-pairs of symmetric group algebras and their intermediate defect 4 blocks

a

43

ock

Loewyverify

f

l defect

Journal of Algebra 288 (2005) 505–526

www.elsevier.com/locate/jalgebr

[3 : 2]-pairs of symmetric group algebrasand their intermediate defect 4 blocks

Stuart Martina,∗, Kai Meng Tanb

a DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UKb Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 1175

Received 23 June 2004

Available online 17 March 2005

Communicated by Gordon Jones

Abstract

In this paper we study[3 : 2]-pairs of symmetric group algebras and their ‘intermediate’ blin detail. The aim is to understand how one block of a[3 : 2]-pair can inherit two properties—the Ext-quiver being bipartite and the principal indecomposable modules having a commonlength 7—from the other. We establish some sufficient conditions for this inheritance, andthese conditions for some special blocks. 2005 Elsevier Inc. All rights reserved.

1. Introduction

Let Sn denote the symmetric group onn letters, andk an algebraically closed field ocharacteristicp ( 5). In [10], we showed that the principal blocks ofkS3p+r (0 r < p)have the following common properties:

(P1) The Ext-quiver is bipartite.

This work incorporates a revised version of part of the second author’s doctoral thesis [K.M. Tan, Smalblocks of symmetric group algebras, PhD thesis, University of Cambridge, 1998].

* Corresponding author.
E-mail addresses:[email protected] (S. Martin), [email protected] (K.M. Tan).
0021-8693/$ – see front matter 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.jalgebra.2005.01.036

506 S. Martin, K.M. Tan / Journal of Algebra 288 (2005) 505–526

.

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ain6. Wesor

quire.ory of

ith

(P2) The principal indecomposable modules have common Loewy length equal to 7

We also studied in detail how a defect 3 block in a[3 : 1]-pair may inherit these twoproperties from the other, establishing some sufficient conditions for this to hold.

In this paper, we study[3 : 2]-pairs of symmetric group algebras and their ‘intermediablock in detail, with the aim of understanding how one block of a[3 : 2]-pair can inheritproperties (P1) and (P2) from the other.

Our approach is as follows. We begin by giving a short account of the representheory we require. In Section 3, we present the known background on[3 : 2]-pairs as shownin [9]. In Section 4, we study the so-called ‘intermediate blocks’ of[3 : 2]-pairs, followedby obtaining new information about the[3 : 2]-pairs in Section 5. We are then able to obtsome sufficient conditions for the inheritance of properties (P1) and (P2) in Sectionconclude by verifying in Section 7 these sufficient conditions for the defect 3 blockBi

(1 i p) havingp-core (p + i − 2, i − 1), thus showing that (P1) and (P2) hold fthese blocks.

2. Preliminaries

In this section, we give a brief account of the representation theory which we reFor more detailed accounts, we refer the reader to [2,3] for the representation thesymmetric groups, and to [8] for general theory of group representations.

Firstly, the following notations will be used in this paper:

(1) the projective cover of a moduleM will be denoted byP(M), andΩ(M) will denotethe submodule ofP(M) satisfyingP(M)/Ω(M) ∼= M ;

(2) a filtrationM = M0 ⊃ M1 ⊃ M2 ⊃ · · · ⊃ Mr = 0 will be denoted by a matrix withrrows, where theith row is the factorMi−1/Mi ;

(3) the multiplicity of a simple moduleS as a composition factor of a moduleM will bedenoted by[M : S];

(4) M ∼ M ′ means the two modulesM andM ′ have the same composition factors (wmultiplicities).

The following easy lemma on general modules will be used often in this paper.

Lemma 2.1. SupposeM is a module having a composition series, andN is a submoduleof M such thatN/ rad(N) ∼= nS for some simple moduleS. If L is a submodule ofM suchthat [L : S] = [M : S], thenN ⊆ L.

In particular, if [M : S] = [N : S], thenN contains(as a subset) every submoduleK ofM such thatK/ rad(K) ∼= kS.

Proof. If N L, then 1 [N/(N ∩ L) : S] = [(N + L)/L : S] [M/L : S] = 0, a con-
tradiction.

S. Martin, K.M. Tan / Journal of Algebra 288 (2005) 505–526 507

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Recall that the Specht modulesSλ, asλ runs through the set of partitions ofn, give acomplete list of mutually non-isomorphic simple modules ofSn in zero characteristic. Inpositive characteristicp, the Specht moduleSλ has a simple headDλ if the partitionλ isp-regular; all composition factorsDµ of its radical satisfyµ λ. If λ is p-singular, thenall composition factorsDµ of Sλ satisfyµ λ. Asλ runs through thep-regular partitionsthe set of simple heads is a complete list of mutually non-isomorphic simple modukSn.

Two Specht modulesSλ andSµ of kSn lie in the same block if, and only if,λ andµ

have the samep-core (Nakayama’s Conjecture). The Branching Rule provides a Spectration for the restricted Specht moduleSλ↓Sn−1 and the induced Specht moduleSλ↑Sn+1.The Specht moduleSµ is a factor in this filtration if, and only if,µ can be obtained fromλby removing or adding a node. A factorSα lies above another factorSβ in this filtration ifα β.

The Ext-quiver of ak-algebraA is a finite directed graph whose vertex set is labeby the (isomorphism classes of) simpleA-modules and the number of edges fromS1 to S2

is given by dimk Ext1A(S1, S2). Since the simple modules of symmetric group algebrasself-dual, all edges in its Ext-quiver are directed two-ways. Recall that a (directed)is termed bipartite if there is a partition of its vertex set into two parts such that thereedge between any two vertices in the same part. If the Ext-quiver ofA is bipartite, then oneof the consequences is that ifP(S) is the projective cover of a simpleA-module andS′is a simple module occurring in an odd (respectively even) Loewy layer ofP(S), then allcomposition factors ofP(S) isomorphic toS′ lie in odd (respectively even) Loewy layer(of P(S).

A key ingredient in our presentation is Kleshchev’s work [5–7] on the restricted simoduleDλ↓B . Let B be a block ofkSn and letB be a block ofkSn−1 such that the coreof the latter has an abacus display having the same number of beads on each cothat ofB, except the (i − 1)th andith columns, where respectively there is one bead mthan and one bead less than that ofB. Let λ be ap-regular partition ofB. A bead lyingon theith column andj th row of the abacus display ofλ is normal if the position on itsleft is unoccupied and, for alll > j , counting the beads between the (j + 1)th andlth row(both inclusive), those lying on the (i − 1)th column is not more than those lying on tith column. A normal bead isgoodif it is the highest (in the abacus display) normal beWe may move a normal bead ofλ one position to its left and obtain ap-regular partitionλ of B. The multiplicity of the simple moduleDλ as a composition factor ofDλ↓B is one

more than the number of normal beads below the normal bead moved to obtainDλ . If thenormal bead moved is good, then the simple moduleDλ obtained is the head and socleDλ↓B . In particular,Dλ↓B is either zero (when there is no normal bead forλ), or has asimple head and a simple socle (and thus indecomposable). Moreover, two non-isomsimple modules ofkSn may not give non-zero isomorphic simple heads when restrito B.

Schaper [11] has a formula for calculating an upper bound of each entry in the dposition matrix of a block of symmetric group algebras. In fact his upper bound is nonif and only if the entry is non-zero. If a symmetric group block has the property that
entry in its decomposition matrix is at most 1, then the formula determines the decomposi-


r the

aving

nings

isplay.

[9] on

wing

s thef of

stab-

etimes

y

tion matrix completely. For one version of this formula suited to our method, we refereader to [9].

Let us recall also the non-standard notation used in [10] for weight 3 partitions hp-coreτ .

Definition 2.2. Let τ be ap-core havingr parts. Letb be a fixed integer not less thar + 3p. Any weight 3 partitionλ having coreτ may be represented on the abacus havb beads. Thenλ may be denoted using the〈〉-notation, defined as follows: if the abacudisplay havingb beads ofλ has

(1) one bead of weight 3 on columni, then denoteλ by 〈i〉;(2) one bead of weight 2 on columni and one bead of weight 1 on columnj , then denote

λ by 〈i, j 〉;(3) three beads of weight 1 on column(s)i, j andk, then denoteλ by 〈i, j, k〉.

It is clear that the〈〉-notation depends on the number of beads used in the abacus dThere is usually a natural choice for the number of beads used. For example, if thep-coreof the defect 3 blockB hasr parts, then we usually use the〈〉-notation with 3p + r beadsto denote the partitions ofB.

3. Background

In this section we remind the reader of the results of Russell and the first authordefect 3 blocks and, in particular,[3 : 2]-pairs.

In [9], a proof that the defect 3 blocks of symmetric group algebras have the folloproperties is announced.

(X1) Each entry of its decomposition matrix is either 0 or 1.(X2) The Ext1-space between any two simple modules is at most one-dimensional.(X3) Every simple module does not extend itself.

However, James and Mathas [4] found gaps in the proof of property (X1), and aproof of properties (X2) and (X3) relies on the validity of (X1), this renders the proothese three properties incomplete.

In April 2004, Fayers [1] produced a complete proof of property (X1), thereby elishing the truth of [9].

Throughout this paper, we shall often make use of these three properties, somwithout comment.

For the rest of this paper,B will be denote a defect 3 block ofkSn which forms a[3 : 2]-pair with B, a defect 3 block ofkSn−2, with the ith column of the abacus displaof the core ofB having two beads more than that ofB.

We begin by introducing some (non-standard) terminology.


is

is

-

ns

Definition 3.1. With respect to the[3 : 2]-pairB andB, we call

(1) a partitionλ of B is exceptionalif we can move more than two beads on theith columnof its abacus display to their respective preceding positions on the(i − 1)th column.Otherwise, it isnon-exceptional;

(2) a Specht moduleSλ of B is exceptionalif and only if λ is exceptional;(3) a simple moduleDλ of B is exceptionalif Dλ↓B is not semisimple. Otherwise, it

non-exceptional;(4) a partitionλ of B is exceptionalif we can move more than two beads on the (i − 1)th

column of its abacus display to their respective succeeding positions on theith column.Otherwise, it isnon-exceptional;

(5) a Specht moduleSλ of B is exceptionalif, and only if, λ is exceptional;(6) a simple moduleDλ of B is exceptionalif Dλ↑B is not semisimple. Otherwise, it

non-exceptional.

There are four exceptional Specht modules ofB, denoted asSα , Sβ , Sγ andSδ , whosecorresponding partitions have the following (i − 1)th andith columns in their abacus displays:

i−1 i

......

• •• −− •− •− •α

i−1 i

......

• •− •• −− •− •β

i−1 i

......

• •− •− •• −− •γ

i−1 i

......

• •− •− •− •• −δ

Similarly, there are four exceptional Specht modules ofB, denoted asSα , Sβ , Sγ andSδ , whose corresponding partitions have the following (i − 1)th andith columns in theirabacus displays:

i−1 i

......

• •• −• −• −− •α

i−1 i

......

• •• −• −− •• −

β

i−1 i

......

• •• −− •• −• −γ

i−1 i

......

• •− •• −• −• −

δ

The partitionsα andα are alwaysp-regular. So are the conjugate partitionsδ′ and δ′.The diagrams below show the dependence ofp-regularity among the exceptional partitio
and their conjugates.


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e

,

tpecht

β p-regular ⇒ γ p-regular ⇒ δ p-regular⇐⇒ ⇐⇒ ⇐⇒

δ p-regular ⇒ γ p-regular ⇒ β p-regular

γ ′ p-regular ⇒ β ′ p-regular ⇒ α′ p-regular⇐⇒ ⇐⇒ ⇐⇒

α′ p-regular ⇒ β ′ p-regular ⇒ γ ′ p-regular

Restriction of the exceptional Specht modules ofB and induction of the exceptionaSpecht modules ofB give the following:

(C1) Sα↓B ∼ 2(Sα ⊕ Sβ ⊕ Sγ

);(C2) Sβ↓B ∼ 2

(Sα ⊕ Sβ ⊕ Sδ

);(C3) Sγ ↓B ∼ 2

(Sα ⊕ Sγ ⊕ Sδ

);(C4) Sδ↓B ∼ 2

(Sβ ⊕ Sγ ⊕ Sδ

);(D1) Sα↑B ∼ 2

(Sα ⊕ Sβ ⊕ Sγ

);(D2) Sβ↑B ∼ 2

(Sα ⊕ Sβ ⊕ Sδ

);(D3) Sγ ↑B ∼ 2

(Sα ⊕ Sγ ⊕ Sδ

);(D4) Sδ↑B ∼ 2

(Sβ ⊕ Sγ ⊕ Sδ

).

There is only one exceptional simple module ofB, namelyDα . All other simple mod-ules ofB remain semisimple when restricted toB. In fact, ifDλ is a non-exceptional simplmodule ofB, then there is a unique simple moduleDλ of B such thatDλ↓B

∼= 2Dλ. Sim-ilarly, there is only one exceptional simple module ofB, namelyDα . All other simplemodules ofB remain semisimple when induced toB. If Dµ is a non-exceptional simplmodule ofB, then there is a unique simple moduleDµ of B such thatDµ↑B ∼= 2Dµ. Therestricted moduleDα↓B has six copies ofDα , and onlyDα occurs in its head. Similarlythe induced moduleDα↑B has six copies ofDα , and onlyDα occurs in its head.

The exceptional simple moduleDα is a composition factor ofSα , Sβ , Sγ andSδ , andin no other Specht module isDα a composition factor. The projective coverP(Dα) has aSpecht filtration

0⊂ M3 ⊂ M2 ⊂ M1 ⊂ P(Dα)

such thatP(Dα)/M1 ∼= Sα , M1/M2 ∼= Sβ , M2/M3 ∼= Sγ andM3 ∼= Sδ .Similarly, Dα is a composition factor ofSα , Sβ , Sγ andSδ , and in no other Spech

module is it a composition factor. By analogy its projective cover has a similar Sfiltration.

We may also note the following.


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at

ent

ht

y:

(1) If β is p-regular (equivalent toδ beingp-regular), thenDβ↓B∼= 2Dδ andDδ↑B ∼=

2Dβ . Moreover,Dβ occurs in bothSγ andSδ .(2) If γ is p-regular (equivalent toγ beingp-regular), thenDγ ↓B

∼= 2Dγ andDγ ↑B ∼=2Dγ . Moreover,Dγ andDγ occur inSδ andSδ , respectively.

(3) If δ is p-regular (equivalent toβ beingp-regular), thenDδ↓B∼= 2Dβ andDβ↑B ∼=

2Dδ . Moreover,Dβ occur in bothSγ andSδ .

4. The intermediate block B

Given any partition ofB, there are beads on theith column of its abacus display whicmay be moved to their respective preceding positions on the(i −1)th column. Moving oneof these beads corresponds to restricting the associated Specht module ofB to a defect 4blockB. The abacus display of thep-core ofB has one bead more on the(i −1)th columnand one bead less on theith column than that ofB. Similarly, moving one of the beads othe (i − 1)th column in the abacus display of a partition ofB to its succeeding positioon theith column corresponds to inducing the associated Specht module ofB to B. Inthis section, we study this defect 4 blockB with the aim of understanding our[3 : 2]-pairbetter.

We first note thatB is the ‘intermediate’ block between the blocksB andB when werestrict or induce modules of one to the other, in the following sense.

Lemma 4.1. Let M be a B-module. ThenM↓B∼= (M↓B)↓B . Similarly, if N is a B-

module, thenN↑B ∼= (N↑B)↑B .

Proof. We know thatM↓Sn−2∼= (M↓Sn−1)↓Sn−2. Using the Branching Rule, we see th

the summands ofM↓Sn−1 which do not lie inB vanish when they are restricted toB. Thus,(M↓Sn−1)↓B

∼= (M↓B)↓B . The first statement now follows easily. A similar argumapplies for the second statement.

With exactly six exceptions, every Specht module ofB restricts to a unique Specmodule ofB (or gives zero) and induces to a unique Specht module ofB (or gives zero).The exceptional Specht modules will be denoted asSα , Sβ , Sγ , Sδ , Sε andSκ , and theircorresponding partitions have the following (i−1)th andith columns in the abacus displa

i−1 i

......

• •• −• −− •− •

i−1 i

......

• •• −− •• −− •

i−1 i

......

• •− •• −• −− •

i−1 i

......

• •• −− •− •• −

i−1 i

......

• •− •• −− •• −

i−1 i

......

• •− •− •• −• −

α β γ δ ε κ


cht

a also

nt

These Specht modules ofB have the following relationships with the exceptional Spemodules ofB andB:

(E1) Sα↓B ∼ Sα ⊕ Sβ; Sα↑B ∼ Sα ⊕ Sβ;(E2) Sβ↓B ∼ Sα ⊕ Sγ ; Sβ↑B ∼ Sα ⊕ Sγ ;(E3) Sγ ↓B ∼ Sα ⊕ Sδ; Sγ ↑B ∼ Sβ ⊕ Sγ ;(E4) Sδ↓B ∼ Sβ ⊕ Sγ ; Sδ↑B ∼ Sα ⊕ Sδ;(E5) Sε↓B ∼ Sβ ⊕ Sδ; Sε↑B ∼ Sβ ⊕ Sδ;(E6) Sκ↓B ∼ Sγ ⊕ Sδ; Sκ↑B ∼ Sγ ⊕ Sδ;(F1) Sα↑B ∼ Sα ⊕ Sβ ⊕ Sγ ; Sα↓B ∼ Sα ⊕ Sβ ⊕ Sδ;(F2) Sβ↑B ∼ Sα ⊕ Sδ ⊕ Sε; Sβ↓B ∼ Sα ⊕ Sγ ⊕ Sε;(F3) Sγ ↑B ∼ Sβ ⊕ Sδ ⊕ Sκ ; Sγ ↓B ∼ Sβ ⊕ Sγ ⊕ Sκ ;(F4) Sδ↑B ∼ Sγ ⊕ Sε ⊕ Sκ ; Sδ↓B ∼ Sδ ⊕ Sε ⊕ Sκ .

For each non-exceptional Specht moduleSλ of B with Sλ↑B ∼ 2Sλ, there exist twoSpecht modulesSλ andSµ of B (with λ > µ) such thatSλ↑B ∼ Sλ ⊕Sµ ∼ Sλ↓B , Sλ↓B

∼=Sµ↓B

∼= Sλ andSλ↑B ∼= Sµ↑B ∼= Sλ.

Lemma 4.2. SupposeDλ is a non-exceptional simple module ofB. Then(Dλ↑B)↓B∼=

2Dλ.Similarly, if Dλ is a non-exceptional simple module ofB, then(Dλ↓B)↑B ∼= 2Dλ.

Proof. By the Branching Rule, we see that, for a non-exceptional Specht moduleSλ of B,(Sλ↑B)↓B ∼ 2Sλ. Thus using induction, we can show that, forλ > α, (Dλ↑B)↓B ∼ 2Dλ,

and hence that(Dλ↑B)↓B∼= 2Dλ, since the simple modules ofB do not self-extend. Now

using relationships (F1), (E1)–(E3), we see(Sα↑B)↓B ∼ 3Sα ⊕ Sβ ⊕ Sγ ⊕ Sδ , so that

(Dα↑B)↓B ∼ Sα ⊕ Sβ ⊕ Sγ ⊕ Sδ ⊕ 2Dα . Noting thatDα is a composition factor ofSβ ,

Sγ andSδ , and using relationships (E1)–(E6) and (F1)–(F4), we see that the lemmholds for λ ∈ β, γ , δ. Since the remaining Specht modules ofB do not haveDα as acomposition factor, the lemma holds for allλ = α inductively. An analogous argumeapplies toDλ. Proposition 4.3. For each non-exceptional simple moduleDλ of B, with Dλ↓B

∼= 2Dλ

there exists a unique simple moduleDλ of B such that

(1) Dλ↑B ∼= Dλ;


-

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e

de

(2)

ion of

both

s

e

(2) Dλ↓B is non-simple, has head and socle both isomorphic toDλ, and all the composition factorsDµ in its heart satisfyDµ↑B = 0;

(3) Dλ↓B∼= Dλ;

(4) Dλ↑B is non-simple, has head and socle both isomorphic toDλ, and all the composition factorsDµ in its heart satisfyDµ↓B = 0;

(5) Dλ↑B ∼= Dλ↓B .

Proof. Kleshchev’s results on restricted simple modules [6] show thatDλ↓B has a simple

head and simple socle which are isomorphic, toDλ say. By Frobenius reciprocity,Dλ↑B

has head and socle both isomorphic toDλ. Using Lemma 4.2, we have(Dλ↓B)↑B ∼=2Dλ, so thatDλ↓B is not simple, as otherwise(Dλ↓B)↑B will be indecomposable. Henc

Dλ occurs at least twice inDλ↓B . Looking at Lemma 4.2 again, we can then conclustatements (1) and (2).

Now, Kleshchev’s results on the socles of restricted simple modules [6] show thatDλ ∼=soc(Dλ↓B ) and thatDλ ∼= soc(Dλ↑B). Similar arguments to those used in (1) andshow that (3) and (4) also hold.

For (5), using Lemma 4.1 and Frobenius reciprocity, we see that the dimensHom(Dλ↓B,Dλ↑B) ∼= Hom(Dλ,Dλ↑B) is 2. Since the multiplicities ofDλ as compo-

sition factors ofDλ↓B andDλ↑B are both 2, and they occur as the head and socle of

modules, this implies that there is an isomorphism fromDλ↓B to Dλ↑B . Corollary 4.4. LetDλ be a non-exceptional simple module ofB, with Dλ↓B

∼= 2Dλ. If M

is anyB-module, then, for any nonnegative integerr , we have

Extr(M↑B,Dλ

) ∼= Extr(M↓B ,Dλ

).

Proof. SinceDλ↓B∼= Dλ↑B , we have

Extr(M↑B,Dλ

) ∼= Extr(M,Dλ↓B

) ∼= Extr(M,Dλ↑B

) ∼= Extr(M↓B ,Dλ

).

Lemma 4.5. We have the following multiplicities involving exceptional simple module:

[Dα↓B : Dα

] = 3; [Dα↑B : Dα

] = 3; [Dα↑B : Dα

] = 2; [Dα↓B : Dα

] = 2.

Hence,Dα is a composition factor ofSα , Sβ , Sγ , Sδ , Sε andSκ , all with multiplicity 1. Inno other Specht module canDα be a composition factor.

Proof. Using Proposition 4.3 and relationship (E1), we see that the multiplicities[Dα↑B :Dα] and[Dα↓B : Dα] are as stated. Relationships (E2)–(E6) now tell us thatDα occurs at

most once in each of the exceptional Specht modules ofB. For example,[Sβ↓B : Dα] = 2by relationship (E2), and since[Dα↓B : Dα] = 2, we see thatDα can at most occur onc

in Sβ . Kleshchev’s result on restricted simple modules [7] shows that[Dα↓B : Dα] = 3.


n)

ow,

ties

d and

thisy for

-

g

at

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t they

Thus, together with relationships (F1)–(F4), we see thatDα must occur exactly once ieach of the remaining exceptional Specht modules ofB. For example, relationship (F3shows that[Sβ ⊕ Sγ ⊕ Sκ : Dα] 3; this forcesDα to occur exactly once in each ofSβ ,Sγ andSκ . The last multiplicity[Dα↑B : Dα] can now be seen easily, say using (F1). Nif Dα is a composition factor ofSλ, thenDα is a composition factor ofSλ↓B . But if Sλ

is not exceptional, thenSλ↓B∼= Sλ is also not exceptional, and thus will not haveDα as a

composition factor. Proposition 4.6. LetDλ be a non-exceptional simple module ofB, with Dλ ∼= soc(Dλ↓B)

andDλ↓B∼= Dλ. The following table gives a complete list of possibilities of multiplici

of Dλ, Dλ andDλ as composition factors of some modules which have simple heasimple socle both isomorphic to an exceptional simple module:

[P(Dα) : Dλ] [P(Dα) : Dλ] [P(Dα) : Dλ] [Dα↑B : Dλ] [Dα↓B : Dλ] [Dα↑B : Dλ] [Dα↓B

: Dλ]0 0 0 0 0 0 01 3 3 1 0 1 02 6 2 2 0 0 23 3 1 0 1 0 14 6 4 1 1 1 14 12 4 4 0 0 4

Proof. Again, we rely heavily on relationships (E1)–(E6) and (F1)–(F4) in provingproposition. We will only justify the entries in the second row. Similar arguments applall other rows. Since[P(Dα) : Dλ] = 1, there is exactly one Specht module amongSα , Sβ ,Sγ andSδ of whichDλ is a composition factor. SinceDλ↓B

∼= Dλ, we see that the multi

plicities of Dλ as a composition factor ofSα , Sβ , Sγ , Sδ , Sε andSκ are at most 1, usinrelationships (E1)–(E6). For ease of reference, we will assume thatDλ is a compositionfactor Sγ . Relationship (E1) then tells us thatDλ is not a composition factor ofDα↓B .

Furthermore, (E1)–(E6) show thatDλ is a composition factor ofSβ , Sδ andSκ , and isnot a composition factor ofSα , Sγ andSε . This shows that[P(Dα) : Dλ] = 3. Moreover,relationship (F1) yields[Dα↑B : Dλ] = 1. Proceeding in this way, using (F2), we find thDλ is not a composition factor ofSβ , andDλ is not a composition factor ofDα↓B . Rela-tionships (F1)–(F4) then show thatDλ is a composition factor ofSα , Sγ andSδ . Hence,[P(Dα) : Dλ] = 3. Relationship (E1) then shows that[Dα↑B : Dλ] = 1. Lemma 4.7. The modulesDα↑B andDα↓B have a common Loewy length3.

Proof. Using Proposition 4.6 we see that the multiplicity of a non-exceptional simple mule as a composition factor ofDα↑B (or Dα↓B ) is at most 1. SinceDα↑B andDα↓B areboth indecomposable, non-simple (by Lemma 4.5) and self-dual, this implies tha
have a common Loewy length 3.


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ead

t, if

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d

ld

s

5. [3 : 2]-pairs

We now turn our attention back to the blocksB and B. With the information abouthe intermediate blockB made available in the previous section, we are able to studblocksB andB in more detail.

We begin with the observation that bothDα↓B andDα↑B are direct sums of two isomorphic indecomposable modules.

Lemma 5.1. The restricted simple moduleDα↓B is a direct sum of two isomorphic moules, each of which has head and socle isomorphic toDα and another copy ofDα in itsheart. Similarly,Dα↑B is a direct sum of two isomorphic modules, each of which has hand socle isomorphic toDα and another copy ofDα in its heart.

Proof. It is clear from Proposition 4.6, Lemma 4.5 and Frobenius reciprocity tha[P(Dα) : Dλ] = 1, thenP(Dλ)↓B

∼= 2P(Dλ) (whereDλ↓B∼= 2Dλ), so thatDα↓B is

necessarily a direct sum of two isomorphic modules. Each summand is self-dual, hacopies ofDα , with only Dα occurring in its head and socle. If it is indecomposable thewill have a simple head and a simple socle both isomorphic toDα and another copy ofDα

in its heart. If it is decomposable, then it must be a direct sum ofDα and another moduleMsay, which has a simple head and simple socle isomorphic toDα . Since[Dα↓B : Dλ] = 4

from Proposition 4.6, we see thatDλ occurs twice inM . But Dα is a submodule ofDα↓B ,so thatDα↓B is a submodule ofDα↓B

∼= 2Dα ⊕ 2M . Now,Dα↓B is indecomposable an

has only a copy ofDλ, and thus cannot be a submodule ofM , which is a contradiction.Finally it remains to show the existence of such aDλ. Now, if δ is p-regular, then

certainlyDδ only occurs once inP(Dα), while if δ is p-singular, thenα′ is p-regular, sothatSα has a simple socle which occurs exactly once inP(Dα).

Similar arguments hold forDα↑B . Notation. We shall denote the indecomposable direct summand ofDα↓B by L5 and thatof Dα↑B by L5.

It is clear thatL5 has Loewy length greater than or equal to 5, sinceDα does not extenditself. Also L5 has a submodule isomorphic toDα↑B . Analogous statements also hofor L5.

Proposition 5.2. The projective moduleP(Dα) has exactly(not just up to isomorphism)four submodules with simple headDα , namelyDα , Dα↑B , L5 andP(Dα) where the firstthree are embedded intoP(Dα), and they are totally ordered by inclusion.

Proof. Any proper submodule ofP(Dα) with simple headDα is contained inL5 byLemma 2.1 (withM = rad(P (Dα))). ThusDα andDα↑B are contained properly inL5.Applying Lemma 2.1 again withM = rad(L5) then showsDα ⊂ Dα↑B . The submodule
listed above are the only four such since dimEnd(P (Dα)) = 4.


ry

dt

cteds

at

Similarly, viewingDα , Dα↓B , L5 as submodules ofP(Dα), these are the only propesubmodules ofP(Dα) having head isomorphic toDα , and they are totally ordered binclusion.

Lemma 5.3. The Loewy length ofL5 is at least5, and that ofP(Dα) is at least7. Ananalogous statement holds forP(Dα).

Proof. Since there are no self-extensions, then the Loewy length ofL5 is at least 5, byLemma 5.1. Also, viewingL5 as a submodule ofP(Dα), the head ofL5 is lying in orbelow the third Loewy layer ofP(Dα). HenceP(Dα) has Loewy length at least 7.Lemma 5.4.

(1) Sκ↑B has simple socleDα and, ifγ andδ arep-regular, simple headDγ .(2) Sδ↑B has socle2Dα and, ifβ is regular, head2Dβ .

Analogous statements hold forSκ↓B andSδ↓B .

Proof. (1) Note thatSκ has a simple socleDα , so that by Frobenius reciprocity anProposition 4.3,Sκ↑B has a simple socleDα . By the Branching Rule,Sκ↑B has a Spechfiltration of the form

Sγ

Sδ .

If γ andδ arep-regular, then from the Specht filtration, we see that its head containsDγ ,and possiblyDδ , which we shall proceed to rule out. By Kleshchev’s results on restrisimple modules,Dκ is a composition factor ofDγ ↓B , and relationship (F4) then showthatDκ is not a composition factor ofDδ↓B . By Frobenius reciprocity, this implies thDδ does not occur in the head ofSκ↑B .

(2) This follows from Frobenius reciprocity, sinceSδ has a simple socleDα and, ifβ isregular, a simple headδ. Corollary 5.5. Let 0⊂ M3 ⊂ M2 ⊂ M1 ⊂ P(Dα) be the Specht filtration ofP(Dα).

(1) If δ is p-regular, thenM3 = Sδ .(2) If γ andδ arep-regular, thenM2 = Sκ↑B .(3) If β is p-regular, then2M1 = Sδ↑B .

Analogous statements hold for the Specht filtration ofP(Dα).

In this corollary, we are viewingSδ andSκ↑B as submodules ofP(Dα), andSδ↑B as a
submodule of 2P(Dα).


rations

s

r

Proof. By Lemmas 2.1 and 5.4, if the respectivep-regular conditions hold, then

Sδ ⊆ M3, Sκ↑B ⊆ M2, Sδ↑B ⊆ 2M1.

Equality holds for each instance, since the modules on both sides have Specht filtwith the same factors (hence the same composition length).Corollary 5.6. Supposeβ is p-regular, andExt1(Dα,Dλ) = 0. Thenλ = β or Dλ occursin the second Loewy layer ofSα .

An analogous statement holds in the blockB.

Proof. By Lemma 5.4(2) and Corollary 5.5(3),M1 has headDβ , and the statement followimmediately. Lemma 5.7. Suppose the exceptional partitions are allp-regular. ThenDγ lies in or belowthe third Loewy layer ofP(Dα), andDδ lies in or below the fourth Loewy layer ofP(Dα).

An analogous statement holds forP(Dα).

Proof. This follows from Lemma 5.4(1), Corollaries 5.5(2) and 5.6.We now give a sufficient condition for non-zero extensions betweenDα (or Dα) and a

simple module.

Lemma 5.8. Suppose[P(Dα) : Dλ] = 3. ThenExt1(Dα,Dλ) = 0.An analogous statement holds forDα .

Proof. By Lemma 4.7,Dα↑B has Loewy length 3. By Proposition 4.6, we see thatDλ isa composition factor ofDα↑B , and hence lies in its semisimple heart. This shows thatDα

extendsDλ. The following technical results will be required in the last section.

Lemma 5.9. SupposeP(Dα) has a subquotient with simple headDλ and simple socleDµ,where[P(Dα) : Dλ] = 2= [P(Dα) : Dµ] + 1. Then

(1) Ext1(Dα↑B,Dµ) = 0,(2) Ext1(Dα,Dλ) = 0,

whereDλ↓B∼= 2Dλ.

Proof. Let 0 ⊆ N ⊆ M ⊆ P(Dα) such thatM/N has simple headDλ and simple so-cleDµ.

(1) We may assume thatM has simple headDλ. By Proposition 4.6,Dλ does not occu
in Dα↑B , so thatM ⊆ Ω(Dα↑B) by Lemma 2.1. SinceDµ does not lie in the head


t

rf

.,

ite

of M , and hence ofΩ(Dα↑B) (since [Ω(Dα↑B) : Dµ] = [M : Dµ]), we see thaExt1(Dα↑B,Dµ) = 0.

(2) We may assume thatP(Dα)/N has simple socleDµ. Note thatDµ does not occuin N and L5, the latter by Proposition 4.6. HenceL5 ⊆ N by the dual version oLemma 2.1. SinceDλ does not lie in the socle ofP(Dα)/N , and hence ofP(Dα)/L5(since[P(Dα)/L5 : Dλ] = [P(Dα)/N : Dλ]), we see that

Ext1(2Dλ,Dα

) ∼= Ext1(Dλ,2L5

) = 0. Similarly, if P(Dα) has a subquotient with simple headDλ and simple socleDµ, where

[P(Dα) : Dλ] = 2 = [P(Dα) : Dµ] + 1, then Ext1(Dα↓B ,Dµ) = 0 = Ext1(Dα,Dλ) = 0,

whereDλ↑B ∼= 2Dλ.

Lemma 5.10. Supposeγ andδ arep-regular, and

[Sγ : Dλ] = [Sδ : Dλ] = 1= [P(Dα) : Dλ

] − 2.

ThenDλ lies in or below the third Loewy layer ofSδ .An analogous statement holds forSδ .

Proof. By Lemma 5.4(1),Sκ↑B has a simple headDγ and a simple socleDα . Let M bethe submodule ofP(Dα) that is isomorphic to

(Sκ↑B + Ω(L5)

)/Ω(L5) ⊆ P(Dα)/Ω(L5) ∼= L5 ⊆ P(Dα).

ThenM has a simple socleDα and a simple headDγ , sinceM ∼= Sκ↑B/(Ω(L5)∩Sκ↑B).Furthermore,[M : Dλ] > 0, as[Sκ↑B : Dλ] = 2 > 1 = [Ω(L5) : Dλ] by Proposition 4.6Viewing Sκ↑B as a submodule ofP(Dα), we haveM ⊆ Sκ↑B by Lemma 2.1. In factM Sκ↑B so thatM ⊆ rad(Sκ↑B). Now, viewingSδ as a submodule ofSκ↑B , we seethatM ⊆ Sδ by Lemma 2.1 (applied to rad(Sκ↑B)). The lemma thus follows.

6. Some sufficient conditions

In this section, we show thatB inherits the properties that its Ext-quiver is bipartand its principal indecomposable modules have a common Loewy length fromB if the[3 : 2]-pair has the following conditions:

(Y1) If Dλ is a non-exceptional simple module ofB, then[P(Dα) : Dλ] 3.(Y2) If [P(Dα) : Dλ] is odd, then Ext1(Dα↑B,Dλ) = 0.(Y3) If [P(Dα) : Dλ] = 2, with Dλ↓B

∼= 2Dλ, then Ext1(Dα,Dλ) = 0 if and only if

Ext1(Dα,Dλ) = 0.


e.6

e

each

e

w

urth

While these conditions appear to be imposed on the blockB, they are equivalent to thanalogous conditions imposed on the blockB, which can be seen using Proposition 4and Corollary 4.4.

Remark. If (Y1) holds, then

(1) the last two possibilities in Proposition 4.6 do not occur;(2) [L5 : Dλ] = [Dα↑B : Dλ] + 1 for all composition factorsDλ of L5;(3) [L5 : Dλ] = [Dα↓B : Dλ] + 1 for all composition factorsDλ of L5;(4) [P(Dα) : Dµ] = [L5 : Dµ] + 1 for all composition factorsDµ of P(Dα);(5) [P(Dα) : Dµ] = [L5 : Dµ] + 1 for all composition factorsDµ of P(Dα).

We now proceed to extract more information about[3 : 2]-pairs satisfying one or morconditions. We will merely state and prove the results forB, and it is not difficult to seethat analogous results and proofs also hold forB, usually by interchangingDα with Dα ,L5 with L5, Dλ with Dλ, restriction with induction, andB with B.

Lemma 6.1. Suppose(Y1) holds. ThenL5 has Loewy length5.

Proof. The copy ofDα lying in the heart ofL5 must lie in the third Loewy layer ofL5,sinceDα↑B is a quotient ofL5 and has Loewy length 3. LetL3 be the submodule ofL5isomorphic toDα↑B . The head ofL3, being the copy ofDα lying in the heart ofL5, liesin the third Loewy layer ofL5. Each composition factor in the heart ofL3 occurs twicein L5, once in its second Loewy layer and once below the third Loewy layer. Sincecomposition factor ofL5 which does not occur inL3 only occurs once inL5, andL5 isself-dual, this implies that it must lie in the second or third Loewy layer ofL5. ThusL5has Loewy length 5.

Out of the proof we get an immediate corollary.

Corollary 6.2. Suppose(Y1) holds and[L5 : Dλ] = 2. ThenDλ occurs once each in thsecond and fourth Loewy layers ofL5.

SinceP(Dα) has a quotient isomorphic toL5, the two copies ofDα lying in the heart ofP(Dα) lies in the third and fifth Loewy layer ofP(Dα), by Lemma 6.1. Now, viewingL5as a submodule ofP(Dα), we see that head ofL5 lies in the third Loewy layer ofP(Dα)

and the copy ofDα in its heart lies in the fifth Loewy layer ofP(Dα). Each compositionfactor occurring twice inL5 occurs thrice inP(Dα), in the second, the fourth and belothe fifth Loewy layers ofP(Dα). Each composition factor occurring once inL5 occurstwice in P(Dα), once in the second or third Loewy layer and once in or below the foLoewy layer. Each composition factor ofP(Dα) which is not a composition factor ofL5occurs only once. Together with the fact thatP(Dα) is self-dual, we have:

α
Proposition 6.3. Suppose(Y1) holds. Then the Loewy length ofP(D ) is 7.


rs of

the

tion

of

e as-its

Corollary 6.4. Suppose(Y1) holds and[P(Dα) : Dλ] = m.

(1) If m = 3, thenDλ occurs once each in the second, fourth, and sixth Loewy layeP(Dα).

(2) If m = 2, thenDλ occurs once in the second or third Loewy layer, and once infourth or fifth Loewy layer ofP(Dα).

(3) If m = 1, thenDλ occurs in the second, third or fourth Loewy layer ofP(Dα).

Also,Dα occurs once each in the first, third, fifth and seventh Loewy layers ofP(Dα).

Lemma 6.5. Suppose(Y2) holds, and[P(Dα) : Dλ] = 1. ThenExt1(Dα,Dλ) = 0.

Proof. By (Y2), we have Ext1(Dα↑B,Dλ) = 0. By Proposition 4.6,[Dα↑B : Dλ] = 0.The result thus follows. Corollary 6.6. Suppose(Y1) and (Y2) hold, and[P(Dα) : Dλ] = 1. ThenDλ lies in thethird or fourth Loewy layer ofP(Dα).

Condition (Y3) allows us to refine this corollary.

Lemma 6.7. Suppose(Y1)–(Y3) hold, and[P(Dα) : Dµ] = 1. ThenDµ lies in the fourthLoewy layer ofP(Dα).

Proof. If Dµ lies in the third Loewy layer ofP(Dα), then there exists a quotientM ofP(Dα) having a simple socleDµ and Loewy length 3. There is at least one composifactorDλ of the heart ofM which has multiplicity 2 inP(Dα) (otherwise, all compositionfactors of the heart ofM multiplicity 3 by Lemma 6.5, and they all occur in the heartDα↑B by Proposition 4.6; the existence ofM then implies that Ext1(Dα↑B,Dµ) = 0, con-tradicting (Y2)). By self-duality ofP(Dα) and Lemma 5.9(2), Ext1(Dα,Dλ) = 0 (whereDλ↓B

∼= 2Dλ). But this contradicts (Y3), sinceDλ lies in the second Loewy layer ofM ,and hence ofP(Dα).

We shall now state the main theorem of this section.

Theorem 6.8. Suppose(Y1)–(Y3) hold, andB has the properties that itsExt-quiver isbipartite and its principal indecomposable modules have a common Loewy length7. ThenB inherits these properties fromB.

The remainder of this section will be devoted to proving the above theorem. Wsume that (Y1)–(Y3) hold, andB has the properties that its Ext-quiver is bipartite and
principal indecomposable modules have a common Loewy length 7.


f

the

les ofond

e

,

the

ny of

Let Λ1, Λ2 be a partition of the simple modules ofB displaying the bipartite nature othe Ext-quiver ofB, with Dα ∈ Λ1. For j ∈ 1,2, define a subsetΛj of simple modulesof B by

Λj = Dλ

∣∣ soc(Dλ↑B

)∼= 2Dλ for someλ ∈ Λj

.

Proposition 6.9. The partitionΛ1,Λ2 defined above displays the bipartite nature ofExt-quiver ofB.

Proof. If suffices to show thatDα does not extend any simple module contained inΛ1.By Corollaries 6.4 and 6.6,Dα extendsDµ only if [P(Dα) : Dµ] = 3 or 2 (equivalently[P(Dα) : Dµ] = 1 or 2). The analogue of Lemma 6.7 shows that those simple moduthe first case are contained inΛ2, while (Y3) ensures the simple modules of the seccase are also contained inΛ2. Proposition 6.10. Assume that the hypotheses of Theorem6.8hold. Let[P(Dα) : Dλ] = 3.ThenP(Dλ) has Loewy length7.

Proof. Let Dλ↓B∼= 2Dλ. By Proposition 4.6,Dα occurs once inP(Dλ), so it is lying in

the fourth Loewy layer ofP(Dλ) by Lemma 6.7. For 2 j 6, letMj be the direct sum

of simple modules lying in thej th Loewy layer ofP(Dλ) and not isomorphic toDα , andlet Mj be such thatMj↑B ∼= 2Mj . Also, for 2 l 4, letNl be the direct sum of simplmodules lying in thelth Loewy layer ofL5 and not isomorphic toDα . By Frobeniusreciprocity,P(Dλ)↑B ∼= 2P(Dλ), and Corollary 6.4 tells us thatDα lies in the secondfourth and sixth Loewy layers ofP(Dλ). We have

P(Dλ

) =

Dλ

M2M3

Dα ⊕ M4M5M6

Dλ

and P(Dλ

) =

Dλ

M2M3

L5 ⊕ M4M5M6Dλ

=

Dλ

Dα ⊕ M2N2 ⊕ M3

Dα ⊕ N3 ⊕ M4N4Dα

M5M6Dλ

.

Now, if the Loewy length ofP(Dλ) is greater than 7, then by the bipartite nature ofExt-quiver ofB, it has to be 9, with some composition factor ofM5 lying in the seventhLoewy layer, directly below the copy ofDα in the sixth Loewy layer. But this will meathat there is no copy ofDα lying in the second socle layer, contradicting the self-dualitP(Dλ). HenceP(Dλ) has Loewy length 7. Proposition 6.11. Assume that the hypotheses of Theorem6.8hold. Let[P(Dα) : Dλ] = 1.
ThenP(Dλ) has Loewy length7.


ewyof

ich is

cleeef

eoft of

Proof. Using Lemma 6.7, we see thatDα is lying in the fourth Loewy layer ofP(Dλ).By self-duality ofP(Dλ), we see that its Loewy length is at least 7. Suppose its Lolength is greater than 7; then it is at least 9, by the bipartite nature of the Ext-quiverB.Thus there exists a submoduleM of P(Dλ) having a simple head (isomorphic toDµ say)and the Loewy length at least 7. IfM has no composition factor isomorphic toDα , thenM↓B is a B-module having no projective summand and Loewy length at least 7, whimpossible. HenceM has a composition factor isomorphic toDα . In fact, looking at theLoewy structure ofM , we find thatDα has to lie in the second Loewy layer ofM , so thatDµ necessarily extendsDα . Proposition 6.10 now tells us that[P(Dα) : Dµ] = 2. But thisimplies thatP(Dλ) has a quotientN (having Loewy length 4) which has a simple soDα and a copy ofDµ in its third Loewy layer. SinceN is isomorphic to a submodulof P(Dα) andDλ lies in the fourth Loewy layer ofP(Dα), this further shows that theris a copy ofDµ in the sixth Loewy layer ofP(Dα). But this is impossible, in view oCorollary 6.4.

We need the next two lemmas to prove the next proposition.

Lemma 6.12. Suppose(Y1) holds. LetM be aB-module with simple headDα and simplesocleDλ, and having Loewy length 4, where[P(Dα) : Dλ] = 2. ThenDα is a composi-tion factor of the heart ofM .

Proof. We show[M : Dα] 2, and for this, it suffices to showL5 ⊆ Ω(M), or equiva-lently, M is not a quotient ofP(Dα)/L5.

We know thatDλ occurs exactly once in the first three Loewy layers ofP(Dα) byCorollary 6.4(2), and thatL5 ⊆ rad2(P (Dα)) sinceDα does not self-extend. Thus, as walso have[P(Dα)/L5 : Dλ] = 1, we see thatDλ occurs in the first three Loewy layersP(Dα)/L5, and soDλ may only occur in the first three Loewy layers of any quotienP(Dα)/L5. Hence,M is not a quotient ofP(Dα)/L5. Lemma 6.13. Suppose(Y1) holds. LetM be a submodule ofP(Dα) having a simple headDλ and Loewy length3, where[P(Dα) : Dλ] = 2. Then[M : Dµ] = 0 for all Dµ (µ = λ)

such that[P(Dα) : Dµ] = 2.

Proof. We first show thatM is a submodule ofL5. Otherwise,Dλ occurs twice inM +L5.SinceDλ occurs in the first three socle layers of bothM andL5, the two copies ofDλ arefound in the first three socle layers ofM + L5. But this contradicts Corollary 6.4.

Thus,M is a submodule ofL5. Now, sinceL5 is self-dual and[L5 : Dλ] = 1 = [L5 :Dµ], we see thatM cannot have a composition factor isomorphic toDµ. Proposition 6.14. Assume that the hypotheses of Theorem6.8holds. Let

[P(Dα) : Dλ

] = 2.

ThenP(Dλ) has Loewy length7.


rdand

eoewy

ley

us,

o-.

n

e

of

Proof. We have two cases to consider:

(1) Ext1(Dα,Dλ) = 0.(2) Ext1(Dα,Dλ) = 0.

In the first case,Dα lies in the third and fifth Loewy layers ofP(Dλ). By self-duality ofP(Dλ), we see that its Loewy length is at least 7. SupposeP(Dλ) has Loewy length greatethan 7. Then it has a submoduleM whose head (isomorphic toDµ say) lies in the seconLoewy layer ofP(Dλ) and whose Loewy length is at least 7. Using Propositions 6.106.11, we conclude that[P(Dα) : Dµ] = 2. Also, Ext1(Dα,Dµ) = 0. NowDα must occuras a composition factor ofM , otherwiseM↓B would be aB-module having no projectivsummand and Loewy length at least 7, which is impossible. By considering the Lstructure ofM , we see that, in fact,Dα lies in the second and/or fourth Loewy layers ofM .If Dα lies in the second Loewy layer ofM , we will have a quotientN of P(Dλ) havinga simple socleDα , a copy ofDµ in its heart and Loewy length 3. But this is impossibby Lemma 6.13. On the other hand, ifDα lies in the fourth but not in the second Loewlayer ofM , we will have a quotientN ′ of M having a simple headDµ, a simple socleDα ,a heart which does not haveDα as a composition factor and Loewy length 4. ButN ′ isisomorphic to a submodule ofP(Dα) which is again impossible by Lemma 6.12. Ththe Loewy length ofP(Dλ) is 7.

In the second case, the composition factors in the second Loewy layer ofP(Dλ) doesnot extendDα by Proposition 6.9. Thus rad(P (Dλ)) has Loewy length at most 6, by Propsition 6.11 and the first case of this proof, so thatP(Dλ) has Loewy length at most 7Suppose for a contradiction that the Loewy length ofP(Dλ) is less than 7. It must thebe 5, with a copy ofDα each occurring in its second and fourth Loewy layers. LetM be aquotient ofP(Dλ) having a simple socleDα and Loewy length 4. Using Lemma 6.12, wsee thatM has another copy ofDα lying in its heart. HenceM has the following Loewystructure:

Dλ

Dα ⊕ N2N3Dα

.

Now, M↓B has head isomorphic to 2Dα ⊕ 2Dλ and socle isomorphic to four copies

Dα by Frobenius reciprocity. SinceP(Dλ)↓B∼= 2P(Dα) ⊕ 2P(Dλ), we see thatM↓B

is, in fact, a direct sum 2P(Dα) ⊕ 2M , whereM has head isomorphic toDλ and socleisomorphic toDα , and has another copy ofDα lying in its heart. SinceM is a submoduleof P(Dα), we can then conclude thatM has Loewy length 4. In fact, ifNj↓B

∼= 2Nj forj ∈ 2,3, M has the following Loewy structure:

Dλ

Dα ⊕ N2N ′

3,

Dα


layer

n

6.9–

ction

cks,

d

es have

onalt

h

whereN ′3 has a submodule isomorphic to that ofN3. Let Ω(M) have the following socle

structure:

T4T3T2Dλ

(soTj , which may be zero, lies in thej th socle layer ofP(Dλ)). It is not difficult to seethatΩ(M)↓B

∼= 2Ω(M). Let Tj↓B∼= 2Tj . ThenTj is thej th socle layer ofΩ(M). If T4

and henceT4 are both non-zero, then the latter must in fact lie in the second Loewyof P(Dλ). Hence, regardless whetherT4 is zero or not, sinceP(Dλ) has Loewy length 7and the Ext-quiver ofB is bipartite, we must have the socle ofM extending a compositiofactor of T3. But this contradicts the self-duality ofP(Dλ).

Thus, the main theorem of this section (Theorem 6.8) follows from Propositions6.11 and 6.14.

7. Examples

We conclude this paper by verifying the sufficient conditions listed in the last sefor certain[3 : 2]-pairs.

Let Bi (1 i p) be the defect 3 block ofkS4p+2i−3 with p-core(p + i − 2, i − 1).We use the〈〉-notation with 3p+2 beads to denote the partitions belonging to these bloand to avoid confusion, we include a subscripti to indicate that the partition belongs toBi .For example,〈1,2〉2 is the partition ofB2 whose abacus display of 3p+2 beads has a beaof weight 2 on the first column, and a bead of weight 1 on second column.

The blockB1 is the principal block ofkS4p−1, and this block is shown in [10] to havthe properties that its Ext-quiver is bipartite and its principal indecomposable modulea common Loewy length 7. The blocksBi andBi−1 (2 i p) form a[3 : 2]-pair, and wedenote the exceptional partition with respect to this[3 : 2]-pair byαi , βi , γi , δi , αi , βi , γi

andδi (again, the subscripti is included to avoid confusion). We note that these exceptipartitions are allp-regular, except forβ2 andδ2. Furthermore,α′

i is alsop-regular, excepfor α′

p.

Proposition 7.1. The[3 : 2]-pair Bi andBi−1 satisfy(Y1)–(Y3).

As a corollary, we have

Theorem 7.2. The blocksBi (1 i p) have the properties that theirExt-quivers arebipartite, and their principal indecomposable modules have a common Loewy lengt7.

The remainder of this paper will be devoted to proving Proposition 7.1.


n

,

r

nd

-

.

r

e,

Firstly, by calculating the decomposition matrix ofBi using Schaper’s formula, we caeasily verify that (Y1) is satisfied.

Now, by Proposition 4.6, if[P(Dαi ) : Dλ] > 0, andDλ↓Bi−1∼= 2Dλ, then[P(Dαi ) :

Dλ] + [P(Dαi ) : Dλ] = 4.By Corollary 4.4, we have Ext1(Dα↑B,Dλ) ∼= Ext1(Dα↓B ,Dλ), whereDλ↓B

∼= 2Dλ.Thus, to verify (Y2), it suffices to show that Ext1(Dαi ↑Bi ,Dλ) = 0 whenever[P(Dαi ) :Dλ] = 1, and Ext1(Dαi ↓Bi−1,D

µ) = 0 whenever[P(Dαi ) : Dµ] = 1.Let [P(Dαi ) : Dλ] = 1. From the decomposition matrix ofBi , we see[Sβi : Dλ] = 0.

If [Sγi : Dλ] or [Sδi : Dλ] = 1, then [Sκi ↑Bi : Dλ] = 1 by Corollary 5.5(2). ThusExt1(Dαi ↑Bi ,Dλ) = 0 by Lemmas 5.4(1) and 5.9(1). If[Sαi : Dλ = 1], theni p − 1,andDλ is the socle ofSαi . In this case, since[Sαi : Dβi+1] = 1 = [P(Dαi ) : Dβi+1] − 1,Ext1(Dαi ↑Bi ,Dλ) = 0 by Lemma 5.9(1).

Now let [P(Dαi ) : Dµ] = 1. From the decomposition matrix ofBi−1, we see that eithe

• [Sγi : Dµ] = 1; or• [Sδi : Dµ] = 1 andi 3.

In both of these cases, Ext1(Dαi ↓Bi−1,Dµ) = 0 by the analogues of Lemmas 5.4(1) a

5.9(1).For (Y3), we show a stronger version—if[P(Dαi ) : Dλ] = 2, then Ext1(Dαi ,Dλ) =

0= Ext1(Dαi ,Dλ), whereDλ↓Bi−1∼= 2Dλ. By Corollary 5.6, we only need to look atDµ

with [P(Dαi ) : Dµ] = 2 = [Sαi : Dµ] + 1, and if i 3, Dλ with [P(Dαi ) : Dλ] = 2 =[Sαi : Dλ] + 1. Note that we can (and shall) assume that the Ext-quiver ofBi−1 is bipartitewhen we are determining ifDαi could extendDµ.

If [P(Dαi ) : Dµ] = 2= [Sαi : Dµ] + 1, then either

• [Sβi : Dµ] = 1 or [Sγi : Dµ] = 1; or• [Sδi : Dµ] = 1.

In the first case, relationships (D1) and (D3) and Proposition 4.6 show[Sαi : Dµ] = 1,whereDµ↑B ∼= 2Dµ. As Sαp is simple, isomorphic toDαp (from the decomposition matrix of Bp), we see that this case does not occur fori = p. Thus i < p, so thatα′

i isp-regular andSαi has a simple socle, and hence Ext1(Dαi ,Dµ) = 0 by Lemma 5.9(2)In the second case, there is a copy ofDµ lying in the fifth Loewy layer ofP(Dαi ),by Lemma 5.7 and Corollary 6.4. By induction the Ext-quiver ofBi−1 is bipartite soExt1(Dαi ,Dµ) = 0.

If [P(Dαi ) : Dλ] = 2 = [Sαi : Dλ] + 1, then from the decomposition matrix ofBi , wesee thati < p andλ = βi+1 or 〈i, i + 1〉i or, if i = p − 2, 〈i〉i . Moreover,[Sγi+1 : Dλ] =[Sδi+1 : Dλ] = 1 = [P(Dαi+1) : Dλ] − 2. ThusDλ lies in or below the third Loewy layeof Sδi+1 = Sαi by Lemma 5.10. Hence Ext1(Dαi ,Dλ) = 0 by Corollary 5.6 fori > 2. Fori = 2, Sβ2 is simple, isomorphic toDα2. Thus if Ext1(Dα2,Dν) = 0, then eitherν = γ2 orDν occurs in the second Loewy layer ofSα2 by Lemma 5.4(1) and Corollary 5.5(2). Henc
Ext1(Dαi ,Dλ) = 0 for i = 2 as well. Finally, we show that Ext1(Dα2,Dγ2) = 0. This is


2)

e alge-

l. 682,

Appl.,

) 599–

(1995)

ngew.

es and

ol. 84,

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because[Sγ2 : D〈1,1,3〉1] = 1= [P(Dα2) : D〈1,1,3〉1], so that the analogue of Lemma 5.9(applies.

This completes the proof of Proposition 7.1.

References

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[6] A.S. Kleshchev, Branching rules for modular representations of symmetric groups II, J. Reine AMath. 459 (1995) 163–212.

[7] A.S. Kleshchev, Branching rules for modular representations of symmetric groups III: Some corollaria problem of Mullineux, J. London Math. Soc. (2) 54 (1996) 25–38.

[8] P. Landrock, Finite Group Algebras and their Modules, London Math. Soc. Lecture Note Ser., vCambridge Univ. Press, 1983.

[9] S. Martin, L. Russell, Defect 3 blocks of symmetric group algebras, J. Algebra 213 (1999) 304–339.[10] S. Martin, K.M. Tan, Defect 3 blocks of symmetric group algebras I, J. Algebra 237 (2001) 95–120.[11] K.D. Schaper, Charakterformuln für Weyl-Moduln und Specht-Moduln in Primcharakteristik, Diploma

Bonn, 1981.

-pairs of symmetric group algebras and their intermediate defect 4 blocks

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