =2 n and let P ∈ Syl ( G ) ∩ Syl ( G ′ ) . Then, Sc( G × G ′ , ∆ P ) induces a splendid Moritaequivalence between B ( kG ) and B ( kG ′ ) .Proof. The groups G and G ′ have a common central subgroup Z ≤ P of order 2 suchthat ¯ G := G/Z ∼ = PGL ( p f ) and ¯ G ′ := G ′ /Z ∼ = PGL ( p ′ f ′ ) have a common Sylow 2-subgroup ¯ P := P/Z isomorphic to D n − (see [ABG70, p.4]). Hence it follows from[KL20a, Theorem 1.1(6)] that Sc( ¯ G × ¯ G ′ , ∆ ¯ P ) induces a splendid Morita equivalencebetween B ( k ¯ G ) and B ( k ¯ G ′ ). Therefore Sc( G × G ′ , ∆ P ) induces a Morita equivalencebetween B ( kG ) and B ( kG ′ ) by [KL20b, Proposition 3.3(b)]. The claim follows. (cid:3) Proposition 5.3.
Let G := SU ± ( p f ) and G ′ := SU ± ( p ′ f ′ ) with p f − = 4( p ′ f ′ − =2 n and let P ∈ Syl ( G ) ∩ Syl ( G ′ ) . Then, Sc( G × G ′ , ∆ P ) induces a splendid Moritaequivalence between B ( kG ) and B ( kG ′ ) .Proof. Again, the groups G and G ′ have a common central subgroup Z ≤ P of order 2such that ¯ G := G/Z ∼ = PGL ( p f ) and ¯ G ′ := G ′ /Z ∼ = PGL ( p ′ f ′ ) have a common Sylow 2-subgroup ¯ P := P/Z isomorphic to D n − (see [ABG70, p.4]). Hence the assertion followsfrom the same argument as in the proof of Proposition 5.2, where [KL20a, Theorem 1.1(6)]is replaced by [KL20a, Theorem 1.1(5)]. (cid:3) Proposition 5.4. G := PSL ( p f ) and G ′ := PSL ( p ′ f ′ ) with p f + 1) = 4( p ′ f ′ + 1) =2 n and let P ∈ Syl ( G ) ∩ Syl ( G ′ ) . Then, Sc( G × G ′ , ∆ P ) induces a splendid Moritaequivalence between B ( kG ) and B ( kG ′ ) .Proof. First, we claim that M := Sc( G × G ′ , ∆ P ) induces a stable equivalence of Moritatype between B ( kG ) and B ( kG ′ ). Let z be the unique involution in Z := Z ( P ), and set C := C G ( z ), C ′ := C G ′ ( z ), C := C/O ′ ( C ), C ′ := C ′ /O ′ ( C ′ ) and P := P O ′ ( C ) /O ′ ( C ) ∼ = P ∼ = P O ′ ( C ′ ) /O ′ ( C ′ ) (and we identify the two groups). Then, by [ABG70, Proposition4(iii), p.21] and Theorem 3.1, we obtain that C ∼ = SL ± ( p f ) ⋊ C d for an odd d with d | f and C ′ ∼ = SL ± ( p ′ f ′ ) ⋊ C d ′ for an odd d ′ with d ′ | f ′ . We can consider that B ( kC ) = B ( kC ), B ( kC ′ ) = B ( kC ′ ) and P ∈ Syl ( C ) ∩ Syl ( C ′ ).Hence it follows from Proposition 5.2 and Lemma 5.1 that Sc( C × C ′ , ∆ P ) induces aMorita equivalence between B ( kC ) and B ( kC ′ ). Thus, M Z := Sc( C × C ′ , ∆ P ) inducesa Morita equivalence between B ( kC ) and B ( kC ′ ) by [KL20b, Proposition 3.3(b)]. Onthe other hand, F P ( G ) = F P ( G ′ ) by [CG12, Theorem 5.3]. Hence it follows from [KL20a,Lemma 3.2] that M Z | M (∆ Z ) and therefore by [KT19, Theorem 1.2], M Z = M (∆ Z ).Thus, again the gluing method of [KL20a, Lemma 4.1] implies the claim. ext we claim that the stable equivalence of Morita type between B ( kG ) and B ( kG ′ )induced by M is actually a Morita equivalence. Since Aut( P ) is a 2-group, N G ( P ) = P × O ′ ( C G ( P )), so that N G ( P ) ≤ C and we can consider the Green correspondences f := f ( G,P,C ) and f ′ := f ( G ′ ,P,C ′ ) . Then, it follows from [Erd79, (3.4)] that we can consider B ( kC ) = B ( kC ′ ) and f ( S ) = f ′ ( S ′ )for all three simple kG -modules S and S ′ in B ( kG ) and B ( kG ′ ), respectively, where S corresponds to S ′ . Thus, [Lin18, Theorem 4.14.10] yields that M induces a Moritaequivalence between B ( kG ) and B ( kG ′ ), which is automatically splendid. (cid:3) Proposition 5.5.
Let G := PSU ( p f ) and G ′ := PSU ( p ′ f ′ ) with p f − = 4( p ′ f ′ − =2 n and let P ∈ Syl ( G ) ∩ Syl ( G ′ ) . Then, Sc( G × G ′ , ∆ P ) induces a splendid Moritaequivalence between B ( kG ) and B ( kG ′ ) .Proof. The same arguments as in the proof of Proposition 5.4 yield the result. Moreprecisely, in this case C ∼ = SU ± ( p f ) and C ′ ∼ = SU ± ( p ′ f ′ ) so that it follows from Propo-sition 5.3 that the Scott module M Z := Sc( C × C ′ , ∆ P ) induces a Morita equivalencebetween B ( kC ) and B ( kC ′ ), and [Erd79, (3.4)] is replaced by [Erd79, (4.10)]. (cid:3) Finally we deal with the groups of type (ab) , that is of the form PGL ∗ ( p f ). This caserequires more involved arguments. However, the proof of [KL20a, Proposition 5.4] –showing that the principal blocks of PGL ( q ) and PGL ( q ′ ) with a common dihedralSylow 2-subgroup and q ≡ q ′ ≡ ( q ) is anextension of degree two of PSL ( q ). Proposition 5.6.
Let G := PGL ∗ ( p f ) , G ′ := PGL ∗ ( p ′ f ′ ) with p f − = 2( p ′ f ′ − = 2 n and let P ∈ Syl ( G ) ∩ Syl ( G ′ ) . Then, Sc( G × G ′ , ∆ P ) induces a splendid Moritaequivalence between B ( kG ) and B ( kG ′ ) .Proof. Set B := B ( kG ), B ′ := B ( kG ′ ) and M := Sc( G × G ′ , ∆ P ). By the definitionof G and G ′ in Section 1, there are normal subgroups N ⊳ G and N ′ ⊳ G ′ with G hasa normal subgroup N | G/N | = 2, | G ′ /N ′ | = 2 and N ∼ = PSL ( p f ), N ′ ∼ = PSL ( p ′ f ′ ).Hence there is Q ∈ Syl ( N ) ∩ Syl ( N ′ ) such that Q ∼ = D n − (see [ABG70]). Recall that F P ( G ) = F P ( G ′ ) by [CG12, Theorem 5.3].First, we claim that(5) M realizes a stable equivalence of Morita type between B and B ′ . Let z be the unique involution in Z ( P ). Set C := C G ( z ) and C ′ := C G ′ ( z ). We knowthat z ∈ Q ∈ Syl ( N ) ∩ Syl ( N ′ ). Now recall that C N ( z ) and C N ′ ( z ) are both 2-nilpotentby [Bra66, Lemma (7A)]. Hence, as | G/N | = 2 = | G ′ /N ′ | , C and C ′ are also 2-nilpotent.Set C := C/O ′ ( C ), C ′ := C ′ /O ′ ( C ′ ), P := [ P O ′ ( C )] /O ′ ( C ) ∼ = [ P O ′ ( C ′ )] /O ′ ( C ′ ).Obviously, C ∼ = C ′ ∼ = P ∼ = P . Hence, [KL20a, Lemma 3.1] implies that Sc( C × C ′ , ∆ P )induces a Morita equivalence between B ( kC ) and B ( kC ′ ) and [KL20a, Lemma 3.2]yields Sc( C × C ′ , ∆ P ) (cid:12)(cid:12) M (∆ h z i )However, as M is Brauer indecomposable by [KT19], we have Sc( C × C ′ , ∆ P ) = M (∆ h z i ).Since, by [ABG70, Proposition 1.1(iii), p.10], all involutions in G are G -conjugate andall involutions in G ′ are G ′ -conjugate, (5) follows from [KL20a, Lemma 4.1] as we havealready seen in the proof of Lemma 4.1. econd, in order to prove that the stable equivalence realized by M is in fact a Moritaequivalence, by [Lin18, Theorem 4.14.10] it suffices to prove that all simple B -modulesare mapped to simple B ′ -modules ( ∗ ). However, to do this it is enough to note that in thestatement of [KL20a, Proposition 5.4] the groups PGL ( q ) and PGL ( q ′ ) can be replacedwith G = PGL ∗ ( p f ) and G ′ = PGL ∗ ( p ′ f ′ ) and the proof of [KL20a, Proposition 5.4]as well as the proof of the case q ≡ ∗ ). (This is because thearguments involved only rely on the facts that PSL ( q ) is normal of index 2 in PGL ( q )and ℓ ( B (PGL ( q ))) = 2, which is also true for G and G ′ .) (cid:3) Proof of Theorem 1.1
We can now prove Theorem 1.1.
Proof of Theorem 1.1.
Part (b) is given by the case-by-case analysis of Section 5.Hence it remains to prove (a).To start with, B ( kG ) = Sc( G × [ G/O ′ ( G )] , ∆ P ) (seen as a ( kG, k [ G/O ′ ( G )])-bimodule)induces a splendid Morita equivalence between B ( kG ) and B ( k [ G/O ′ ( G )]) =: ¯ B , be-cause O ′ ( G ) acts trivially on the principal block. Furthermore, if G ′ denotes one ofthe groups listed in Theorem 1.1(a) and there is a splendid Morita equivalence between B ( k [ G/O ′ ( G )]) and B ( kG ′ ) realized by the Scott module Sc([ G/O ′ ( G )] × G ′ , ∆ P ), thencomposing both equivalences, we obtain a splendid Morita equivalence between B ( kG )and B ( kG ′ ) realized bySc( G × [ G/O ′ ( G )] , ∆ P ) ⊗ ¯ B Sc([
G/O ′ ( G )] × G ′ , ∆ P ) ∼ = Sc( G × G ′ , ∆ P ) . Therefore, we may assume that O ′ ( G ) = 1 and so G must be of type (x), where (x)denotes one of the seven families of groups (bb), (ba1), (ba2), (ab), (aa1), (aa2), (aa3) ofTheorem 3.1. Claim 1: B := B ( kG ) is splendidly Morita equivalent to the principal block B ′ := B ( kG ′ ) for some group G ′ of type (bb), (ba1), (ba2), (ab), (aa1), (aa2) listed in Theo-rem 1.1(a) with P ∈ Syl ( G ) ∩ Syl ( G ′ ) and the splendid Morita equivalence is realizedby Sc( G × G ′ , ∆ P ).Here we emphasize that the lists of groups in Theorem 3.1 and in the statement of Theo-rem 1.1(a) are not the same, hence we use different fonts to distinguish them. We proveClaim 1 through a case-by-case analysis as follows. • Suppose that G is of type (bb) . Then G = P by Theorem 3.1(bb). Then, we may take G ′ := P , that is G ′ of type (bb) . Obviously B = B ′ = kP and Sc( P × P, ∆ P ) = kP kP kP induces a splendid Morita equivalence between B and B ′ , as required. • Suppose that G is of type (ba1). Then G = SL ± ( p f ) ⋊ C d where 4( p f + 1) = 2 n and d is an odd divisor of f . We take G ′ := SL ± ( p ′ f ′ ), that is of type (ba1) , and we may assumethat we have chosen P such that P ∈ Syl ( G ) ∩ Syl ( G ′ ). Then, by Frattini’s argument G = N G ( P ) G ′ = C G ( P ) P G ′ = C G ( P ) G ′ and it follows from Lemma 5.1 (i.e. [KL20a,Theorem 2.2(b)]) that 1 B kG B ′ = Sc( G × G ′ , ∆ P )induces a splendid Morita equivalence between B and B ′ . • Suppose that G is of type (ba2) . Then G = SU ± ( p f ) ⋊ C d where 4( p f − = 2 n and d isan odd divisor of f . We take G ′ := SU ± ( p ′ f ′ ), that is of type (ba2) , and we may assumethat we have chosen P such that P ∈ Syl ( G ) ∩ Syl ( G ′ ). Then the same arguments as n case (ba1) yield the claim. • Suppose that G is of type (ab) . Then G = PGL ∗ ( p f ) ⋊ C d where 2( p f − = 2 n and d is an odd divisor of f . We take G ′ := PGL ∗ ( p ′ f ′ ), that is of type (ab) , and wemay assume that we have chosen P such that P ∈ Syl ( G ) ∩ Syl ( G ′ ). Then the samearguments as in case (ba1) yield the claim. • Suppose that G is of type (aa1) . Then G = PSL ( p f ) .H where 4( p f + 1) = 2 n and H ≤ C (3 ,p f − × C d for an odd divisor d of f . We take G ′ := PSL ( p ′ f ′ ), that is of type (aa1) , and we may assume that we have chosen P such that P ∈ Syl ( G ) ∩ Syl ( G ′ ). Thenthe same arguments as in case (ba1) yield the claim, where C d is replaced by H . • Suppose that G is of type (aa2) . Then G = PSU ( p f ) .H where 4( p f − = 2 n and H ≤ C (3 ,p f +1) × C d for an odd divisor d of f . We take G ′ := PSU ( p ′ f ′ ), that is of type (aa2) , and we may assume that we have chosen P such that P ∈ Syl ( G ) ∩ Syl ( G ′ ). Thenthe same arguments as in case (ba1) yield the claim, where C d is replaced by H . • Suppose that G is of type (aa3) . Then G = M by Theorem 3.1(aa3) and n = 4. Wetake G ′ := PSL (3), so that P ∈ Syl ( G ) ∩ Syl ( G ′ ) and G ′ is of type (aa1) . Moreover, byProposition 4.2, Sc( G × G ′ , ∆ P ) induces a splendid Morita equivalence between B and B ′ , as required.Furthermore, the fact that the group G ′ in Claim 1 is independent of the choice of p and f for types (ba1), (ba2), (ab), (aa1), (aa2) follows directly from Part (b). Hence itonly remains to prove the following claim. Claim 2.
The principal blocks of the groups listed in the different cases of Theorem 1.1(a)are mutually not splendidly Morita equivalent.So let B := B ( kG ) for G of type (x) with (x) ∈ { (bb), (ba1), (ba2), (ab), (aa1), (aa2) } asin Theorem 1.1(a). It is enough to show that B is not Morita equivalent to B ′ := B ( kG ′ )for G ′ of type (y) = (x) and (y) ∈ { (bb), (ba1), (ba2), (ab), (aa1), (aa2) } .Now, type (bb) is the unique case in which ℓ ( B ) = 1, so we can assume that G is notof type (bb) . Next, assume that ℓ ( B ) = 3. Then, by [Ols75, table on p. 231], G is oftype (aa1) or (aa2) . However, the principal blocks of groups of type (aa1) and (aa2) arenever Morita equivalent because their 2-decomposition matrices are different by [Erd90,SD(2 B ) , p.299 and SD(2 A ) , p.298]. Therefore, it only remains to consider the case ℓ ( B ) = 2. Then, by [Ols75, table on p. 231], G is of type (ab) or (ba) (i.e. (ba1) or (ba2) ).Then, by looking at k ( B )s (see [Ols75, the table on p.231]), we obtain that the principalblocks of groups of type (ab) and (ba) are never Morita equivalent. Hence we can alsoassume that G is not of type (ab) , so that we may assume that G is of type (ba1) and G ′ is of type (ba2) , that is G = SL ± ( p f ) with 4( p f + 1) = 2 n G ′ = SU ± ( p ′ f ′ ) with 4( p ′ f ′ − = 2 n and we may identify a Sylow 2-subgroup P of G and G ′ . Since G has a central involution,say z , set Z := h z i , G := G/Z ∼ = PGL ( p f ), B := B ( kG ), P := P/Z and note that P ∼ = D n − . We also have Z ≤ G ′ , hence we can set G ′ := G ′ /Z ∼ = PGL ( p ′ f ′ ) and B ′ := B ( kG ′ ). Then, it follows from the condition on p f and [KL20a, Corollary 8.1(f)] (see also[Erd90, D(2 B ), p.295]) and from the condition on p ′ f ′ and [KL20a, Corollary 8.1(e)] (see Erd90, D(2 A ), p.294]), respectively, that C B = (cid:18) n − + 1 (cid:19) and C B ′ = (cid:18) n − n − n − n − + 1 (cid:19) . Now suppose that B and B ′ are Morita equivalent. Then, C B = C B ′ . Since Z is a centralsubgroup of G and G ′ of order 2, [NT88, Theorem 5.8.11] implies that C B = 2 C B and C B ′ = 2 C B ′ . Thus C B = C B ′ , a contradiction. Claim 2 follows. (cid:3) Acknowledgment.
The authors are grateful to Burkhard K¨ulshammer for useful conversa-tions.
References [Alp76] J.L. Alperin,
Isomorphic blocks , J. Algebra (1976), 694–698.[ABG70] J.L. Alperin, R. Brauer, D. Gorenstein , Finite groups with quasi-dihedral and wreathedSylow -subgroups , Trans. Amer. Math. Soc. (1970), 1–261.[BC87] D.J. Benson, J.F. Carlson , Diagrammatic methods for modular representations andcohomology , Comm. Algebra (1987), 53–121.[Bra66] R. Brauer , Some applications of the theory of blocks of characters of finite groups III , J. Al-gebra (1966), 225–255.[Atlas] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, R.A. Wilson , Atlas of FiniteGroups , Clarendon Press, Oxford, 1985.[CEKL11]
D.A. Craven, C.W. Eaton, R. Kessar, M. Linckelmann , The structure of blocks witha Klein four defect group , Math. Z. (2011), 441–476.[CG12]
D.A. Craven, A. Glesser , Fusion systems on small p -groups , Trans. Amer. Math. Soc. (2012), 5945–5967.[Dad77] E.C. Dade, Remarks on isomorphic blocks , J. Algebra (1977), 254–258.[Erd79] K. Erdmann , On -blocks with semidihedral defect groups , Trans. Amer. Math. Soc. (1979), 267–287.[Erd90] K. Erdmann , Blocks of Tame Representation Type and Related Algebras . Lecture Notes inMathematics, vol. 1428 , Springer-Verlag, Berlin, 1990.[GAP]
The GAP Group , GAP — Groups, Algorithms, and Programming, Version 4.8.4, , 2016.[Go69] D. Gorenstein,
Finite groups the centralizers of whose involutions have normal 2-complements ,Canad. J. Math. (1969), 335–357.[HM76] W. Hamernik, G.O. Michler , On vertices of simple modules in p -solvable groups .Mitt. Math. Sem. Giessen (1976), 147–162.[HK00] G. Hiss, R. Kessar , Scopes reduction and Morita equivalence classes of blocks in finiteclassical groups , J. Algebra (2000), 378–423.[HK05]
G. Hiss, R. Kessar , Scopes reduction and Morita equivalence classes of blocks in finiteclassical groups II , J. Algebra (2005), 522–563.[KL20a]
S. Koshitani, C. Lassueur , Splendid Morita equivalences for principal -blocks with dihedraldefect groups , Math. Z. (2020), 639–666.[KL20b] S. Koshitani, C. Lassueur , Splendid Morita equivalences for principal -blocks with gener-alised quaternion defect groups , J. Algebra (2020), 523–533.[KT19] S. Koshitani, ˙I. Tuvay , The Brauer indecomposability of Scott modules with semidihedralvertex , preprint, arXiv:1908.05536v2[Lan83]
P. Landrock , Finite Group Algebras and their Modules , London Math. Soc. Lecture NoteSeries, vol.84 , Cambridge Univ. Press, Cambridge, 1983.[Lin96]
M. Linckelmann , The isomorphism problem for cyclic blocks and their source algebras ,Invent. Math. (1996), 265–283.[Lin01]
M. Linckelmann , On splendid derived and stable equivalences between blocks of finite groups ,J. Algebra (2001), 819–843.[Lin18]
M. Linckelmann , The Block Theory of Finite Group Algebras, Volumes 1 and 2 , LondonMath. Soc. Student Texts and , Cambridge Univ. Press, Cambridge, 2018, NT88]
H. Nagao, Y. Tsushima , Representations of Finite Groups , Academic Press, New York,1988.[Ols75]
J.B. Olsson , On -blocks with quaternion and quasidihedral defect groups , J. Algebra (1975), 212–241.[Pui94] L. Puig , On Joanna Scopes’ criterion of equivalence for blocks of symmetric groups , AlgebraColloq. (1994), 25–55.[Pui99] L. Puig , On the Local Structure of Morita and Rickard Equivalences between Brauer Blocks ,Birkh¨auser, Basel, 1999.[Sch83]
G. J.A. Schneider , The vertices of the simple modules of M over a field of characteristic (1983), 189–200.[Th´e85] J. Th´evenaz , Relative projective covers and almost split sequences , Comm. Algebra (1985),1535–1554.[Th´e95] J. Th´evenaz , G -Algebras and Modular Representation Theory . Clarendon Press, Oxford,1995.[ModAtl] R. Wilson, J. Thackray, R. Parker, F. Noeske, J. M¨uller, F. L¨ubeck, C. Jansen,G. Hiss, T. Breuer , The Modular Atlas Project , . Center for Frontier Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan.
Email address : [email protected] FB Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany.
Email address : [email protected] Institut f¨ur Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universit¨atHannover, Welfengarten 1, 30167 Hannover, Germany.
Email address : [email protected]@math.uni-hannover.de