Specht modules in the Auslander–Reiten quiver

Journal of Algebra 397 (2014) 343–364

Contents lists available at ScienceDirect

Journal of Algebra www.elsevier.com/locate/jalgebra

Specht modules in the Auslander–Reiten quiver Susanne Danz a , Karin Erdmann b,∗ a b

Department of Mathematics, University of Kaiserslautern, P.O. Box 3049, 67653 Kaiserslautern, Germany Mathematical Institute, University of Oxford, ROQ, Oxford OX2 6GG, UK

a r t i c l e

i n f o

Article history: Received 7 May 2012 Available online 28 September 2013 Communicated by Michel Broué MSC: 16E99 16G70 20C20 20C30

a b s t r a c t Let A be a symmetric algebra over an algebraically closed field. We study the position of indecomposable A-modules with small socles or heads in Auslander–Reiten components of tree class A ∞ . We apply the results to Specht modules when A is a block of a group algebra of a symmetric group. In particular, we show that if the block has weight 2 then all Specht modules are quasi-simple, that is, they lie ‘at the ends’ of their components. © 2013 Elsevier Inc. All rights reserved.

Keywords: Auslander–Reiten quiver Quasi-length Specht module Symmetric group

1. Introduction Given a finite-dimensional algebra A over an algebraically closed field F of characteristic p  0, one says that A has finite (representation) type if there are only finitely many isomorphism classes of indecomposable A-modules; otherwise A is said to have infinite (representation) type, and one then distinguishes further between tame and wild type. Roughly speaking, the isomorphism classes of indecomposable modules of a tame algebra can still be classified, whereas this is no longer possible for algebras of wild type, so that the representation theory of the latter is generally rather poorly understood. It is, therefore, desirable to have additional structural invariants that at least enable us to distinguish particular classes of indecomposable modules of wild algebras. The Auslander–Reiten quiver is an important homological invariant that can be attached to every finite-dimensional algebra, and it provides part of a presentation of the module category of the

*

Corresponding author. E-mail addresses: [email protected] (S. Danz), [email protected] (K. Erdmann).

0021-8693/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jalgebra.2013.08.019

344

S. Danz, K. Erdmann / Journal of Algebra 397 (2014) 343–364

algebra. For algebras of wild representation type, very often the stable part of an Auslander–Reiten component has tree class A ∞ . If one labels the rows of such a component then one has a new length function: if M is indecomposable and not projective then we say that M has quasi-length d if M belongs to the d-th row of its stable component. We remark that, in particular, if A is a block of a group algebra of a finite group and if A has wild type then the stable parts of all its Auslander–Reiten components have tree class A ∞ , by [12]. Moreover, the representation type of a block of a group algebra can be read off from its defect groups; for details we refer to the introduction of [11]. In general, one might expect that distinguished classes of modules in stable Auslander–Reiten components of tree class A ∞ have small quasi-lengths. Kawata [22], for instance, has studied the positions of simple A-modules in such components, in the case where A is a symmetric F -algebra, that is, A ∼ = Hom F ( A , F ) as ( A , A )-bimodules. Moreover, Bessenrodt–Uno [5] have proved that when A is the group algebra of a symmetric group, the simple A-modules in blocks of wild type have always quasi-length 1. In fact, Bessenrodt–Uno’s result is more general, since it also covers wild blocks of alternating groups, and wild blocks of the covering groups of the symmetric and alternating groups, and as well blocks of wild type for Hecke algebras of symmetric groups. In [24] it was shown that the group algebra F Sn of the symmetric group Sn as well as the associated Hecke algebra are cellular algebras in the sense of Graham–Lehrer [16]. In the theory of cellular algebras the cell modules play a central role: many of these have unique simple quotients, which in turn give rise to a full set of non-isomorphic simple modules. The cell modules of F Sn are precisely the Specht modules. Furthermore, the Specht modules are the p-modular reductions of the simple Sn -modules in characteristic 0. Motivated by the results in [5], the main aim of the present paper is to prove that Specht modules of symmetric groups that belong to wild blocks with small defect groups have quasi-length 1. This will be achieved in Section 6. Along the way we will derive a number of results that should be of independent interest, since they cover more general classes of finite-dimensional algebras and modules: in Section 3 we study F -algebras A that are symmetric and have some additional structural properties. In fact, most blocks of group algebras of symmetric groups, or of other cellular algebras satisfy the hypotheses on the algebra. In Lemma 3.7 we consider non-projective indecomposable A-modules that lie in an Auslander–Reiten component of tree class A ∞ , and have a unique simple submodule (or a unique simple quotient module). We prove that such a module has quasi-length 1, unless possibly the algebra A has a simple branch module; for the definition, see 2.2. In Theorem 6.1 we then apply this general result to Specht modules of symmetric groups. As a main result we show that every Specht module in a block of weight w = 2 < p has quasi-length 1. We also show in Theorem 6.2 that if p = 3 then all Specht modules in a block of weight w = 3 have quasi-length 1; this needs a more detailed analysis. We remark that blocks of symmetric groups of weight at most 1 have finite type, and 2-blocks of weight 2 have tame type. For these blocks the positions of Specht modules in the stable part of the Auslander–Reiten quiver are known; for finite type it can be deduced from [17], and tame type was done in [21]. Furthermore, in Section 5 we consider blocks of symmetric groups of a fixed (arbitrary) weight w  3. By work of Scopes [28] and Kleshchev [23], these blocks are related by sequences of functors or, in terms of combinatorics, by a sequence of [ w : k] pairs (for details see [28]). In Corollary 5.4 we compare Specht modules belonging to wild blocks of different symmetric groups that correspond to each other under such functors; we give a sufficient condition for these Specht modules to have the same quasi-length. The present paper is organized as follows: we begin, in Section 2, by briefly summarizing the notational background needed throughout. In Section 3 we then examine the quasi-lengths of indecomposable modules over symmetric F -algebras that lie in stable Auslander–Reiten components of tree class A ∞ . In Section 4 we recall some results concerning partial Morita equivalences between module categories of finite-dimensional F -algebras. In Section 5 we apply these to group algebras of symmetric groups. Finally, in Section 6, we prove that Specht modules in blocks of weight 2 for p > 2 have quasi-length 1, and also that Specht modules in any 3-block of weight 3 have quasi-length 1.

S. Danz, K. Erdmann / Journal of Algebra 397 (2014) 343–364

345

For general background information about representations of finite-dimensional algebras we refer the reader to [1,2,25]. 2. Preliminaries 2.1. Notation (1) We fix an algebraically closed field F of arbitrary characteristic. We also fix a finitedimensional symmetric F -algebra A, that is, A is as an ( A , A )-bimodule isomorphic to its F -linear dual Hom F ( A , F ). We also assume throughout that, as an algebra, A is indecomposable but not of finite type (thus, in particular, is not simple). We consider finite-dimensional left A-modules. The category of these is denoted by A-mod. Moreover, we denote the stable module category of finite-dimensional A-modules by A-mod. That is, the objects of A-mod coincide with those of A-mod, and the morphisms between two objects M and N in A-mod are equivalence classes of A-homomorphisms M −→ N modulo A-homomorphisms that factor through projective A-modules. If M is an A-module and D is a simple A-module, we denote by [ M : D ] the composition multiplicity of D in M. (2) If M is an A-module then we denote the radical and the socle of M by Rad( M ) and Soc( M ), respectively. Moreover, we call M / Rad( M ) the head of M, and we denote it by Hd( M ). As usual, we denote by Radi ( M ) and Soci ( M ) (i  0) the higher radical and socle powers, respectively. Recall that Rad0 ( M ) = M and Soc0 ( M ) = 0. We write P ( M ) for a projective cover of M. Recall that, since we are assuming A to be a symmetric F -algebra, every projective A-module has isomorphic head and socle. Suppose that M has Loewy length l  1 with Loewy layers Radi −1 ( M )/ Radi ( M ) ∼ = D i1 ⊕ · · · ⊕ D iri , for i = 1, . . . , l, some r i ∈ N, and simple A-modules D i1 , . . . , D iri . Then we write

⎡ ⎢

M =⎣

D 11 ⊕ · · · ⊕ D 1r1

⎤

⎥ .. ⎦. . D l1 ⊕ · · · ⊕ D lrl

We call M rigid if the Loewy series of M and the socle series of M coincide. (3) For an A-module M we denote by l( M ), s( M ) and t( M ), respectively, the (composition) lengths of M, Soc( M ), and Hd( M ), respectively. By our assumption on A, we have Soc( P ) ⊆ Rad( P ), for every projective A-module P . We define the heart of such a projective A-module P as Ht( P ) := Rad( P )/ Soc( P ). We also mention the following easy, but useful, lemma. 2.1. Lemma. Let M = 0 be a rigid A-module of Loewy length m. If N is an A-submodule of M of Loewy length n  m then we have N ⊆ Radm−n ( M ). Proof. Since M is rigid of Loewy length m, we have Radi ( M ) = Socm−i ( M ), for every i = 0, . . . , m. Moreover, by [25, Thm. 1.8.18], we have Socm−i ( M ) = {x ∈ M | Radm−i ( A )x = 0}, for i = 0, . . . , m. So if N is an A-submodule of M of Loewy length n and if x ∈ N then we get Radn ( A )x ⊆ Radn ( N ) = 0, so that x ∈ Radm−n ( M ). 2 2.2. The Ext-quiver We fix a set of representatives { D 1 , . . . , D r } for the isomorphism classes of simple A-modules. 2.2. Remark/Definition. (a) The Ext-quiver of A is defined as the directed graph with vertices labelled by { D 1 , . . . , D r } and such that, for i , j ∈ {1, . . . , r }, the number of arrows from D i to D j equals

346

S. Danz, K. Erdmann / Journal of Algebra 397 (2014) 343–364

dim F (Ext1 ( D i , D j )) where Ext1 ( D i , D j ) := Ext1A ( D i , D j ). Note that if P i and P j are the projective covers of D i and D j , respectively, then













Rad( P i )/ Rad2 ( P i ) : D j = dim F Ext1 ( D i , D j ) = Soc2 ( P j )/ Soc( P j ) : D i ;

see, for example, [11, I.6.3]. (b) Suppose that the algebra A has infinite representation type, that is, there are infinitely many isomorphism classes of indecomposable A-modules. Suppose further that, for i , j ∈ {1, . . . , r }, we have Ext1 ( D i , D j ) ∼ = Ext1 ( D j , D i ). Then we call a simple A-module D a branch A-module if

• there is precisely one j ∈ {1, . . . , r } with Ext1 ( D , D j ) ∼ = F and D  D j , • and Ext1 ( D , D i ) = 0, for all i ∈ {1, . . . , r }  { j }. When no confusion about the underlying algebra is possible we just say D is a branch module. Note that, by part (a), D is a branch A-module if and only if Rad( P ( D ))/ Rad2 ( P ( D )) ∼ = D j and D j  D. So, in this case, D j is the unique neighbour of D in the Ext-quiver of A. 2.3. Remark. Suppose that the Ext-quiver of A had one of the following shapes:

•

•

•

Then A would have to be of finite representation type (which is not the case, by our hypotheses); this can be deduced from [1, Prop. 2.3 in IV.2]. We will also use the following consequence: suppose P = 0 is a projective A-module. Then P has Loewy length at least 3. Namely, otherwise P would have length 2, and the quiver of A would be a vertex with one loop, or just one vertex, which is not possible, since A is of infinite type. Many prominent classes of finite-dimensional algebras, such as cellular algebras, have an antiinvolution, which then gives rise to a duality (−)∗ on A-mod. More precisely, if ι : A −→ A is an anti-involution and M is an A-module then the F -linear dual M ∗ := Hom F ( M , F ) carries a left A-module structure via



 (a · f )(m) := f ι(a)m a ∈ A, f ∈ M ∗, m ∈ M . If this occurs then we say that A has a duality, and we call an A-module M self-dual, or say that

(−)∗ fixes M if M ∼ = M ∗ . Note that, for any two A-modules M 1 and M 2 , one then has Ext1 ( M 1 , M 2 ) ∼ = 1 ∗ ∗ Ext ( M 2 , M 1 ). Furthermore, if D is simple and self-dual then so is the projective cover P ( D ), and, as it is easy to see, its heart Ht( P ( D )) is self-dual as well. In particular, whenever G is a finite group, the group algebra F G has a duality that is induced by the anti-involution of F G mapping g ∈ G to g −1 . A similar duality exists for Hecke algebras of symmetric groups. 2.4. Proposition. Assume A has a duality fixing all simple modules. Assume also that the projective indecomposable modules are rigid of common Loewy length l  2. Let M be an A-module of Loewy length 2 such that Rad( M ) = Soc( M ). (a) For every simple factor module D of M, there are a simple submodule D of M and a uniserial subquotient X of M with Soc( X ) ∼ = D and Hd( X ) ∼ = D . In this case, Ext1 ( D , D ) ∼ = Ext1 ( D , D ) = 0. (b) For every simple submodule D of M, there are a simple factor module D of M and a uniserial subquotient X of M with Hd( X ) ∼ = D and Soc( X ) ∼ = D. In this case, Ext1 ( D , D ) ∼ = Ext1 ( D , D ) = 0.

S. Danz, K. Erdmann / Journal of Algebra 397 (2014) 343–364

347

Proof. We prove assertion (a); assertion (b) follows by a similar argument. Let D 1 , . . . , D n be simple A-modules and let 1  m < n be such that Rad( M ) = D 1 ⊕ · · · ⊕ D m and Hd( M ) = D m+1 ⊕ · · · ⊕ D n . Suppose first that m = 1, in which case M has a simple socle. Then, for each m + 1 = 2  i  n, there is clearly a uniserial submodule of M that has length 2, socle D 1 , and head isomorphic to D i . Now let m be arbitrary. Let further I := P ( D 1 ) ⊕ · · · ⊕ P ( D m ), which is an injective hull for M. That is, there is a short exact sequence

0 −→ M −→ I −→ Ω −1 ( M ) −→ 0 of A-modules. By our assumption, I has Loewy length l and is rigid. So, by Lemma 2.1, we deduce that Soc( M ) = Soc( I ), and every composition factor of Hd( M ) occurs as a composition factor of Radl−2 ( I )/ Radl−1 ( I ). Hence, for every m + 1  i  n, there is some 1  j  m with













M ∩ Radl−2 P ( D j ) / Radl−1 P ( D j ) : D i = 0.

Let M be the image of M under the canonical projection I −→ P ( D j ). Then M is a submodule of I of Loewy length 2, with socle isomorphic to D j , and with a factor module isomorphic to D i . By the first part of the proof, we already know that M has a submodule of length 2, with socle isomorphic to D j , and with head isomorphic to D i . This completes the proof of (a). 2 2.3. The Auslander–Reiten quiver We recall briefly the basic facts concerning Auslander–Reiten sequences of A-modules. Many of these do hold in much greater generality. For details we refer the reader to [2, Chapter IV] or [3, Chapter 4]. 2.5. Definition. Let M be an indecomposable A-module. An Auslander–Reiten (AR) sequence for M is a non-split short exact sequence f

g

0 −→ N −→ E −→ M −→ 0

(1)

of A-modules satisfying the following conditions: (i) N is indecomposable, and (ii) for each A-module X and each A-homomorphism h : X −→ M that is not a split epimorphism, there is an A-homomorphism h : X −→ E with h = g ◦ h . 2.6. Remark. (a) Given an indecomposable non-projective A-module M, there is an AR-sequence (1) terminating at M, and it is unique up to isomorphism of short exact sequences. Since A is a symmetric F -algebra, one has N ∼ = Ω 2 ( M ), by [1, IV.3.8]. Moreover, there is also an AR-sequence starting at M, and this sequence is also unique up to isomorphism of short exact sequences. The end term of such a sequence is isomorphic to Ω −2 ( M ). (b) Given an indecomposable projective A-module P , then P is also injective, and there is an Auslander–Reiten sequence of the form

0 −→ Rad( P ) −→ P ⊕ Ht( P ) −→ P / Soc( P ) −→ 0.

(2)

Furthermore, up to isomorphism of short exact sequences, (2) is the unique AR-sequence in which P occurs. One calls (2) the standard sequence corresponding to P . (c) If 0 −→ N −→ E −→ M −→ 0 is an AR-sequence of A-modules then there are projective A-modules Q and R such that

348

S. Danz, K. Erdmann / Journal of Algebra 397 (2014) 343–364

0 −→ Ω( N ) −→ Ω( E ) ⊕ Q −→ Ω( M ) −→ 0,

(3)

0 −→ Ω −1 ( N ) −→ Ω −1 ( E ) ⊕ R −→ Ω −1 ( M ) −→ 0

(4)

and

are AR-sequences of A-modules. We have Q = 0, unless (3) is a standard sequence; and R = 0, unless (4) is a standard sequence. (d) We also remark that condition (ii) in Definition 2.5 is equivalent to requiring that, for each A-homomorphism k : N −→ X that is not a split monomorphism, there is an A-homomorphism k : E −→ X with k = k ◦ f . 2.7. Remark/Definition. (a) The Auslander–Reiten (AR) quiver of A is the directed graph Γ ( A ) whose vertices are the isomorphism classes of indecomposable A-modules. As usual, we identify an isomorphism class with a particular representative. To define the arrows in Γ ( A ), let M and N be indecomposable A-modules. Then one has F -vector spaces

 R( M , N ) := f ∈ Hom A ( M , N ) f not an isomorphism





  R2 ( M , N ) := Span F g ◦ f f ∈ R M , M , g ∈ R M , N , M indecomposable A-module , and the homomorphisms in R( M , N )  R2 ( M , N ) are the irreducible maps from M to N. Note that an irreducible map is either injective or surjective, but not bijective. In Γ ( A ) one then has dim F (R( M , N )/R2 ( M , N )) arrows from M to N. The stable Auslander–Reiten quiver Γs ( A ) is obtained from Γ ( A ) by removing the vertices corresponding to isomorphism classes of projective modules, and all arrows attached to these vertices. (b) By C. Riedtmann’s structure theorem [27], every connected component Θ of Γs ( A ) has the form Z T /Π where T is a directed tree and Π is an admissible group of automorphisms of Z T . Let T be the undirected tree associated to T . This tree is unique, and it is called the tree class of Θ . If A is a block of a group algebra of wild representation type then, by [12], all components of Γs ( A ) have tree class A ∞ . That is, the component is isomorphic to Z T /Π where T is a straight line

•

•

•

•

...

and Π is either the identity group, or it is the cyclic group generated by the Heller operator Ω 2k , for some k ∈ N0 . In the second case Z T /Π is a tube of rank k. 2.8. Definition. Let M be an indecomposable non-projective A-module, and assume the component Θ of Γs ( A ) containing M has tree class A ∞ . Then one says that M has quasi-length 1, or that M lies at the end of Θ , if M has only one predecessor in Θ . In general, one defines the quasi-length ql( M ) of M in Θ to be the least integer l  1 such that there is a sectional path of length l − 1 from a module with quasi-length 1 to M. If in addition, for an indecomposable projective A-module P , both Rad( P ) and P / Soc( P ) belong to Θ then one also says that P belongs to Θ , and defines the quasi-length of P as the quasi-length of Rad( P ). 3. Modules with small socle or head 3.1. Hypotheses. From now on we will assume that all indecomposable A-modules under consideration belong to stable AR-components of A that have tree class A ∞ , and we also assume that every simple A-module in such a component has quasi-length 1. Furthermore, as before, we suppose that the algebra A is symmetric, indecomposable, and not of finite type. We also suppose that A has a duality fixing the simple modules.

S. Danz, K. Erdmann / Journal of Algebra 397 (2014) 343–364

349

3.2. Remark. Let A and B be symmetric F -algebras all of whose stable AR-components satisfy Hypotheses 3.1, and suppose that A and B are stably equivalent, via a functor Φ : A-mod −→ B-mod. Then Φ induces a graph isomorphism between Γs ( A ) and Γs ( B ). This follows, for example, from [1, X.1.3.]. In particular, if A = B then this applies to Φ ∈ {Ω, Ω −1 }, as well as to a functor induced by tensoring with a 1-dimensional module in the case of group algebras or Hopf algebras. Suppose that D is a simple A-module. Then D is in a component of tree class A ∞ if and only if Rad( P ) is, where P = P ( D ). If so then ql(Rad( P )) = ql( P / Soc( P )) = ql( D ) = 1; this follows from Hypotheses 3.1 and the fact that Rad( P ) ∼ = Ω( D ) and P / Soc( P ) ∼ = Ω −1 ( D ). The following exploits the fact that A is of infinite type. 3.3. Lemma. Let D and D be simple A-modules. Then one has (a) Rad( P ( D ))  P ( D )/ Soc( P ( D )). (a ) Ω 2 ( D )  D . (b) Rad( P ( D )) and P ( D )/ Soc( P ( D )) are not simple. Proof. Assume that Rad( P ( D )) ∼ = P ( D )/ Soc( P ( D )). Then Rad( P ( D ))/ Rad2 ( P ( D )) ∼ = Hd( P ( D )) ∼ = D, so that in the Ext-quiver of A there is only one arrow starting at D , namely D → D. Moreover, we get Soc2 ( P ( D ))/ Soc( P ( D )) ∼ = Soc( P ( D )) ∼ = D . Since P ( D ) is self-dual, we deduce Rad( P ( D ))/ Rad2 ( P ( D )) ∼ = D . Hence, D → D is the only arrow in the Ext-quiver of A starting at D. Since the Ext-quiver of A is connected, it must be of the form

•

•

But, by Remark 2.3, this is impossible, since A is not of finite type. This proves (a). Assertion (a ) follows immediately from (a) and the fact that Ω( D ) ∼ = Rad( P ( D )) and Ω −1 ( D ) ∼ = P ( D )/ Soc( P ( D )). To complete the proof, note that if Rad( P ( D )) were simple then the Ext-quiver of A would have only one vertex and one loop, and A would thus be of finite type, by Remark 2.3, which is not the case. If P ( D )/ Soc( P ( D )) were simple then also Rad( P ( D )) would be simple, which we have just excluded. 2 Let M be a non-projective indecomposable A-module. Recall that we are assuming Hypotheses 3.1, and let n = ql( M ). In [21, Lemma 3.2], Jost gives a formula for the lengths s( M ) and t( M ), and also l( M ) in terms of data associated to the wing of Γs ( A ) spanned by M. The modules in this part of Γs ( A ) are labelled as M i j , where i = 1, . . . , n and j = 1, . . . , n − i + 1, and there is an arrow from M i j to M kl if and only if (k, l) = (i + 1, j ) or (k, l) = (i − 1, j + 1). The ingredient of Jost’s formulae is the following lemma, which can be proved using the defining properties of AR-sequences, see [21, Lemma 3.1]. f

g

3.4. Lemma. Suppose that there is an AR-sequence 0 −→ M −→ X −→ M −→ 0 of A-modules. (a) The sequence 0 −→ Soc( M ) −→ Soc( X ) −→ Soc( M ) −→ 0 is exact if and only if M is not simple. (b) The sequence 0 −→ Hd( M ) −→ Hd( X ) −→ Hd( M ) −→ 0 is exact if and only if M is not simple. With our Hypotheses 3.1 the formulae of Jost can be refined. To do so, we fix some further notation, which is chosen in accordance with [21]. 3.5. Notation. (a) Let ql( M ) = n, as in [21, Lemma 3.2], and let B = { M 1 , . . . , M n } be the set of modules of quasi-length 1 in the wing of Γs ( A ) spanned by M. We choose notation such that M i = Ω −(2i −2) ( M 1 ). In Jost’s notation mentioned above, M i is called M 1i , and M is M n1 . (b) Furthermore, let J1 be the set of integers 2  i  n such that M i is not simple and not of the form P / Soc( P ) for any indecomposable projective A-module P , and let J2 be the set of integers

350

S. Danz, K. Erdmann / Journal of Algebra 397 (2014) 343–364

2  j  n such that M j is of the form P ( D )/ Soc( P ( D )) where D is simple and not a branch module, in the sense of Remark/Definition 2.2. (c) Dually, let J1 be the set of integers 1  j  n − 1 such that M j is not simple and not of the form Rad( P ), for any indecomposable projective A-module P , and let J2 be the set of integers 1  i  n − 1 such that M i is of the form Rad( P ( D )) where D is simple and not a branch module, in the sense of Remark/Definition 2.2. (d) Let also P be the set of indecomposable projective A-modules P such that P / Soc( P ) ∼ = M j , for some j  2. Then we define I to be the set of integers 1  i  n such that Rad( P )  M i  P / Soc( P ), for all P ∈ P . 3.6. Lemma. With Notation 3.5, one has





(a) s( M ) = s( M 1 ) +  i ∈J1 s( M i ) + i ∈J2 (s( M i ) − 1); (b) t( M ) = t( M n ) + i ∈J t( M i ) + i ∈J (t( M i ) − 1); (c) l( M ) =



i ∈I

l( M i ) +

1 

2

P ∈P

l(Ht( P )).

Proof. We only prove (a); parts (b) and (c) are proved similarly. By [21, Lemma 3.2(i)],

s( M ) =

n 

s ( M i ) − | E | − |P |

(5)

i =1

where E is the set of isomorphism classes of simple modules in the wing spanned by M n−1,2 . By our Hypotheses 3.1, E is a subset of B  { M 1 }, and since, trivially, s( M i ) = 1 if M i is simple, we can cancel the simple modules of the form M i from (5). Furthermore, if P ∈ P then P / Soc( P ) ∼ = Mi , for some i  2. So if P / Soc( P ) ∼ = M i and s( M i ) = 1 then we can also cancel M i and P from (5). By Lemma 3.3(b), this cancellation does not interfere with the previous cancellation. Now, s( P / Soc( P )) = 1 if and only if Soc( P ) is a branch module. This implies (a). 2 We will now apply this to A-modules with certain socle structures, and we will show that these have small quasi-lengths. 3.7. Lemma. Assume Hypotheses 3.1. Suppose that M is a non-projective indecomposable A-module with simple socle, and that M is not isomorphic to the heart of any projective indecomposable A-module. Then one has: (a) ql( M )  3. (b) If ql( M ) = 3 then Soc( M ) is the unique neighbour of a branch A-module D in the Ext-quiver of A. Moreover, Ht( P ( D )) is then isomorphic to a submodule of M. (c) If ql( M ) = 2 then Hd( M ) has a composition factor D that is a branch A-module, and 0 −→ Ω 2 ( D ) −→ M −→ D −→ 0 is an AR-sequence of A-modules. Proof. We apply Lemma 3.6 to M. To prove (a) (and part of (b)), we assume that n = ql( M )  3, and will show that this forces n = 3. By Lemma 3.6(a), we must have s( M ) = 1 = s( M 1 ), and J1 = ∅ = J2 . That is, for 2  i  n, the module M i is either simple, or of the form P ( D )/ Soc( P ( D )) where D is a simple branch A-module. Assume now that M i , for some 2  i < n, is simple. Then, by Lemma 3.3(a ), M i +1 cannot be simple. So M i +1 ∼ = P ( D )/ Soc( P ( D )), for some branch A-module D, and hence M i ∼ = Rad( D ), contradicting Lemma 3.3(b). Therefore, M 2 ∼ = P ( D )/ Soc( P ( D )), for a simple (branch) A-module D . Then, by Lemma 3.3(a), the module M 3 cannot be of the form P ( D )/ Soc( P ( D )), for any simple A-module D. Hence M 3 must be simple. But in the case where n > 3 this was just excluded, and we must have n = 3, proving (a).

S. Danz, K. Erdmann / Journal of Algebra 397 (2014) 343–364

351

Suppose that ql( M ) = 3. Then, by Lemma 3.6(a) and our previous considerations, there are a simple A-module D and a projective indecomposable A-module P = P ( D ) such that the wing in Γs ( A ) spanned by M has the following shape:

M f5

f6

Ht( P )

U

f1

Rad( P )

f2

f4

P

f3

f7

f8

P / Soc( P )

D

Each of the indecomposable modules drawn here must have a simple socle. More precisely, there is a simple A-module D

such that Soc( P / Soc( P )) ∼ = Soc(Ht( P )). Moreover, Soc(Rad( P )) ∼ = D

∼ = D . Each of the maps in the above picture is irreducible and thus either injective or surjective. By comparing dimensions, we see that each of the maps f 4 , f 5 , f 6 , and f 7 has to be injective. In particular, Soc( M ) ∼ = Soc(Ht( P )) ∼ = D

, and M has a submodule isomorphic to Ht( P ). This settles the proof of (b). Now suppose that ql( M ) = 2. Since M is not isomorphic to the heart of any projective indecomposable A-module, there must be an AR-sequence

0 −→ Ω 2 ( D ) −→ M −→ D −→ 0. Since M has a simple socle, Lemma 3.6 forces D to be simple. Moreover, Soc(Ω 2 ( D )) ∼ = Soc( M ) is simple, so that Hd(Ω( D )) is simple as well. Moreover, Hd(Ω( D ))  D. For otherwise the Ext-quiver of A would consist of one vertex and one loop only, which is impossible, by Remark 2.3. Hence D is a branch A-module in the sense of Remark/Definition 2.2, proving (c). 2 3.8. Lemma. Assume Hypotheses 3.1. Suppose further that the heart of every projective indecomposable A-module has composition length at least 4. Let M be an indecomposable A-module of Loewy length 2 such that Hd( M ) = D 1 ⊕ D 2 and Rad( M ) = D 3 ⊕ D 4 , for pairwise non-isomorphic simple A-modules D 1 , . . . , D 4 none of which has a branch module as a neighbour in the Ext-quiver of A. Then ql( M )  2. Proof. We apply Lemma 3.6(c). This gives

l( M ) = 4 =

 i ∈I

l( M i ) +





l Ht( P ) .

P ∈P

If |P | > 0 then |P | = 1, n = 2, and M must be isomorphic to the heart of some indecomposable projective module. But this is not possible, since, by Remark 2.3, the heart of every projective A-module is self-dual, whereas M is not. So we have |P | = 0, so that |I | = n  4. If we had n = 4 then all M i would be simple, which contradicts Lemma 3.3. Assume now that n = 3. Then two of the M i must be simple, and the third has length 2. Again by Lemma 3.3, the only possibility is that M 1 and M 3 are simple, and M 2 has length 2. Comparing dimensions, each of the two irreducible maps between M 1 and M must be injective, and hence M 1 ∼ = D 3 or M 1 ∼ = D 4 . We claim that M 1 is a branch module: − 1 ∼ we have Ω ( M 1 ) = Ω( M 2 ) and Soc(Ω( M 2 )) ∼ = Hd( M 2 ), which is simple. Therefore, the second socle of P ( M 1 ) is simple, that is, M 1 is a branch module. Note that the second socle of P ( M 1 ) cannot be isomorphic to M 1 , by Remark 2.3. By Proposition 2.4, M 1 is a neighbour of D 1 or D 2 in the Ext-quiver of A. But this contradicts our hypotheses, and the assertion follows. 2 3.9. Lemma. Keep the set-up and the assumptions from Lemma 3.8. Suppose further that, for 1  i  2, there is (up to isomorphism) precisely one simple A-module D 3  D i  D 4 such that the second Loewy layer of P ( D i ) has composition factors D 3 , D 4 , and D i , each with multiplicity 1. Then ql( M ) = 1.

352

S. Danz, K. Erdmann / Journal of Algebra 397 (2014) 343–364

Proof. By Lemma 3.8, we know that ql( M )  2. Assume that ql( M ) = 2. Note that M is not self-dual, and thus is not isomorphic to the heart of any projective A-module, by Remark 2.3. Hence there are indecomposable A-modules M 1 and M 2 and an AR-sequence

0 −→ M 1 −→ M −→ M 2 −→ 0. We may label the composition factors of M in such a way that we are in one of the following three cases: (i) Soc( M 1 ) = Rad( M 1 ) ∼ = D2. = D 3 ⊕ D 4 and Hd( M 1 ) ∼ = D 1 , and M 2 ∼ ∼ D 3 , and Soc( M 2 ) = Rad( M 2 ) ∼ (ii) M 1 = = D 4 and Hd( M 2 ) ∼ = D1 ⊕ D2. (iii) Soc( M 1 ) = Rad( M 1 ) ∼ = D 4 , Hd( M 1 ) ∼ = D 1 , and Soc( M 2 ) = Rad( M 2 ) ∼ = D 3 , Hd( M 2 ) ∼ = D2. We will show that each of these cases leads to a contradiction. So assume (i). By our assumptions, there is a simple A-module D 2 such that Ext1 ( D 2 , D 2 ) and Ext1 ( D 2 , D i ) are non-zero (i = 3, 4), so that D 2 , D 3 , D 4 are composition factors of Hd(Ω( M 2 )), and thus of Soc(Ω 2 ( M 2 )) ∼ = Soc( M 1 ) ∼ = Soc( M ), which is not the case. This excludes possibility (i), and (ii) is treated similarly. Lastly, assume (iii). It suffices to show that Hd(Ω( M 2 )) has at least two composition factors. Then we know that these also occur in Soc(Ω 2 ( M 2 )) ∼ = Soc( M 1 ), and we have again reached a contradiction. We have P ( M 2 ) = P ( D 2 ), and by our hypothesis, there is a simple A-module D 2 with

Rad( P ( D 2 ))/ Rad2 ( P ( D 2 )) ∼ = D 3 ⊕ D 4 ⊕ D 2 . Thus the factor module P := P ( D 2 )/ Rad2 ( P ( D 2 )) has Loewy length 2, with socle isomorphic to D 3 ⊕ D 4 ⊕ D 2 and head isomorphic to D 2 . We identify D 3 , D 4 , and D 2 with actual submodules of P . Then the factor module X := P /( D 4 ⊕ D 2 ) has length 2, with head isomorphic to D 2 and socle isomorphic to D 3 . Since X is also a factor module of P ( D 2 ) = P ( M 2 ), this forces X ∼ = M 2 . So we get non-split short exact sequences

0 −→ D 2 −→ P / D 4 −→ M 2 −→ 0, 0 −→ D 4 −→ P / D 2 −→ M 2 −→ 0, so that Ext1 ( M 2 , D 4 ) = 0 = Ext1 ( M 2 , D 2 ). Therefore,









Hom A Ω( M 2 ), D 4 = 0 = Hom A Ω( M 2 ), D 2 , and D 4 and D 2 occur in Hd(Ω( M 2 )), as claimed. This completes the proof of the lemma.

2

If there is a module of quasi-length 2 with simple socle, one can sometimes deduce that there must be a uniserial module of length 3 of a certain form. 3.10. Proposition. Assume Hypotheses 3.1. Let U be a non-projective A-module of Loewy length 2 with simple socle S, and suppose that U belongs to a stable component of Γs ( A ) of tree class A ∞ . If ql(U ) = 2 then the following hold: (a) There is an irreducible map U −→ D, where D is a branch A-module that is joined to S in the Ext-quiver of A. (b) There is a uniserial A-module of length 3, with composition factors S , D , S, from head to socle; here D denotes the branch module in part (a). Proof. Part (a) follows from Lemma 3.7(c). So we have an AR-sequence g

f

0 −→ U −→ D ⊕ X −→ V −→ 0 of A-modules. We will show that X has a uniserial subquotient of the form stated in (b).

(6)

S. Danz, K. Erdmann / Journal of Algebra 397 (2014) 343–364

353

Since ql(U ) = 2, our general Hypotheses 3.1 show that the sequence (6) cannot be a standard sequence. Therefore X is indecomposable and non-projective. Since D ∼ = D ∗ , it follows that the duality ∼ fixes the AR-sequence (6) and interchanges the end terms, that is, V = U ∗ and X ∼ = X ∗. U ⊕ D where U is semisimple. Then, by duality, we have Rad ( V) ∼ Write U / S ∼ = 1 = U 1∗ ⊕ D. Note 1 that [U 1 : D ] = [U 1∗ : D ] = 0, by Remark/Definition 2.2, since U 1∗ ⊕ D is isomorphic to a direct summand of Rad( P ( S ))/ Rad2 ( P ( S )). We now construct the desired uniserial module in three steps.

(1) Let U 1∗ be the submodule of V described above. There is a submodule of Soc( X ) isomorphic to U 1∗ : the inclusion map ι : U 1∗ −→ V factors through f , by the Auslander–Reiten property. Thus there is an A-homomorphism ψ : U 1∗ −→ D ⊕ X such that ι = f ◦ ψ . Since [U 1∗ : D ] = 0, the image of ψ is contained in X , and ψ is injective, since ι is. Since U 1∗ is semisimple, ψ(U 1∗ ) is contained in Soc( X ). (2) We claim that ψ(U 1∗ ) ∩ g (U ) = 0. Let m = ψ(x) ∈ g (U ) with x ∈ U 1∗ . We have f (m) = 0, since m ∈ g (U ) = ker( f ). So





0 = f ψ(x) = ι(x) and x = 0, since ι is injective. Thus m = 0. (3) By the above, U has a factor module isomorphic to U 1 ; let π : U −→ U 1 be an epimorphism. Then π factors through g, by the Auslander–Reiten property. Since D does not occur in U 1 , there is an A-homomorphism η : X −→ U 1 with π = η ◦ g 2 , where g 2 is the component of g mapping into X . We can now define the required module: let X := ker(η). Then Rad( X ) ⊆ X , since im(η) is semisimple. So we have a chain of submodules





0 ⊂ ψ U 1∗ ⊂ Soc( X ) ⊂ Rad( X ) ⊂ X ⊂ X with quotients, in the same order, isomorphic to: U 1∗ , S , D , S , U 1 . We define M := X /ψ(U 1∗ ), which has composition factors D with multiplicity 1, and S with multiplicity 2. It remains to show that M is uniserial with head and socle isomorphic to S. For this, we will prove the following assertions: (i) M has a uniserial submodule with head isomorphic to D and socle isomorphic to S; (ii) M has a uniserial quotient with head isomorphic to S and socle isomorphic to D. We first prove (i). Let Y ⊂ U be the kernel of π . Then Y is uniserial of length 2, with Hd(Y ) ∼ =D and Soc(Y ) ∼ = S. Namely, Y has these composition factors, and 0 = Soc(Y ) ⊆ Soc(U ) = S, which is simple. Let g 1 : U −→ D, g 2 : U −→ X , f 1 : D −→ V , and f 2 : X −→ V be the obvious homomorphisms induced by g and f , respectively. All these maps are irreducible and, by comparing dimensions, we deduce that f 1 and g 2 are injective, and f 2 and g 1 are surjective. We have proved that the intersection g (U ) ∩ ψ(U 1∗ ) = g 2 (U ) ∩ ψ(U 1∗ ) is 0, so g 2 (Y ) ∩ ψ(U 1∗ ) = 0 as well. Thus g 2 induces an injection g 2 : Y −→ X /ψ(U 1∗ ). Now we only need to show that g 2 (Y ) is contained in X . Suppose not, so that ( g 2 (Y ) + X )/ X is non-zero. But then D ∼ = Hd( g 2 (Y )) occurs as a composition factor in X / X . But X/X ∼ = U 1 , and [U 1 : D ] = 0, a contradiction. Hence g¯2 (Y ) is a uniserial submodule of M, of the required structure. This settles (i). (ii) From the information in (3), recalling X = ker(η), we get exact sequences that form a commutative diagram

354

S. Danz, K. Erdmann / Journal of Algebra 397 (2014) 343–364

0

0

⏐ ⏐ 

⏐ ⏐  gˆ

0 −−−−→ Y −−−−→ D ⊕ X

⏐ ⏐ 

⏐ ⏐ 

g 

g=

1 g2

ˆf

−−−−→ W −−−−→ 0 ⏐ γ⏐  f =( f 1 , f 2 )

0 −−−−→ U −−−−→ D ⊕ X −−−−−−→ V −−−−→ 0

⏐ ⏐

⏐ ⏐

π

(0,η) id

U 1 −−−−→

⏐ ⏐  0

U1

⏐ ⏐  0

Here the maps gˆ , ˆf in the top vertical row are the restrictions of g and f , respectively. The map γ is induced by the top left commutative square. In fact, γ is an isomorphism; to see this, apply the Snake Lemma to the two vertical short exact sequences. Let Z := V /ι(U 1∗ ). This is uniserial of length 2, with head S and socle D (namely, it is a quotient of V and therefore has a simple head, and it has length 2). Let ρ : V → Z be the canonical epimorphism. Let ˆf 2 be the restriction of f 2 to X , which is a component of ˆf . We have ˆf 2 (ψ(U 1∗ )) = f 2 (ψ(U 1∗ )) = ι(U 1∗ ) = ker(ρ ). Consider the composite

f 2 := ρ ◦ ˆf 2 : X −→ Z . The submodule ψ(U 1∗ ) is mapped to 0 by f 2 . Hence f 2 induces an epimorphism from M onto Z .

2

3.11. Proposition. Assume Hypotheses 3.1. Assume also that, for each branch A-module D and its neighbour S in the Ext-quiver, both P ( D ) and P ( S ) are rigid, with the same Loewy length. Then there is no A-module U with ql(U ) = 2 that satisfies the hypotheses in Proposition 3.10. Proof. Assume, for a contradiction, that U as in Proposition 3.10 exists. So there are a branch A-module D and an AR-sequence

0 −→ Ω 2 ( D ) −→ U −→ D −→ 0.

(7)

The neighbour of D in the Ext-quiver of A is the socle S of U . Recall that, by the definition of a branch module in Remark/Definition 2.2(b), we have D  S. By Proposition 3.10, there is a uniserial A-module M with composition factors S , D , S. This module is a submodule of the projective A-module P := P ( S ). Our strategy now is to show that P does not have such a uniserial submodule and hence get a contradiction. First of all, we apply Ω −1 to the AR-sequence (7), and obtain the standard sequence

0 −→ Ω( D ) −→ P ( D ) ⊕ H −→ Ω −1 ( D ) −→ 0 where H := Ht( P ( D )), so that, in particular, Ω( H ) ∼ = U . Since D is a branch module with unique neighbour S in the Ext-quiver, we must have Hd( H ) ∼ = S. As mentioned in Remark 2.3, H is self-dual,

S. Danz, K. Erdmann / Journal of Algebra 397 (2014) 343–364

355

so that also Soc( H ) ∼ = Hd( H ) ∼ = S. Since P is a projective cover of H and Ω( H ) ∼ = U , there is a short exact sequence s

h

0 −→ U −→ P −→ H −→ 0 . Let l be the common Loewy length of P = P ( S ) and P ( D ), and recall that both of these modules are rigid. So H has Loewy length l − 2. Claim. We have s(U ) = Radl−2 ( P ) and h(Radl−3 ( P )) = Soc( H ) ∼ = S. From this we then deduce that Radl−3 ( P )/ Radl−2 ( P ) ∼ = S, since s(U ) = ker(h). To prove the claim, note, firstly, that s(U ) ⊆ Radl−2 ( P ), by Lemma 2.1. On the other hand, we have h(Radl−2 ( P )) ⊆ Radl−2 (h( P )) = Radl−2 ( H ) = 0. Thus Radl−2 ( P ) = s(U ). Moreover, we have h(Radl−3 ( P )) ⊆ Radl−3 ( H ) = Soc( H ) ∼ = S, which is simple. So either h(Radl−3 ( P )) = Soc( H ) or h(Radl−3 ( P )) = 0. But, by what we have just shown, in the latter case we would get Radl−3 ( P ) ⊆ ker(h) = Radl−2 ( P ), a contradiction. This settles the claim. Now consider an inclusion γ : M −→ P , where M is the uniserial module of length 3. Then γ ( M ) ⊆ Radl−3 ( P ), by Lemma 2.1. Moreover, γ ( M )  Radl−2 ( P ), since Rad(γ ( M )) has length 2 and Rad(Radl−2 ( P )) = Soc( P ) is simple. By the above claim, we also know that Radl−2 ( P ) is the unique maximal submodule of Radl−3 ( P ). Hence γ ( M ) = Radl−3 ( P ). This implies Rad( P )/ Rad2 ( P ) ∼ = Radl−2 ( P )/ Radl−1 ( P ) ∼ = D. Consequently, both S and D are branch modules, and they are the unique neighbours of each other. Since the Ext-quiver of A is connected, it has only two vertices, labelled by D and S. As we have seen in Remark 2.3, this cannot happen, since A is not of finite type. This completes the proof of the theorem. 2

3.12. Remark. Proposition 3.11 will be a key ingredient in the proof of Theorem 6.1 on weight-2 blocks of symmetric groups. The conditions on P ( D ) and P ( S ) in Proposition 3.11 imply that P ( S ) has a ‘waist’, that is, P ( S ) is not uniserial, but has some simple Loewy layer other than socle or head. From Proposition 3.10 and from the proof of Proposition 3.11 we deduce that the second socle layer of P ( S ) has length > 1 and the third socle layer is simple. By duality, the second radical layer of P ( S ) has length > 1 and the third radical layer is simple. We learnt that Fayers [14] recently proved that projective modules for blocks of symmetric groups of weight w = 2 do not have a waist. 4. Morita equivalences Let A and B be symmetric F -algebras. Throughout this subsection, we suppose that

Φ : A-mod −→ B-mod and Ψ : B-mod −→ A-mod are F -linear functors such that Φ is left adjoint to Ψ . Thus, for every A-module M and every B-module N, we have an F -isomorphism



 ψ M , N : Hom B Φ( M ), N −→ Hom A M , Ψ ( N ) , and we set η M := ψ M ,Φ( M ) (idΦ( M ) ) and ε N := (ψΨ ( N ), N )−1 (idΨ ( N ) ). Then η := (η M ) M ∈ A -mod : Id A -mod −→ Ψ ◦ Φ and ε := (ε N ) N ∈ B -mod : Φ ◦ Ψ −→ Id B -mod are natural transformations, called the unit and the counit, respectively, of the adjunction (Φ, Ψ ). Let { D 1 , . . . , D r } be a set of representatives for the isomorphism classes of simple A-modules, and we fix a subset I ⊆ {1, . . . , r } such that,

356

S. Danz, K. Erdmann / Journal of Algebra 397 (2014) 343–364

• for every i ∈ I , the B-module Φ( D i ) is simple, and • for ever i ∈ I , both η D i and εΦ( D i ) are isomorphisms. Denote by C I the full subcategory of A-mod such that every composition factor of a module in C I is isomorphic to one of the simple modules in { D i | i ∈ I }. Furthermore, denote by D I the full subcategory of B-mod such that every composition factor of a module in D I is isomorphic to one of the simple modules in {Φ( D i ) | i ∈ I }. 4.1. Proposition. With the above notation, the restrictions of the functors Φ and Ψ , respectively, induce an equivalence between C I and D I . Proof. The result follows using the arguments given by Scopes in the proof of [28, Theorem 4.2].

2

4.2. Remark. We will now show that C I is actually equivalent to the module category of a symmetric F -algebra. In particular, there exists a concept of Auslander–Reiten theory for modules in C I . (a) Let e be an idempotent in A. Then one has a pair (Λ, Γ ) of adjoint functors

Λ : e Ae-mod −→ A-mod,

N −→ Ae ⊗e Ae N ,

Γ : A-mod −→ e Ae-mod,

M −→ eM .

We call a simple A-module D regular if e D = 0, in which case e D is then a simple e Ae-module, see [18, Section 6.2]. Otherwise, we call D singular. For any A-module M, we denote by ( M ) the largest A-submodule of M all of whose composition factors are singular. Similarly, we denote by Σ( M ) the smallest submodule M of M such that the factor module M / M has only singular composition factors. In this way, we obtain a functor

 : A-mod −→ A-mod, M −→ M /( M ). (b) Now let {e i | i ∈ I } be a set of mutually orthogonal idempotents in A such  that, for i ∈ I , the A-module Ae i is a projective cover of the simple A-module D i . We set e I := i ∈ I e i , which is again an idempotent in A. 4.3. Theorem. (See [8, Thm. 3.1d].) Let M be the full subcategory of A-mod whose objects are the A-modules M such that ( M ) = 0 and Σ( M ) = M. Then the restrictions of the functors Γ and  ◦ Λ induce an equivalence between the categories M and e I Ae I -mod. 4.4. Remark. Since the full subcategory M of A-mod is equivalent to the module category e I Ae I -mod and since e I Ae I is a (symmetric) F -algebra, for every module M in M there is, in M, an AR-sequence ending in M. One might therefore ask how the AR-sequences for M in A-mod and those in M are related. Note that the objects in M are precisely those A-modules M such that all composition factors of Hd( M ) and all composition factors of Soc( M ) are regular. Thus, in particular, given a non-projective indecomposable A-module belonging to M and a projective cover P of M in A-mod, the module P belongs to M as well. This follows from the fact that Soc( P ) ∼ = Hd( P ) ∼ = Hd( M ). However, the AR-translate Ω 2 ( M ) need not be in M. So the AR-sequences for M in A-mod can be different from those in M. Nevertheless, imposing further conditions on M we get the following: 4.5. Proposition. Let M be a non-projective indecomposable A-module belonging to M, and suppose that Ω 2 ( M ) belongs to M as well. Then any AR-sequence for M in A-mod restricts to an AR-sequence for M in M, and every AR-sequence for M in M is obtained in this way.

S. Danz, K. Erdmann / Journal of Algebra 397 (2014) 343–364

357

Proof. Let f

g

0 −→ Ω 2 ( M ) −→ E −→ M −→ 0

(8)

be an AR-sequence of A-modules. Then, by our hypothesis, Ω 2 ( M ) belongs to M. Recall that an A-module belongs to M if and only if all of its simple submodules and all of its simple factor modules are regular. Since (8) is exact, it follows that E must belong to M. So (8) is a non-split exact sequence in M. Moreover, if h : X −→ M is a morphism in M that is not a split epimorphism in M then h : X −→ M is not a split epimorphism in A-mod either, since M is a full subcategory of A-mod. Thus h factors through E, proving that (8) is indeed an AR-sequence in M. Let, conversely, f

g

0 −→ N −→ E −→ M −→ 0

(9)

be another AR-sequence in M. Then (8) and (9) are isomorphic as sequences in M, and thus in A-mod. Since (8) is an AR-sequence of A-modules, so is (9), and the proof of the proposition is complete. 2 We also mention the following result, which is a consequence of Theorem 4.3. 4.6. Corollary. The category C I is equivalent to e I Ae I -mod, and the category D I is equivalent to Φ(e I ) B Φ(e I )-mod. 5. Symmetric groups Throughout this section, let Sn be the symmetric group of degree n  1. Moreover, we assume from now on that the field F has positive characteristic p. We will briefly summarize some of the basic properties concerning the representation theory of the group algebra F Sn and its blocks. For further background, we refer the reader to [19,20]. Given a partition μ of n, we denote by S μ the Specht F Sn -module corresponding to μ. If μ is p-regular, that is, if every part of μ occurs with multiplicity less than p then the unique simple quotient module of S μ will be denoted by D μ . For every partition μ of n, we denote the conjugate

partition by μ . Then S μ ∼ = ( S μ )∗ ⊗ sgn where sgn denotes the sign representation of F Sn . Recall that blocks B of F Sn are parametrized by pairs ( w , κ ), where w is a non-negative integer and where κ is a p-regular partition of n − p w. One calls w the (p-)weight and κ the (p-)core of B. We sometimes write B = B (κ ). The defect groups of B are then the Sn -conjugates of the Sylow p-subgroups of S p w , by [20, Theorem 6.2.39]. In particular, B is of wild representation type if and only if p = 2 and w  3, or p > 2 and w  2; see the introduction of [11]. We also recall that to each block B of weight w of F Sn we can assign its p-content, which is a sequence γ = (γ0 , . . . , γ p −1 ) of non-negative integers with γ0 + · · · + γ p −1 = n − p w. Here γi is the number of nodes with p-residue i in the Young diagram of any partition in the block B. 5.1. The Scopes equivalence We want to apply the results about partial Morita equivalences from Section 4 to the situation where the F -algebras are p-blocks of symmetric groups of the same weight. This can be done via the higher divided power functors, introduced by Kleshchev. We record the facts needed later in this paper, and we refer the reader to [23, Chapters 11.1, 11.2] for more background information, and for precise definitions of the notions of i-addable/removable and i-normal/conormal nodes of Young diagrams. Let B be a block of F Sn with p-content γ = (γ0 , . . . , γ p −1 ), and let i ∈ {0, . . . , p − 1}. Let further k  0. For every partition μ of n, we denote by [μ] its Young diagram.

358

S. Danz, K. Erdmann / Journal of Algebra 397 (2014) 343–364

5.1. Remark. (a) Assume that F Sn−k has a block B with p-content γ¯ = (γ0 , . . . , γi −1 , γi − k, γi +1 , . . . , γ p −1 ). Then one has F -linear functors

eki : B-mod −→ B-mod,

and

f ik : B-mod −→ B-mod,

where, for a B-module M, one defines

eki ( M ) := B · ResSn ( M ), S

n−k

and, for a B-module N, one defines

f ik ( N ) := B · IndSn ( N ). S

n−k

Then eki and f ik are two-sided adjoint to each other. (b) Furthermore, one has the higher divided power functors (k)

ei

: B-mod −→ B-mod,

where eki is naturally equivalent to (k)

ei

(k)



(k)

k! e i

and

(k)

fi

: B-mod −→ B-mod

, and f ik is naturally equivalent to



k!

(k)

fi

. The functors

and f i are also F -linear and two-sided adjoint to each other. (c) For every B-module M, one defines





εi ( M ) := max m  0 em i ( M ) = 0 ,

and





ϕi ( M ) := max l  0 f il ( M ) = 0 .

If D λ is a simple B-module then ε := εi ( D λ ) equals the number of i-normal nodes of [λ], and ¯ (ε ) ei ( D λ ) ∼ = D λ , where [λ¯ ] is obtained by removing all i-normal nodes from [λ]. Analogously, ϕ := ˆ ϕi ( D λ ) equals the number of i-conormal nodes of [λ], and f i(ϕ ) ( D λ ) ∼ = D λ where [λˆ ] is obtained by

adding all i-conormal nodes to [λ]. A proof for this is given by Kleshchev in [23, L. 11.2.12, Theorem 11.2.10, Theorem 11.2.11].

5.2. Definition. With the above notation, suppose that B and B have common weight w  0. Suppose further that there is some k  1 such that B and B form a [ w : k]-pair, in the sense of [28]. Let I be the set of p-regular partitions of n such that λ ∈ I if and only if







¯

εi D λ = k = ϕi D λ . ¯

¯

Note that D λ ∼ = e i ( D λ ) and D λ ∼ = f i ( D λ ). Let further C I be the full subcategory of B-mod whose objects are the B-modules all of whose composition factors are labelled by partitions in I . Analogously, D I denotes the full subcategory of B-mod whose objects are the B-modules whose composi¯ where λ ∈ I . tion factors are labelled by partitions λ (k)

(k)

The next result is now a consequence of Proposition 4.1: 5.3. Proposition. Keep the notation of Definition 5.2. (k)

(a) The functors e i (k)

(k)

and f i

(k)

induce an equivalence between the categories C I and D I . In particular, e i

and

f i preserve indecomposability, simplicity, and Loewy structures of modules in C I and their correspondents in D I .

S. Danz, K. Erdmann / Journal of Algebra 397 (2014) 343–364

359

(b) Let S μ be a Specht module in C I such that [μ] has precisely k removable nodes of residue i and no

(k) ¯ ] is obtained by removing all i-removable nodes from i-addable nodes. Then e i ( S μ ) ∼ = S μ¯ where [μ [ μ] .

Proof. Part (a) follows from Proposition 4.1. If S μ is as in (b) then the ordinary branching rules for Specht modules [20, Theorem 2.4.3] show that eki ( S μ ) has a Specht filtration with k! factors all of which are isomorphic to S μ¯ . We can proceed as in the proof of [28, Lemma 3.1], and get eki ( S μ ) ∼ =



k!

S μ¯ . On the other hand, eki ( S μ ) ∼ =



(k)

k! e i

(k) ( S μ ). Thus S μ¯ ∼ = e i ( S μ ), as claimed. 2

Together with Corollary 4.6 and Theorem 3.2, we also have 5.4. Corollary. With the notation as in Definition 5.2, suppose that S μ is a Specht F Sn -module belonging to B. Suppose further that B has wild representation type. If both S μ and Ω 2 ( S μ ) are objects in C I then (k) ql( S μ ) = ql(e i ( S μ )). 5.5. Remark. (a) In the situation of Proposition 5.3, we will use the notation

M ←→i N , (k)

whenever M is a module in C I and N = e i ( M ) is its correspondent in D I .

(c) We emphasize that in the case where C I = B-mod and D I = B-mod, the blocks B and B are (k) (k) Scopes equivalent in the sense of [28], and the functors e i and f i are precisely the functors used by Scopes in [28, Sec. 4]. 5.2. Blocks of small weight In Section 6, we want to study indecomposable Specht F Sn -modules in blocks of wild type and small weight, and determine their quasi-lengths. Amongst the wild blocks of F Sn , the ones of weight 2 are best understood. We now summarize some of the known structural properties of F Sn -blocks of weight 2 over characteristic p > 2 in the next theorem, which will be essential for proving Theorem 6.1. 5.6. Theorem. Let p > 2, and let B be a block of F Sn of weight 2. Then the following hold: (a) The projective indecomposable F Sn -modules belonging to B are rigid, and have common Loewy length 5. (b) The Ext-quiver of B is bipartite. In particular, Ext1 ( D , D ) = 0, for every simple B-module. Furthermore, dim(Ext1 ( D , D ))  1, for all simple B-modules D and D . (c) Each decomposition number of B is at most 1. (d) If S μ is a Specht F Sn -module belonging to B then S μ has a simple head or a simple socle. More precisely, (i) If μ is both p-regular and p-restricted then S μ has both a simple head and a simple socle. In this case, S μ has Loewy length 3. (ii) If μ is neither p-regular nor p-restricted then S μ is simple. (iii) If μ is p-regular but not p-restricted then S μ has a simple head, and S μ has Loewy length at most 2. (iv) If μ is p-restricted but not p-regular then S μ has a simple socle, and S μ has Loewy length at most 2. In any case, S μ has at most five composition factors. Proof. Assertions (a)–(c) as well as the assertion concerning the composition length in (iv) have been proved by Scopes in [29]. The remaining statements in (iv) can be deduced from work of Chuang and Tan [9], and Richards [26]. For explicit proofs, we refer to [6, Proposition 2.2] and [10, Theorem 5.4]. 2

360

S. Danz, K. Erdmann / Journal of Algebra 397 (2014) 343–364

6. Quasi-lengths of Specht modules In this section we will consider Specht F Sn -modules in blocks of weight 2 for p > 2, and in blocks of weight 3 for p = 3. As we have already mentioned at the beginning of Section 5, such blocks are of wild representation type, thus, by [12], each component of their stable AR-quivers has tree class A ∞ . Moreover, by [19, Corollary 13.18] each Specht module in such a block is indecomposable, and, by [20, Theorem 7.1.14], each simple F Sn -module is self-dual. Furthermore, a Specht F Sn -module in a block of weight w > 0 cannot be projective. Finally, by [5], each simple F Sn -module in a block of wild type has quasi-length 1. Hence, Hypotheses 3.1 are satisfied for the blocks considered in the sequel. 6.1. Blocks of weight 2 6.1. Theorem. Let p > 2, and let B be a block of F Sn of weight 2. Then, for every Specht F Sn -module S μ belonging to B, one has ql( S μ ) = 1. Proof. Let S μ be an F Sn -Specht module in B, and let Θ be the connected component of Γs ( B ) containing S μ . If S μ has both a simple head and a simple socle, or is simple, then we know that ql( S μ ) = 1, by [21, Theorem 6.3], or by [5]. So, by Theorem 5.6, we are left to consider the case when S μ has either a simple head or a simple socle but not both, and then we know that S μ has Loewy length 2 and is multiplicity-free. We examine the case where S μ has a simple socle; the other case is dual. Since such a module μ S is not self-dual, it is not isomorphic to the heart of a projective module. So Lemma 3.7 applies, and thus ql( S μ )  3. Assume that ql( S μ ) = 3. Then, by Lemma 3.7 again, there is a projective indecomposable B-module P such that Ht( P ) is isomorphic to a submodule of S μ . Moreover, Ht( P ) has a simple socle D , and [Ht( P ) : D ]  2, since D is also isomorphic to the head of Ht( P ) and Ht( P ) is rigid of Loewy length 3. Thus [ S μ : D ]  2 which, by Theorem 5.6, is impossible. Hence, ql( S μ )  2. Next we assume, for a contradiction, that ql( S μ ) = 2. We take U = S μ . Then the hypotheses of Proposition 3.10 are satisfied. As well, by Theorem 5.6, all projective indecomposable B-modules are rigid and have the same Loewy length. Now Proposition 3.11 gives a contradiction. Therefore, ql( S μ ) = 1. 2 6.2. Some blocks of weight w > 2 In the following, assume F has characteristic 3. We will show: 6.2. Theorem. Let n  1, and let B be a block of F Sn of weight 3. Then every Specht F Sn -module belonging to B has quasi-length 1. By [13], we know that there are twelve Scopes-equivalence classes of symmetric group blocks of 3-weight 3. A set of representatives for these is given by the following blocks: n 9 10 11 13 14 15 19 25

3-core

∅ (1) (2) (12 ) (3, 1) (2, 12 ) (3, 12 ) (4, 2) (22 , 12 ) (5, 3, 12 ) (4, 22 , 12 ) (6, 4, 22 , 12 )

Here, blocks are again parametrized by their 3-cores. Since any other 3-block of weight 3 must be Scopes equivalent to one of the blocks listed, it suffices, by Corollary 5.4, to prove Theorem 6.2

S. Danz, K. Erdmann / Journal of Algebra 397 (2014) 343–364

361

only in the cases where B is one of these. Furthermore, since neighbouring blocks in the above table are equivalent via tensoring with the sign representation, it always suffices to prove the assertion of Theorem 6.2 for one of them. This leaves us to deal with eight blocks, with respective 3-cores: ∅, (1), (2), (3, 1), (3, 12 ), (4, 2), (5, 3, 12 ), and (6, 4, 22 , 12 ). For the proof of Theorem 6.2, we examine these, and we say that a given Specht module S λ in one of these blocks

satisfies the condition (∗) if λ is 3-regular or 3-restricted, or if S λ has at most two composition factors. Note that the decomposition numbers for the relevant blocks are known and can, for instance, be found in [13]. We, moreover, recall from [5, Proposition 4.6] that, for every indecomposable projective module P = P ( D ) in a 3-block of weight 3, we have [Ht( P ) : D ]  2. This will be important when applying Lemma 3.7(b) and Lemma 3.9. We will see that all but three Specht modules in the eight blocks listed above have a simple head 3 2 3 or a simple socle. These remaining modules are S (6,3,2,1 ) in the block B ((3, 12 )), S (8,3 ,2,1 ) in the 2 ( 9, 42 , 3, 15 ) 2 2 block B ((5, 3, 1 )), and S in the block B ((6, 4, 2 , 1 )). It will be helpful to treat these three modules separately, which will be done in the next proposition. For ease of notation, we replace simple modules by their labelling partitions. 3 2 3 2 5 6.3. Proposition. The Specht modules S (6,3,2,1 ) , S (8,3 ,2,1 ) , and S (9,4 ,3,1 ) , respectively, have the following Loewy structures:





3 (6, 42 ) ⊕ (6, 3, 22 , 1) S (6,3,2,1 ) = , (9, 3, 2) ⊕ (6, 4, 22 ) 2 5 S (9,4 ,3,1 ) =







2 3 (8, 5, 4, 2) ⊕ (8, 32 , 22 , 1) S (8,3 ,2,1 ) = , (11, 32 , 2) ⊕ (8, 6, 3, 2)

 (9, 6, 5, 3, 12 ) ⊕ (9, 42 , 32 , 2) . (12, 42 , 3, 12 ) ⊕ (9, 7, 4, 3, 12 )

3 2 3 2 5 In particular, ql( S (6,3,2,1 ) ) = ql( S (8,3 ,2,1 ) ) = ql( S (9,4 ,3,1 ) ) = 1.

Proof. It suffices to prove the assertion concerning the Loewy structures. From [13, Fig. 4] we then deduce that the three Specht modules satisfy the hypotheses of Lemma 3.9, and the assertion concerning the quasi-lengths follows. From the decomposition matrices in [13] we know that the modules 3 2 3 2 5 S (6,3,2,1 ) , S (8,3 ,2,1 ) , and S (9,4 ,3,1 ) , respectively, have the claimed composition factors. We apply the 3 3 Jantzen–Schaper formula [4] to S (6,3,2,1 ) , and get that S (6,3,2,1 ) has precisely two non-zero Jantzen– 2 2 Schaper layers. The second layer has composition factors D (6,4 ) and D (6,3,2 ,1) , and the third layer 2 has composition factors D (9,3,2) and D (6,4,2 ) . Since Jantzen–Schaper layers are self-dual, these two 3 3 non-zero layers must be semisimple. Since S (6,3,2,1 ) is indecomposable, S (6,3,2,1 ) has the claimed Loewy structure. Moreover, 2 5 2 4 2 3 2 4 3 S (9,4 ,3,1 ) ←→1 S (9,4,3 ,1 ) ←→2 S (8,3 ,2,1 ) ←→1 S (7,3 ,1 ) ←→0 S (6,3,2,1 ) , 2 3 2 5 and from Proposition 5.3 we deduce that the Loewy structures of S (8,3 ,2,1 ) and S (9,4 ,3,1 ) are as claimed. 2

We now prove Theorem 6.2. Proof of Theorem 6.2. (1) The blocks of F S9 and F S10 have no simple branch modules, and every Specht module belonging to one of these blocks satisfies condition (∗) above. So Lemma 3.7 implies the assertion of the theorem, for 9  n  10.

362

S. Danz, K. Erdmann / Journal of Algebra 397 (2014) 343–364

(2) By [13, Fig. 4], the principal block B 0 of F S11 contains one simple branch module, namely D (8,3) , whose unique neighbour in the Ext-quiver is F . Every Specht module in B 0 that does not have a composition factor isomorphic to D (8,3) satisfies condition (∗), and has thus quasi-length 1, by Lemma 3.7. The remaining Specht modules in B have the following Loewy structures, which we have obtained by explicitly constructing the relevant modules with the computer, using the computer algebra systems GAP [15] and MAGMA [7]. For notational convenience, we replace simple modules again by their labelling partitions.

⎡

⎤ (4, 3, 22 ) 2 2 ⎢ (11) ⊕ (5, 4, 1 ) ⊕ (5, 3 ) ⎥ 2 ⎢ ⎥ S (4,3,2 ) = ⎢ (8, 2, 1) ⊕ (5, 3, 2, 1) ⎥ , ⎣ ⎦ (11) (8, 3) ⎡

⎤ (5, 3, 2, 1) 3 ⎢ (11) ⎥ S (5,2 ) = ⎣ ⎦, (8, 3) (11) ⎡ ⎤ (8, 2, 1) ⎢ (11) ⎥ S (8,2,1) = ⎣ ⎦, (8, 3) (11)

⎡

⎤ (5, 3, 2, 1) 2 2 ⎢ (11) ⊕ (5, 4, 1 ) ⊕ (5, 3 ) ⎥ ⎢ 2 ⎥ ⎢ (5 , 1) ⊕ (8, 3) ⊕ (6, 3, 12 ) ⎥ (5,3,2,1) S =⎢ ⎥, (11) ⎢ ⎥ ⎣ ⎦ (8, 2, 1) (7, 3, 1) ⎡ ⎤ (7, 3, 1) 8, 2, 1) ⎥ ( ⎢ S (7,3,1) = ⎣ ⎦, (11) (8, 3)  S (8,3) =

 (8, 3) . (11)

Therefore, Lemma 3.7 and [5, Prop. 4.6] show that each of these modules has quasi-length 1. (3) By [13, Fig. 4], the block B ((3, 1)) of F S13 with 3-core (3, 1) does not contain any simple 3 3 3 branch module. For the Specht module S (6,2 ,1) in B ((3, 1)), we get S (6,2 ,1) ←→2 S (5,2 ) , so that

⎡ ⎢ 3 ⎢ S (6,2 ,1) = ⎢ ⎣

(2 )

f 2 ((5, 3, 2, 1)) (2 )

f 2 ((11)) (2 )

f 2 ((8, 3)) (2 )

f 2 ((11))

⎤

⎤ ⎡ (6, 3, 2, 12 ) ⎥ ⎥ ⎢ (12, 1) ⎥ ⎥=⎣ ⎦, (9, 4) ⎦ (12, 1)

3

by (2) and Proposition 5.3. In particular, S (6,2 ,1) has both simple head and simple socle. Every other Specht module in B ((3, 1)) satisfies condition (∗), and has thus simple head or simple socle. Therefore, Lemma 3.7 implies that all Specht modules in B ((3, 1)) must have quasi-length 1. (4) By [13, Fig. 4], the block B ((3, 12 )) of F S14 has no simple branch module. Every Specht mod3 ules in B ((3, 12 )), except S (6,3,2,1 ) , satisfies condition (∗), and has thus simple head or simple socle 3 and therefore quasi-length 1, by Lemma 3.7. By Proposition 6.3, we already know that S (6,3,2,1 ) has quasi-length 1 as well. This settles the case n = 14. (5) By [13, Fig. 4], the block B ((4, 2)) of F S15 has one simple branch module, namely D (7,5,3) with neighbour D (10,5) . The Specht modules that do not have any composition factor isomorphic to 4 3 D (7,5,3) have simple heads or simple socles: since S (7,2 ) ←→0 S (6,2 ,1) , we have

⎡ ⎢ 4 ⎢ S (7,2 ) = ⎢ ⎣

(2 )

f 0 ((6, 3, 2, 12 )) (2 )

f 0 ((12, 1)) (2 )

f 0 ((9, 4)) (2 )

f 0 ((12, 1))

⎤

⎡ ⎤ (7, 32 , 12 ) ⎥ ⎥ ⎢ (13, 2) ⎥ ⎥=⎣ ⎦, (10, 5) ⎦ (13, 2)

S. Danz, K. Erdmann / Journal of Algebra 397 (2014) 343–364

363

by (3) and Proposition 5.3. Every other Specht module without composition factor D (7,5,3) satisfies condition (∗). This leaves us with four Specht modules all of which have D (7,5,3) as a composition factor: S (7,5,3) , S (7,5,2,1) , S (7,4,3,1) , and S (6,5,3,1) . From [13, L. 3, L. 5, L. 8] we deduce that

⎡

⎤ (7, 5, 2, 1) (10, 5) ⎢ ⎥ S (7,5,2,1) = ⎣ ⎦, (7, 5, 3) ⊕ (13, 2) (10, 5)

⎡

⎤ (6, 5, 3, 1) ⎢ (7, 4, 3, 1) ⎥ ⎢ ⎥ S (6,5,3,1) = ⎢ (7, 5, 2, 1) ⎥ , ⎣ (10, 5) ⎦ (7, 5, 3)

and S (7,4,3,1) has a simple head and a simple socle, none of which is a branch module, and D (7,5,3) ∗ occurs only with multiplicity 1 in S (7,4,3,1) . Lastly, the dual Specht module S (7,5,3) has of course ∗ a simple socle isomorphic to D (7,5,3) and D (7,5,3) has multiplicity 1 in S (7,5,3) . Consequently, by Lemma 3.7, we obtain that every Specht module in B ((4, 2)) has quasi-length 1. (6) It remains to deal with the block B ((5, 3, 12 )) of F S19 and the block B ((6, 4, 22 , 12 )) of F S25 . The block B ((5, 3, 12 )) has one simple branch module, namely D (8,6,4,1) , and the block B ((6, 4, 22 , 12 )) 2 2 2 2 2 has two simple branch modules, namely D (9,7,5,2,1 ) and D (6,4 ,3 ,2 ,1) = D (9,7,5,2,1 ) ⊗ sgn. λ 2 Suppose S is a Specht module in B ((5, 3, 1 )) with composition factor D (8,6,4,1) . Then λ ∈ {(8, 6, 4, 1), (8, 6, 3, 2), (8, 5, 4, 2), (7, 6, 4, 2)}. Moreover,

 (8, 6, 3, 2) ←→0 8, 6, 22 ←→1 (7, 5, 2, 1), (8, 6, 4, 1) ←→0 (8, 6, 4) ←→1 (7, 5, 3),

 (8, 5, 4, 2) ←→0 8, 42 , 2 ←→1 (7, 4, 3, 1),

 (7, 6, 4, 2) ←→0 62 , 4, 2 ←→1 (6, 5, 3, 1). ∗

So from (5) and Proposition 5.3 we deduce that each of the modules S (8,6,4,1) , S (8,6,3,2) , S (8,5,4,2) , S (7,6,4,2) has a simple socle, and none of them satisfies the conditions in Lemma 3.7(b) or (c). Thus, by Lemma 3.7, all of these modules have quasi-length 1. Amongst the Specht modules in B ((5, 3, 12 )) 2 3 without composition factor D (8,6,4,1) , only S (8,3 ,2,1 ) does not satisfy condition (∗). So Lemma 3.7 and Proposition 6.3 imply quasi-length 1 for all of these modules as well. Now suppose that S λ is a Specht F S25 -module in the block B ((6, 4, 22 , 12 )) having a composition factor that is a branch module. Then λ is one of the following partitions: (9, 7, 5, 2, 12 ), (6, 4, 33 , 22 , 12 ) = (9, 7, 5, 2, 12 ) , (9, 6, 5, 3, 12 ), (6, 42 , 32 , 2, 13 ) = (9, 6, 5, 3, 12 ) , (8, 7, 5, 3, 12 ), (6, 42 , 32 , 22 , 12 ) = (8, 7, 5, 3, 12 ) , (9, 7, 4, 3, 12 ), (6, 42 , 3, 23 , 12 ) = (9, 7, 4, 3, 12 ) . For each pair of conjugate partitions, the corresponding Specht modules have the same quasi-length, by Remark 3.2. Since































9, 7, 5, 2, 12 ←→1 9, 7, 5, 12 ←→2 (8, 6, 4, 1), 9, 7, 4, 3, 12 ←→1 8, 6, 32 , 1 ←→2 (8, 6, 3, 2), 9, 6, 5, 3, 13 ←→1 9, 52 , 3, 1 ←→2 (8, 5, 4, 2), 8, 7, 5, 3, 12 ←→1 72 , 5, 3, 1 ←→2 (7, 6, 4, 2),

we can apply Lemma 3.7, part (6) above, and Proposition 5.3. We get that each of these modules must 2 5 have quasi-length 1. Except for S (9,4 ,3,1 ) , all Specht modules in B ((6, 4, 22 , 12 )) none of whose composition factors is a branch module satisfy condition (∗). Hence, by Lemma 3.7 and Proposition 6.3, every Specht module in B ((6, 4, 22 , 12 )) has quasi-length 1, and the theorem is proved. 2

364

S. Danz, K. Erdmann / Journal of Algebra 397 (2014) 343–364

Acknowledgments We are grateful to J. Müller for providing us with his GAP routines for constructing Specht modules. We would also like to thank the referee for his or her comments on an earlier version of this article. The first author’s research has been supported by a Marie Curie Intra-European Fellowship (grant PIEF-GA-2008-219543). References [1] M. Auslander, I. Reiten, S. Smalø, Representation Theory of Artin Algebras, Cambridge Stud. Adv. Math., vol. 36, Cambridge University Press, 1995. ´ [2] I. Assem, D. Simson, A. Skowronski, Elements of the Representation Theory of Associative Algebras. 1. Techniques of Representation Theory, London Math. Soc. Stud. Texts, vol. 65, Cambridge University Press, Cambridge, 2006. [3] D.J. Benson, Representations and Cohomology I, Cambridge Stud. Adv. Math., vol. 30, Cambridge University Press, 1991. [4] D.J. Benson, Some remarks on the decomposition numbers for the symmetric groups, in: Proc. Sympos. Pure Math., vol. 47, American Mathematical Society, Providence, 1987, pp. 381–394. [5] C. Bessenrodt, K. Uno, Character relations and simple modules in the Auslander–Reiten graph of the symmetric and alternating groups and their covering groups, Algebr. Represent. Theory 4 (2001) 445–468. [6] D. Bogdanic, Graded Cartan matrices and crystal decomposition numbers for defect 2 blocks of symmetric groups, preprint. [7] W. Bosma, J. Cannon, C. Playoust, The MAGMA algebra system. I. The user language, J. Symbolic Comput. 24 (1997) 235–265. [8] J. Brundan, R. Dipper, A. Kleshchev, Quantum linear groups and representations of GLn (Fq ), in: Mem. Amer. Math. Soc., vol. 706, 2001. [9] J. Chuang, K.M. Tan, On certain blocks of Schur algebras, Bull. Lond. Math. Soc. 33 (2001) 157–167. [10] F. Eisele, Basic orders for defect two blocks of Z p Σn , Comm. Algebra, in press. [11] K. Erdmann, Blocks of Tame Representation Type and Related Algebras, Lecture Notes in Math., vol. 1428, Springer-Verlag, Berlin, 1990. [12] K. Erdmann, On Auslander–Reiten components for group algebras, J. Pure Appl. Algebra 104 (2) (1995) 149–160. [13] M. Fayers, On weight three blocks of symmetric groups in characteristic three, Q. J. Math. 53 (2002) 403–419. [14] M. Fayers, Data for weight 2 blocks of Hecke algebras of type A, preprint, 2012. [15] The GAP Group, GAP-4 – Groups, algorithms and programming, version 4.4.9, 2006. [16] J.J. Graham, G.I. Lehrer, Cellular algebras, Invent. Math. 123 (1996) 1–34. [17] J.A. Green, Walking around the Brauer Tree. Collection of articles dedicated to the memory of Hanna Neumann VI, J. Aust. Math. Soc. 17 (1974) 197–213. [18] J.A. Green, Polynomial Representations of GLn , second corrected and augmented edition, Lecture Notes in Math., vol. 830, Springer-Verlag, Berlin, 2007. With an appendix on Schensted correspondence and Littelmann paths by K. Erdmann, Green and M. Schocker. [19] G.D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Math., vol. 682, Springer-Verlag, New York, 1978. [20] G.D. James, A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia Math. Appl., vol. 16, Addison– Wesley, Reading, 1981. [21] T. Jost, On Specht modules in the Auslander–Reiten quiver, J. Algebra 173 (1995) 281–301. [22] S. Kawata, On Auslander–Reiten components and simple modules for finite group algebras, Osaka J. Math. 34 (3) (1997) 681–688. [23] A. Kleshchev, Linear and Projective Representations of Symmetric Groups, Cambridge University Press, 2005. [24] G.E. Murphy, On the representation theory of the symmetric groups and associated Hecke algebras, J. Algebra 152 (2) (1992) 492–513. [25] H. Nagao, Y. Tsushima, Representations of Finite Groups, Academic Press, San Diego, 1989. [26] M.J. Richards, Some decomposition numbers for Hecke algebras of general linear groups, Math. Proc. Cambridge Philos. Soc. 119 (1996) 383–402. [27] C. Riedtmann, Algebren, Darstellungsköcher, Überlagerungen und zurück, Comment. Math. Helv. 55 (1980) 199–224. [28] J. Scopes, Cartan matrices and Morita equivalence for blocks of the symmetric groups, J. Algebra 142 (1991) 441–455. [29] J. Scopes, Symmetric group blocks of weight two, Q. J. Math. Oxford 46 (1995) 201–234.