Simple modules in the Auslander-Reiten quiver of principal blocks with abelian defect groups
aa r X i v : . [ m a t h . R T ] F e b SIMPLE MODULES IN THE AUSLANDER-REITEN QUIVER OFPRINCIPAL BLOCKS WITH ABELIAN DEFECT GROUPS
SHIGEO KOSHITANI AND CAROLINE LASSUEUR
Abstract.
Given an odd prime p , we investigate the position of simple modules in thestable Auslander-Reiten quiver of the principal block of a finite group with non-cyclicabelian Sylow p -subgroups. In particular, we prove a reduction to finite simple groups.In the case that the characteristic is 3, we prove that simple modules in the principalblock all lie at the end of their components. Introduction
The position of simple modules in the stable Auslander-Reiten quiver of the groupalgebra kG over a field k of characteristic p of a finite group G of order divisible by p is aquestion that was partially investigated in the 1980’s and the 1990’s in a series of articlesby different authors. We refer the reader in particular to [Kaw97, KMU00, KMU01, BU01]and the references therein. The aim of this note is to come back to the following question: Question A.
Let B be a wild p -block of kG . Under which conditions do all simple B -modules lie at the end of their connected components in the stable Auslander-Reiten quiverof kG ? A main reason of interest in this question lies in the fact that a simple kG -module lies atthe end of its component if and only if the heart of its projective cover is indecomposable.In this article, we focus attention on the case in which the principal block B ( kG ) isof wild representation type with abelian defect groups and the prime p is odd, whichamounts to requiring that the p -rank of G is at least 2. The case p = 2 was treated byKawata-Michler-Uno in [KMU00, Theorem 5]. We aim at extending their results and partof their methods to arbitrary primes. Further, we note that the cases when B ( kG ) is offinite or tame representation type are well-understood. In the former case, the distanceof a simple module to the rim of its connected component (a tube of shape ( Z /e Z ) A m )is a function of its position in the Brauer tree of the block, while in the later case theposition of the simple modules in their connected components is given by Erdmann’s work Date : November 11, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Stable Auslander-Reiten quiver, simple modules, principal blocks, abeliandefect.The authors gratefully acknowledge financial support by the funding scheme TU Nachwuchsring Indi-vidual Funding 2016 granted to the second author by the TU Kaiserslautern. The first author was alsosupported by the Japan Society for Promotion of Science (JSPS), Grant-in-Aid for Scientific Research(C)15K04776, 2015–2018. on tame blocks [Erd90].Assuming the field k is algebraically closed we prove the following results: Theorem B.
Let G be a finite group and N E G a normal subgroup such that G/N is solvable of p ′ -order. Let B and b be wild blocks of kG and kN respectively such that B = 1 b . If every simple b -module lies at the end of its connected component in thestable Auslander-Reiten quiver of kN , then every simple B -module lies at the end of itsconnected component in the stable Auslander-Reiten quiver of kG . Theorem C.
Let p be an odd prime. Let G be a finite group with non-cyclic abelianSylow p -subgroups and O p ′ ( G ) = 1 . Write O p ′ ( G ) = Q × H × · · · × H m ( m ≥ ), where Q is an abelian p -group and H i is a non-abelian finite simple group with non-trivial Sylow p -subgroups for each ≤ i ≤ m . Assume that one of the following conditions is satisfied: (i) Q = 1 ; or (ii) Q = 1 and m ≥ ; or (iii) Q = 1 , m = 1 and every simple B ( kH ) -module lies at the end of its connectedcomponent in the stable Auslander-Reiten quiver of kH .Then every simple B ( kG ) -module lies at the end of its connected component in the stableAuslander-Reiten quiver of kG . Corollary D.
Let p be an odd prime. Assume that every simple B ( kH ) -module lies atthe end of its connected component in the stable Auslander-Reiten quiver of kH for everynon-abelian finite simple group H with non-cyclic abelian Sylow p -subgroups. Then everysimple B ( kG ) -module lies at the end of its connected component in the stable Auslander-Reiten quiver of kG for any finite group G with non-cyclic abelian Sylow p -subgroups. We note that if p = 2, then the analogues of Theorem C and Corollary D were essen-tially proven by Kawata-Michler-Uno [KMU00], although not stated in these terms. Asa corollary, we also obtain the equivalent of [KMU00, Theorem 5(a)] for the prime 3. Theorem E.
Assume p = 3 . Let G be a finite group with abelian Sylow -subgroups. If B ( kG ) is a wild - block, then every simple B ( kG ) -module lies at the end of its connectedcomponent in the stable Auslander-Reiten quiver of kG . The paper is organised as follows: in Section 2, we recall the state of knowledge on thesubject and extend a result of Kawata’s [Kaw97, Theorem 1.5] to describe more preciselythe indecomposable summands of the heart of the projective cover of a simple modulenot lying on the rim of its component. In Section 3, we consider groups having a solvablequotient of p ′ -order and prove Theorem B. In Sections 4 and 5, we proceed to a reductionof Question A for principal blocks to the case of finite non-abelian simple groups andprove Theorem C and Corollary D. Finally in Section 6 we deal with the case p = 3 andprove Theorem E. 2. Notation and preliminary results
Throughout, we assume that k is an algebraically closed field of characteristic p > G denote a finite group of order divisible p . All modules are assumed to befinitely generated right modules. For a p -block B , we write 1 B for the corresponding imple modules in the AR-quiver of principal blocks 3 block idempotent and IBr( B ) for the set of isomorphism classes of simple kB - modules.Furthermore, unless otherwise specified, we assume B := B ( kG ), the principal block of kG , is wild. (When the defect groups of B are abelian, we may therefore assume that aSylow p -subgroup of G is non-cyclic, or equivalently that the p -rank of G is at least 2).We denote by k G the trivial kG -module.We let J := J ( kG ) denote the Jacobson radical of kG . For a kG -module U , we define J ( kG ) := kG and for any integer i ≥
0, soc i ( U ) := { u ∈ U | u J i = { }} , then inductivelyfor any i ≥
1, we write L i ( U ) := U J i − /U J i and S i ( U ) := soc i ( U ) / soc i − ( U )for the i -th Loewy layer and the i -th socle layer of U , respectively. We use throughoutwithout further mention the following well-known properties: Lemma 2.1.
Assume N E G of index prime to p . (a) We have J = ˜ J kG = kG ˜ J where ˜ J := J ( kN ) . (b) Let X be a kG -module and Y a kN -module, then for any i ≥ we have L i ( X ) ↓ N = L i ( X ↓ N ) and S i ( X ) ↓ N = S i ( X ↓ N ) and L i ( Y ) ↑ G = L i ( Y ↑ G ) and S i ( Y ) ↑ G = S i ( Y ↑ G ) . Proof.
Part (a) is a well-known result of Villamayor [Vil59] and part (b) follows from (a). (cid:3)
Given a kG -module M , we denote by Ω n ( M ) ( n ∈ Z ) its n -th Heller translate. Given asimple kG -module S , we denote by P ( S ) its projective cover and by H ( P ( S )) the heartof P ( S ), that is H ( P ( S )) = P ( S ) J/ soc( P ( S )).We let Γ s ( kG ), resp. Γ s ( B ), denote the stable Auslander-Reiten quiver of kG , resp. ofthe p -block B , and denote by Γ s ( M ) the connected component of Γ s ( kG ) containing agiven indecomposable kG -module M . Moreover, by convention, we use the terminology AR-component to refer to a connected component of Γ s ( kG ).Erdmann [Erd95] proved that all components of the stable Auslander-Reiten quiverbelonging to a wild block have tree class A ∞ , that is are of the form Z A ∞ or infinite tubes Z A ∞ / h τ a i of rank a , where τ = Ω is the Auslander-Reiten shift.In a component with tree class A ∞ an indecomposable non-projective kG -module M issaid to lie at the end (or on the rim) of its AR-component if the projective-free part ofthe middle term X M of the Auslander-Reiten sequence A ( M ) : 0 −→ Ω ( M ) −→ X M −→ M −→ M is indecomposable. For a non-projective simple kG -module S , theAuslander-Reiten sequence terminating at Ω − ( S ) is of the form A (Ω − ( S )) : 0 −→ Ω( S ) −→ H ( P ( S )) ⊕ P ( S ) −→ Ω − ( S ) −→ standard sequence associated to S . In this set up, clearly a simple module S lies at the end of its component if and only if H ( P ( S )) is indecomposable, and S liesin a tube if and only if S is periodic (i.e. Ω-periodic). S. Koshitani & C. Lassueur
We also recall that for a selfinjective algebra the shape of the components of the stableAuslander-Reiten quiver is an invariant of its Morita equivalence class. By the above, theproperty of lying on the rim of its AR-component for a non-projective simple module isalso invariant under Morita equivalence.Simple kG -modules are known to lie on the rim of their AR -components in the followingcases: Theorem 2.2.
Let B be a wild p -block of kG . Then every simple B -module lies at theend of its AR-component in all of the following cases: (a) G has a non-trivial normal p -subgroup ( [Kaw97, Theorem 2.1] ); (b) G is p -solvable ( [Kaw97, Corollary 2.2] ); (c) G is a perfect finite group of Lie type in the defining characteristic and B has fulldefect ( [KMU01, Theorem] ); (d) G has an abelian Sylow -subgroup and B is the principal -block. [KMU00, The-orem 5] ; (e) G is a symmetric group, an alternating group or a Schur cover of the latter groups,and the defect of B is divisible by p ( [BU01, § ). Moreover, we will use the following computational criterion throughout:
Theorem 2.3 (Kawata’s Criterion on Cartan matrices [Kaw97, Theorem 1.5]) . Let B bea wild p -block of kG . Suppose that there exists a simple B -module S lying on the n -throw from the end of Γ s ( S ) , where n ≥ is minimal with this property. Then there existpairwise non-isomorphic simple B -modules S , . . . , S n with the following properties: (a) For each ≤ i ≤ n, S i ∼ = Ω i − ( S ) and lies at the end of Γ s (Ω( S )) ; (b) The projective covers of P ( S i ) of the simple modules S i (2 ≤ i ≤ n ) are uniserialof length n + 1 with the following Loewy structure: P ( S ) = S S ... S n SS , P ( S ) = S ... S n SS S , · · · , P ( S n ) = S n SS ... S n − S n . The Cartan matrix of B is given by · · · · · · · · ·
01 2 . . . ... ... ...... . . . . . . . . . ... ... ...... . . . · · · · · · · · · ∗ · · · · · · ∗ · · · · · · ... ...... ... ... ... · · · · · · ∗ · · · · · · ∗ , imple modules in the AR-quiver of principal blocks 5 where the columns are labelled by S n , . . . , S , S, . . . in this order.Remark . (a) If the Cartan matrix of a block has the shape of Theorem 2.3(b)above with n = 2, then the simple module S corresponding to the second columnlies on the 2nd row of its AR-component. Indeed in this case P ( S ) = S SS andthe standard sequence associated to S is0 −→ Ω( S ) −→ S ⊕ P ( S ) −→ Ω − ( S ) −→ , so that S lies at the end of its AR-component and S on the 2nd row of its AR-component.A converse to Kawata’s Criterion need not be true in general for an n ≥ F (2) for p = 5 has a simplemodule in the principal block of dimension 875823 lying on the the 2nd row ofits AR-component, and the group 2 .Ru for p = 3 has a faithful simple modulealso lying on the 2nd row. See [KMU01, § S lying on the n -th row of itsAR-component. Corollary 2.5.
With the assumptions and the notation of Theorem 2.3, we have that theheart of the projective cover of the simple module S is decomposable and has a uniserialindecomposable summand of length n − . More precisely H ( P ( S )) = S S ... S n M V , where V = { } is an indecomposable kG -module.Proof. Let S , . . . , S n be the simple kG -modules given by Theorem 2.3 and set H := H ( P ( S )). By the structure of P ( S ) (see Theorem 2.3(b)), there exists a uniserial kG -module W := S S ... S n S .
We may assume that W ⊆ P ( S ) since soc( W ) = S , and hence W ⊆ P ( S ) J since S = S .This implies that W/S ⊆ P ( S ) J/S = H . Namely, there exists a kG -submodule U of H S. Koshitani & C. Lassueur with
H ⊇ U = S S ... S n . On the other hand, by the structure of P ( S n ) (see Theorem 2.3(b)) there is a kG -epimorphism P ( S ) ։ SS S ... S n and hence a kG -epimorphism ψ : H = P ( S ) J/S ։ S S ... S n since S n = S . Set V := ker( ψ ). Since the entry (1 , n ) of the Cartan matrix is equal toone, then by definition of ψ , and by Theorem 2.3(b) we know that soc( U ) ∩ soc( V ) = { } ,hence U ∩ V = { } . Thus we have a direct sum U ⊕ V ⊆ H . Then, by the definitions of U and V and by counting the number of composition factors of H , we obtain U ⊕ V = H .Clearly V = { } by assumption. (cid:3) Finally given a normal subgroup H E G and ˜ b a p -block of H , we will use the group G [˜ b ] defined by Dade [Dad73]. Lemma 2.6 (Dade) . Let H E G such that p G/H | , and let P be a Sylow p -subgroup of H . Let ˜ b := B ( kH ) and B := B ( kG ) be the principal blocks of kH and kG , respectively.Set N := H C G ( P ) . Then the following hold: (a) The block ˜ b is G -invariant. the Frattini argument, that is ”Sylow’s theorem”). (b) N = G [˜ b ] and N E G . (c) If b denotes the principal block of kN , then B = 1 b .Proof. (a) Obvious since ˜ b is the principal block.(b) This follows from [Dad73, Corollary 12.6] since ˜ b is the principal block (see [Dad77,Proof of Lemma 3]). As ˜ b is G -invariant, the fact that G [˜ b ] E G follows from [Dad73,Proposition 2.17].(c) (see also [Kue95, p.303 line 10]) As ˜ b is G -invariant, 1 ˜ b is an idempotent of Z ( kG )and we can write 1 ˜ b = 1 B + 1 B + · · · + 1 B n for an integer n ≥ B , . . . , B n of kG . Thus,1 ˜ b B = 1 B . Namely, 1 B ∈ ˜ b Z ( kG ) ⊆ ˜ b C kG ( H ) =: C. This implies 1 B ∈ Z ( C ) since 1 B ∈ Z ( kG ). Hence it follows from [Kue90, Corollary 4]and part (b) that1 B ∈ C [˜ b ] = Z (˜ b ) ∗ G [˜ b ] (where ∗ denotes the crossed product) ⊆ Z ( kH ) ∗ N ⊆ kN . imple modules in the AR-quiver of principal blocks 7 Thus 1 B ∈ Z ( kN ). On the other hand, since b is the principal block of kN , we have 1 b is G -invariant, so that 1 b ∈ Z ( kG ). Hence as above we can write1 b = 1 B + 1 B ′ + · · · + 1 B ′ t for an integer t ≥ B ′ , · · · , B ′ t of kG . Set˜ e := 1 b − B ∈ Z ( kN ) since 1 B ∈ Z ( kN ). Therefore 1 b = 1 B + ˜ e is a decomposition of 1 b into orthogonal idempotents of Z ( kN ). This implies that ˜ e = 0, and hence 1 b = 1 B . (cid:3) Groups having a solvable quotient of p ′ -order Hypothesis 3.1.
Assume that:(a) G is a finite group of order divisible by p and N E G is a normal subgroup suchthat | G/N | =: q is a prime number with q = p , and we set G/N =: h gN i for anelement g ∈ G \ N .(b) B and b are wild blocks of kG and kN respectively such that 1 B = 1 b . Lemma 3.2.
Assume Hypothesis 3.1 holds. Let ζ ∈ k × be a primitive q -th root of unityin k , and for each ≤ j ≤ q let Z j be the one-dimensional k ( G/N ) -module defined by Z j := h α j i k and α j · gN := ζ j − α j , so that in particular Z = k G/N . The following holds: (a) If S ∈
IBr( B ) is such that S↓ N is not simple, then for each ≤ j ≤ q , S ⊗ k Z j ∼ = S as kG -modules, where we see Z j as a kG -module via inflation. (b) There are integers m ≥ and ℓ ≥ such that IBr( B ) = {S ij | ≤ i ≤ m ; 1 ≤ j ≤ q } G {S i | m + 1 ≤ i ≤ m + ℓ } and IBr( b ) = {T i | ≤ i ≤ m } G {T ij | m + 1 ≤ i ≤ m + ℓ, ≤ j ≤ q } , where for each ≤ i ≤ m and each ≤ j ≤ q , S ij ↓ N = T i and T i ↑ G = S i ⊕ · · · ⊕ S iq , and for each m + 1 ≤ i ≤ m + ℓ and each ≤ j ≤ q , S i ↓ N = T i ⊕ T i ⊕ · · · ⊕ T iq and T ij ↑ G = S i where we may assume that T ij := T i g j − .Moreover, we can assume that for each ≤ j ≤ q , S ij = S i ⊗ k Z j . Proof. (a) Let 1 ≤ j ≤ q . By assumption and Clifford’s theory we have that( S ⊗ k Z j ) ↓ N = S↓ N ⊗ k k N ∼ = S↓ N . = T ⊕ T g ⊕ · · · ⊕ T g q − for some T ∈
IBr( b ). Hence T ↑ G ∼ = S , and T ↑ G ∼ = S ⊗ k Z j for each 1 ≤ j ≤ q .(b) As by Hypothesis 3.1 the quotient G/N is cyclic, the claim follows from the resultof Schur-Clifford [NT88, Chap. 3 Corollary 5.9 and Problem 11(i)]. (cid:3)
Lemma 3.3.
Assume Hypothesis 3.1 holds. Let
S ∈
IBr( B ) . S. Koshitani & C. Lassueur (a) If S↓ N =: T is simple, then P ( S ) ↓ N ∼ = P ( T ) and H ( P ( S )) ↓ N ∼ = H ( P ( T )) . (b) If S↓ N is not simple, then we can write S↓ N = T ⊕ T ⊕ · · · ⊕ T q with T j := T g j − for each ≤ j ≤ q and we have that P ( S ) ↓ N ∼ = P ( T ) ⊕ · · · ⊕ P ( T q ) and H ( P ( S )) ↓ N ∼ = q M j =1 H ( P ( T j )) . Proof. (a) Obviously T = S↓ N = ( P ( S ) /P ( S ) J ) ↓ N = P ( S ) ↓ N / ( P ( S ) J ) ↓ N = P ( S ) ↓ N / ( P ( S ) kG ˜ J ) by Lemma 2.1= P ( S ) ↓ N /P ( S ) ↓ N ˜ J .
Hence the top of P ( S ) ↓ N is T , which implies that P ( S ) ↓ N ∼ = P ( T ). Therefore, H ( P ( S )) ↓ N = ( P ( S ) J/ S ) ↓ N = H ( P ( T )) . (b) Similar to (a). (cid:3) Proposition 3.4.
Assume Hypothesis 3.1 holds. If every simple module T ∈ IBr( b ) liesat the end of its AR-component, then every simple module S ∈ IBr( B ) lies at the end ofits AR-component.Proof. Let S ∈ IBr( B ) be a simple module. First assume that S ↓ N =: T ∈ IBr( b ) issimple. Then by Lemma 3.3(a) H ( P ( S )) ↓ N ∼ = H ( P ( T )) . But by assumption H ( P ( T )) is indecomposable, therefore so is H ( P ( S )).We assume now for the rest of the proof that S ↓ N is not simple. If S lies at the end ofits AR-component, then there is nothing to do. Therefore we now also assume that S lieson the n -th row from the bottom of Γ s ( S ) for an integer n ≥
2, minimal (as in Kawata’sCriterion on Cartan matrices). By Lemma 3.2(b), S ↓ N = T ⊕ · · · ⊕ T q and T j ↑ G = S for each 1 ≤ j ≤ q , where T j := T g j − for 1 ≤ j ≤ q are non-isomorphic simple modules in IBr( b ). We alsoset T := T .Let S , . . . , S n be the simple modules given by Theorem 2.3. Claim 1.
If the modules S ↓ N , . . . , S n ↓ N are all non-simple, then we have a contradiction. Proof of Claim 1.
By assumption and Lemma 3.2, we can write S i ↓ N = T i ⊕ T i ⊕ · · · ⊕ T iq . imple modules in the AR-quiver of principal blocks 9 For each 2 ≤ i ≤ n we define T i ∈ IBr( b ) by T ij = T ig j − , where 1 ≤ j ≤ q . We claim that P ( T ) = T T ... T n T T , P ( T ) = T ... T n T T T , · · · , P ( T n ) = T n T T ... T n − T n . Indeed, we know by Theorem 2.3(b) and Lemma 3.3(b) that P ( T ) ⊕ P ( T ) g ⊕ · · · ⊕ P ( T ) g q − = P ( S ) ↓ N = S S ... S n SS ↓ N = T T g · · · T g q − T T g · · · T g q − · · · T n T ng · · · T ng q − T T g · · · T g q − T T g · · · T g q − , where the boxes mean the Loewy and socle series of the kN -modules. Since the left-hand-side is a direct sum of exactly q indecomposable kN -modules that are h g i -conjugate toeach other, by interchanging the indices of T , . . . , T n , T , we may assume that the PIM P ( T ) has the desired structure. Then automatically the structures of P ( T ) , . . . , P ( T n )are as claimed.Now, using a similar argument as above, we also obtain P ( T ) ⊕ P ( T ) g ⊕ · · · ⊕ P ( T ) g q − = P ( S ) ↓ N = T ⊕ T g ⊕ · · · ⊕ T g q − S S ... S n ↓ N M V ↓ N T ⊕ T g ⊕ · · · ⊕ T g q − , where the last equality holds by Corollary 2.5. Hence we have H ( P ( T )) ⊕ H ( P ( T )) g ⊕ · · · ⊕ H ( P ( T )) g q − = T ⊕ T g ⊕ · · · ⊕ T g q − T ⊕ T g ⊕ · · · ⊕ T g q − ... T n ⊕ T ng ⊕ · · · ⊕ T ng q − ⊕ V ↓ N = T T ... T n ⊕ T T ... T n g ⊕ · · · ⊕ T T ... T n g q − ⊕ V ↓ N since P ( T ) , . . . , P ( T n ) are uniserial by the above.But we are assuming that T , . . . , T n lie at the end of their AR-components, so that H ( P ( T )) , . . . , H ( P ( T n )) are indecomposable. Therefore the right-hand side term in thelater equation has exactly q indecomposable direct summands. This implies that V = { } ,hence a contradiction. Claim 2.
If the modules S ↓ N , . . . , S n ↓ N are all simple, then we have a contradiction. Proof of Claim 2.
Set T i := S i ↓ N for 2 ≤ i ≤ n . We have S ↓ N = T ⊕ T g ⊕ · · · ⊕ T g q − . By the assumption and Lemma 3.2, for each 2 ≤ i ≤ n we can write T i ↑ G = S i ⊕· · ·⊕ S iq with S ij := S i ⊗ k Z j for 1 ≤ j ≤ q . In particular S i = S i for each 2 ≤ i ≤ n . ByTheorem 2.3(b) P ( S ) = S S ... S n SS , so Lemma 3.2(a) implies that P ( S j ) = P ( S ) ⊗ k Z j = S j S j ... S nj SS j for 1 ≤ j ≤ q. imple modules in the AR-quiver of principal blocks 11 This yields that there exists a kG -module S S ... S n ⊕ S S ... S n ⊕ · · · ⊕ S q S q ... S nq S =: W with simple socle isomorphic to S . Therefore W/S has a proper uniserial submodule U := S S ... S n . Now by Corollary 2.5, U |H ( P ( S )), so that by Lemma 3.2(a) S j S j ... S nj = S S ... S n ⊗ k Z j = ( U ⊗ k Z j ) (cid:12)(cid:12)(cid:12) ( H ( P ( S )) ⊗ k Z j ) ∼ = H ( P ( S ⊗ k Z j )) ∼ = H ( P ( S ))for each 1 ≤ j ≤ q . Therefore q = 2 since H ( P ( S )) has exactly two non-projectiveindecomposable direct summands by the assumption that S does not lie at the end of itsAR-component. Notice that this already provides a contradiction in case the characteristicof k is 2, since we assume q = p . So we now assume that p ≥
3. Then, the Loewy andsocle structures of PIMs P ( S ), P ( S i ) and P ( S i ) for 2 ≤ i ≤ n are: SS S ... S n S S ... S n S , S S ... S n SS , S S ... S n SS , S ... S n SS S , S ... S n SS S , · · · , S n SS ... S n − S n , S n SS ... S n − , S n, . Now considering the restrictions S ↓ N and S i ↓ N for 2 ≤ i ≤ n , we obtain by Lemma 3.3that the Loewy and socle structures of the PIMs P ( T ), P ( T g ) and P ( T i ) for each2 ≤ i ≤ n are T T T ... T n T , T g T T ... T n T g , T T ... T n T ⊕ T g T , T ... T n T ⊕ T g T T , · · · , T n T ⊕ T g T T ... T n since T = T g . Now, as the dimension of any PIM for kN is divisible by | N | p =: p a foran integer a ≥
1, and since dim T = dim T g , we have for each 2 ≤ i ≤ n ≡ dim P ( T i ) − dim P ( T ) = dim T i (mod p a ) , so that0 ≡ dim P ( T ) ≡ dim P ( T ) − (dim T + dim T + · · · + dim T n ) = 2 · dim T (mod p a ) . This implies that dim T ≡ p a )since p = 2 (since q = 2). Thus, dim T i ≡ p a ) for any 1 ≤ i ≤ n . Now,looking at the composition factors of PIMs P ( T ) , P ( T g ) , P ( T ) , . . . , P ( T n ), we knowthat IBr( b ) = { T , T g , T , . . . , T n } , which implies that p a | dim T for any T ∈
IBr( b ).Now it follows from Brauer’s result [NT88, Chap.3, Theorem 6.25] that there is a simple T ∈
IBr( b ) such that ν p (dim T ) = a − d ( b ) (where d ( b ) is the defect of b ). Hence we havea contradiction since b is a wild block, i.e. of positive defect. Claim 3. (a) If there is an integer 2 ≤ m ≤ n − S ↓ N , . . . , S m ↓ N are not simple and S m +1 ↓ N is simple, then we have a contradiction.(b) If there is an integer 2 ≤ m ≤ n − S ↓ N , . . . , S m ↓ N are simple and S m +1 ↓ N is not simple, then we have a contradiction. Proof of Claim 3. (a) Set T m +1 := S m +1 ↓ N . By Lemma 3.2 there exists a simple module T m ∈ IBr( b ) with S m ↓ N = T m ⊕ T mg ⊕ · · · ⊕ T mg q − . Then, by Lemma 3.2, T m +1 ↑ G = S m +1 ⊕ S m +1 , ⊕ · · · ⊕ S m +1 ,q where S m +1 ,j := S m +1 ⊗ k Z j for each 1 ≤ j ≤ q and T m ↑ G = S m . By the structure of P ( S ), we have that Ext kG ( S m , S m +1 ) = 0. Therefore by Eckmann- Shapiro’s Lemma wehave that Ext kN ( T m , T m +1 ) = 0. Thus there exists a kN -module with Loewy structure T m T m +1 . So it follows from Lemma 2.1 that T m T m +1 ↑ G = S m S m +1 ⊕ S m +1 , ⊕ · · · ⊕ S m +1 ,q where the right-hand side box is the Loewy and socle series. But P ( S m ) is uniserial byTheorem 2.3(b), so applying again Lemma 2.1, we must have q = 1, which contradictsthe assumption that q is prime.(b) follows in a similar fashion using a dual argument.Altogether, Claims 1-3 prove that the simple modules S , . . . , S n cannot exist, therefore S must lie at the end of its AR-component. (cid:3) As a consequence of the above discussion we obtain Theorem B of the Introduction. imple modules in the AR-quiver of principal blocks 13
Proof of Theorem B.
Because
G/N is solvable of order prime to p , it follows by inductionon | G/N | , that we may assume that | G/N | is a prime distinct from p . Then Proposition 3.4yields the result. (cid:3) The principal block of O p ′ ( G )From now on, we assume that p ≥ G is a finite group with non-cyclic abelian Sylow p -subgroups. Because we consider the principal block only, we assume that O p ′ ( G ) = 1.The structure of O p ′ ( G ) is given by the following well-known result of Fong-Harris [FH93]. Lemma 4.1 ([FH93, 5A–5C]) . Let p be an odd prime. Let G be a finite group with anon-trivial abelian Sylow p -subgroup. Then O p ′ ( G/O p ′ ( G )) ∼ = Q × H × · · · × H m , where m is a non-negative integer (i.e. possibly O p ′ ( G/O p ′ ( G )) ∼ = Q ), Q is an abelian p -group, and for each ≤ i ≤ m , H i is a non-abelian simple group with non-trivial Sylow p -subgroups. Therefore, we fix the notation O p ′ ( G ) = Q × H × · · · × H m , where Q is an abelian p -group, and H , . . . , H m are non-abelian simple groups with non-trivial Sylow p -subgroupsas given by the lemma.4.1. Simple modules in infinite tubes Z A ∞ / h τ a i .Lemma 4.2 ([KMU00, Lemma 5.2] generalised version) . Let H = ˜ H × · · · × ˜ H m ( m ≥ be a finite group such that p | | ˜ H i | for each ≤ i ≤ m . If B ( kH ) is a wild block andcontains a periodic simple module, then m = 1 .Proof. Let S be a simple periodic B ( kH )-module. Then we may write S = S ⊗ k · · ·⊗ k S m where S i is a simple B ( k ˜ H i )-module for each 1 ≤ i ≤ m . Then, by iterating [KMU00,Lemma 2.2], there exists an index 1 ≤ i o ≤ m such that S i is periodic and S j is aprojective k ˜ H j -module for each 1 ≤ j = i o ≤ m . But B ( k ˜ H j ) cannot contain a simpleprojective module, since we assume that p | | ˜ H i | for each 1 ≤ i ≤ m . Hence this forces H = ˜ H i , i.e. m = 1. (cid:3) As a consequence, the existence of simple periodic modules in the principal block lyingin tubes drastically restricts the possible structure of O p ′ ( G ). Corollary 4.3. If B ( kG ) contains a periodic simple module, then O p ′ ( G ) = H is anon-abelian finite simple group with non-cyclic abelian Sylow p -subgroups.Proof. By Lemma 4.2, either O p ′ ( G ) = Q or O p ′ ( G ) = H . But the former cannothappen. Indeed, the indecomposable direct summands of the restriction to O p ′ ( G ) of asimple periodic kG -module are all simple periodic modules, however the unique simple kQ -module is the trivial module, which is not periodic since we assume that B ( kG ) iswild, and hence Q is non-cyclic. The leaves only the possibility O p ′ ( G ) = H , and the p -rank of H must be at least 2 again because we assume that B ( kG ) is wild. (cid:3) This immediately leads to the following reduction to non-abelian simple groups:
Corollary 4.4.
Assume that every periodic simple B ( kH ) -module lies at the end ofits AR-component for every non-abelian finite simple group H with non-cyclic abelianSylow p -subgroups. Then every simple periodic B ( kO p ′ ( G )) -module lies at the end ofits AR-component for any finite group G with O p ′ ( G ) = 1 and non-cyclic abelian Sylow p -subgroups. Simple modules in Z A ∞ -components.Lemma 4.5. Let H = ˜ H × · · · × ˜ H m ( m ≥ be a finite group with abelian Sylow p -subgroups such that p | | ˜ H i | for each ≤ i ≤ m . If B ( kH ) is a wild block containing anon-periodic simple module S not lying at the end of its AR-component, then m = 1 . This lemma and its proof below generalises parts of the proof of [KMU00, Theorem 5(i)].
Proof.
Assume that m ≥
2. Then by Theorem 2.3(b), there exists a simple B ( kH )-module T lying at the end of Γ s (Ω( S )). By Kn¨orr’s Theorem [Kn¨o79, 3.7 Corollary], weknow that the vertices of the simple modules in B ( kH ) are the Sylow p -subgroups of H ,because they are abelian. Now by assumption Γ s ( S ) ∼ = Z A ∞ , which implies that all themodules in Γ s ( S ) and Γ s (Ω( S )) have the Sylow p -subgroups as their vertices by [OU94,Theorem]. So all the modules in Γ s ( S ) and Γ s (Ω( S )) are not projective relatively to thesubgroup N := ˜ H × · · · × ˜ H m − as it does not contain a Sylow p -subgroup of H . Thus, as p = 2, all the simple direct summands of S ↓ N belong to blocks of defect zero by [KMU00,Lemma 1.4]. But B ( kH ) = B ( kN ) ⊗ k B ( k ˜ H m )and there exist a simple B ( kN )-module S and a simple B ( k ˜ H m )-module S m such that S = Inf HN × ˜ H m / × ˜ H m ( S ) ⊗ k Inf HN × ˜ H m /N × ( S m ) . By the above, S is a projective kN -module (indeed S ↓ N = (dim k S m ) S ), henceInf HN × ˜ H m / × ˜ H m ( S )is projective relatively to ˜ H m and therefore so is S seen as the above tensor product. Thiscontradicts the fact that the vertices of S are the Sylow p -subgroups of H . Hence weconclude that S must lie at the end of Γ s ( S ). (cid:3) Proposition 4.6.
Let G be a finite group with O p ′ ( G ) = 1 and non-cyclic abelian Sylow p -subgroups. Assume moreover that one of Conditions (i), (ii), or (iii) of Theorem Cis satisfied. Then every non-periodic simple B ( kO p ′ ( G )) -module lies at the end of itsAR-component.Proof. We have O p ′ ( G ) = Q or O p ′ ( G ) = Q × H × · · · × H m , where Q is an abelian p -group and H i is a non-abelian finite simple group with non-trivial Sylow p -subgroupsfor each 1 ≤ i ≤ m .If (i) holds, that is Q = 1, then by Theorem 2.2(a), all simple B ( kO p ′ ( G ))-modules lieat the end of their AR-components. Therefore, we assume for the rest of the proof that Q = 1.Next if (ii) holds, that is m ≥
2, the claim follows from Lemma 4.5. imple modules in the AR-quiver of principal blocks 15
Finally if (iii) holds, that is O p ′ ( G ) = H , then H must have a non-cyclic Sylow p -subgroup, therefore all simple B ( kO p ′ ( G ))-modules lie at the end of their AR -componentsby assumption. (cid:3) Reduction to O p ′ ( G )We continue assuming that G is a finite group with non-cyclic abelian Sylow p -subgroupssuch that O p ′ ( G ) = 1, unless otherwise stated. We now prove that an answer to Ques-tion A is detected by restriction to the normal subgroup O p ′ ( G ) of G .We set H := O p ′ ( G ), let P ∈ Syl p ( H ) be a Sylow p -subgroup, and set N := HC G ( P ).Moreover we set B := B ( kG ), b := B ( kN ) and ˜ b := B ( kH ). Then N is Dade’s Group G [˜ b ] and N E G , see Lemma 2.6.First of all Question A has an affirmative answer for the group N if and only if it hasan affirmative answer for the group H . Lemma 5.1.
With the above notation, every simple b -module lies at the end of its AR-component if and only if every simple ˜ b -module lies at the end of its AR-component.Proof. By the Alperin-Dade Theorem [Dad77, Theorem], the blocks b and ˜ b are isomorphicas k -algebras, hence Morita equivalent. But for a simple module, lying at the end of itsAR-component is a property preserved by Morita equivalence. (cid:3) Proposition 5.2.
If every simple ˜ b -module lies at the end of its AR-component, thenevery simple B -module lies at the end of its AR-component.Proof. Let S be a simple B -module and let T be a simple direct summand of S ↓ H . Then T is periodic if and only if S is. Therefore Γ s ( S ) ∼ = Z A ∞ if and only if Γ s ( T ) ∼ = Z A ∞ ,and Γ s ( S ) is an infinite tube with tree class A ∞ if and only if Γ s ( T ) is an infinite tubewith tree class A ∞ .In case Γ s ( S ) ∼ = Z A ∞ , then S lies at the end of Γ s ( S ) if and only if T lies at the endof Γ s ( T ) by [KMU00, Lemma 1.5].In case Γ s ( S ) is an infinite tube with tree class A ∞ , then by Corollary 4.3, H is a non-abelian finite simple group with non-cyclic abelian Sylow p -subgroups. Now, by Schreier’sconjecture (now proven by the Classification of Finite Simple Groups, see [ ? , Definition2.1] [GLS3, Theorem 7.1.1]), we know that G/H is a solvable p ′ -subgroup of Out( H ). Nowby Lemma 5.1, we may assume H = N and by Lemma 2.6(c) we have 1 B = 1 b . ThereforeTheorem B implies that S lies at the end of Γ s ( S ) because every simple b -module lies atthe end of its AR-component. (cid:3) As a corollary, we obtain Theorem C of the Introduction.
Proof of Theorem C.
Let G be a finite group with non-cyclic abelian Sylow p -subgroups.As B ( kG ) and B ( kG/O p ′ ( G )) are Morita equivalent, we may assume that O p ′ ( G ) = 1.Therefore, by Proposition 5.2, every simple B ( kG )-module lies at the end of its AR-component if every simple kB ( O p ′ )( G )-module lies at the end of its AR-component. Now if B ( G ) contains a periodic simple module, then by Corollary 4.3 we must havethat O p ′ ( G ) = H is a non-abelian finite simple group with non-cyclic abelian Sylow p -subgroups, then the claim holds by Corollary 4.4. Therefore we may assume that B ( kG ), and hence B ( kO p ′ ( G )), contains no periodic simple module. In this case, if oneof Conditions (i),(ii), or (iii) holds, then the claim follows from Proposition 4.6. (cid:3) Now Corollary D is a direct consequence of Theorem C.6.
Principal -blocks We now fix p := 3, and continue assuming that G is a finite group with non-cyclicSylow 3-subgroups, so that B ( kG ) is wild. We may also assume that O ′ ( G ) = 1.We start by investigating principal 3-blocks of non-abelian finite simple groups withabelian defect group. To this aim, we recall that the list of non-abelian finite simple groupswith abelian Sylow 3-subgroups is known by the classification of finite simple groups andwas determined by Paul Fong (in an unpublished manuscript). Proposition 6.1 ([KY10, Proposition 4.3]) . If G is a non-abelian finite simple groupwith non-cyclic abelian Sylow -subgroup, then G is one of: (i) A , A , M , M , M , HS , O ′ N ; (ii) PSL ( q ) for a prime power q such that || ( q − ; (iii) PSU ( q ) for a prime power q such that || ( q + 1) ; (iv) PSp ( q ) for a prime power q such that | ( q − ; (v) PSp ( q ) for a prime power q such that q > and | ( q + 1) ; (vi) PSL ( q ) for a prime power q such that q > and | ( q + 1) ; (vii) PSU ( q ) for a prime power q such that | ( q − ; (viii) PSL ( q ) for a prime power q such that | ( q + 1) ; (ix) PSU ( q ) for a prime power q such that | ( q − ; or (x) PSL (3 n ) for an integer n ≥ . As a consequence we obtain:
Proposition 6.2. If G is a non-abelian finite simple group with non-cyclic abelian Sy-low -subgroups, then every simple B ( kG ) -modules lies at the end of its component in Γ s ( B ( kG )) .Proof. Let P ∈ Syl ( G ), and set N := N G ( P ) and B := B ( kG ). We go through the listof groups in Proposition 6.1.In case (i), in all cases all simple B -modules lie at the end of their component in Γ s ( B )by Theorem 2.3(b): indeed if G is one of A , M or O ′ N , then one checks from GAP[GAP13] that the Cartan matrix of B has no diagonal entry equal to 2. If G is one of A , M , M , or HS , then one checks from GAP [GAP13] that the Cartan matrix of B does not have the shape of Theorem 2.3(b) either.In case (ii), then the Cartan matrix of B is computed in [Kun00, Table 2] and doesnot satisfy Theorem 2.3(b).Next if G is one of the groups listed in Proposition 6.1(iii),(iv),(vii), or (ix), then itis proven in [KY10, Lemma 3.7] that B is Puig equivalent to B ( kN ). But N has a imple modules in the AR-quiver of principal blocks 17 non-trivial normal Sylow 3-subgroup, therefore all simple B ( kN )-modules lie at the endof their components in Γ s ( B ( kN )) by Theorem 2.2(a), and therefore so do the simple B -modules via the latter Puig (Morita) equivalence.In case (v), the decomposition numbers of B were computed by White and Okuyama-Waki. If q is even then we read from [Whi95, Table II] that each column of the decom-position matrix of B has at least 3 positive entries. If q is odd, then the decompositionmatrix of B is given in [Whi90, Theorem 4.2] up to two parameters α and β . But [OW98,Theorem 2.3] proves that α ∈ { , } . This is enough to see that each column of the de-composition matrix of B has at least 3 positive entries. Therefore in both cases all thediagonal entries of the Cartan matrix of B are at least 3.In case (vi) and (viii), we proceed as follows. For n ∈ { , } fixed, we may regard B ( k PSL n ( q )) as the principal block of SL n ( q ) as 3 ∤ | Z (SL n ( q )) | . Then we check thatthe Cartan matrix of B ( k GL n ( q )) does not satisfy Theorem 2.3(b). To this end weuse the information on the decomposition numbers of B ( k GL n ( q )) provided in [Jam90,Appendix I]. In both cases, it is enough to consider only the square submatrix ∆ n, of the decomposition matrix of B ( k GL n ( q )) whose rows are indexed by the unipotentcharacters. Both in case n = 4 and n = 5, there are five modular characters in theprincipal block (using [FS82]) and∆ , = (4) 1(31) 1 1(2 ) 1 1(21 ) 1 1 1 1(1 ) 1 1 1 ∆ , = (5) 1(32) 1(31 ) 1 1 1(2
1) 1 1 1(1 ) 1 1 1 . (See e.g. [KM00, Proposition 3.1 and Proposition 4.1].) It follows that the Cartan integersof B (GL n ( q )) have lower bounds given by the entries of the following matrices: T ∆ , ∆ , = T ∆ , ∆ , = Therefore the Cartan matrix of B ( k GL n ( q )) cannot satisfy Theorem 2.3(b), and weconclude that all simple B ( k GL n ( q ))-modules lie at the end of their AR-components.Now, from the known values of the unipotent characters of GL n ( q ), we easily check thatthe dimension of the simple modules in B ( k GL n ( q )) are prime to 3, hence they cannot beperiodic by [Car79], as 3 ( a − must divide the dimension of any simple periodic module,where a := the p -rank of the group, but in our case a ≥
2. Therefore every simple B ( k SL n ( q ))-module lies at the end of its AR-component by [KMU00, Lemma 1.5].Finally, if G = PSL (3 n ) for some integer n ≥
2, then the claim follows from Theo-rem 2.2(c) as G is a finite simple group of Lie type in defining characteristic. (cid:3) As a corollary we obtain Theorem E of the Introduction.
Proof of Theorem E.
The claim now follows from Corollary D together with Proposi-tion 6.1. (cid:3)
Acknowledgements.
Both authors gratefully acknowledge financial support and the hospi-tality provided by the Centre Interfacultaire Bernoulli (CIB) of the ´Ecole Polytechnique F´ed´eralede Lausanne during the writing period of this article. The first author is grateful to the hospi-tality of the Department of Mathematics in TU Kaiserslautern.
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E-mail address : [email protected] Caroline Lassueur, FB Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaisers-lautern, Germany.
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