On permutation modules and decomposition numbers for symmetric groups
aa r X i v : . [ m a t h . R T ] A p r On permutation modules anddecomposition numbers for symmetricgroups
Eugenio Giannelli
Abstract
We study the indecomposable summands of the permutation module ob-tained by inducing the trivial F ( S a ≀ S n )-module to the full symmetric group S an for any field F of odd prime characteristic p such that a < p ≤ n . Inparticular we characterize the vertices of such indecomposable summands. Asa corollary we will disprove a modular version of Foulkes’ Conjecture.In the second part of the article we will use this information to give anew description of some columns of the decomposition matrices of symmetricgroups in terms of the ordinary character of the Foulkes module φ ( a n ) . The determination of the decomposition matrices and the study of the modularstructure of permutation modules are two important open problems in the represen-tation theory of symmetric groups.Young permutation modules were deeply studied by James in [11], Klyachko in[14] and Grabmeier in [9]. They completely parametrized the indecomposable sum-mands of such modules (known as Young modules) and developed a Green corre-spondence for those summands. Their original description of the modular structureof Young modules was based on Schur algebras. More recently Erdmann in [5] de-scribed completely the Young modules using only the representation theory of thesymmetric groups. In particular it is proved that the vertex of a given Young moduleof the symmetric group S n of degree n is always conjugate to a Sylow p -subgroup of S λ × S λ × · · · × S λ k , where ( λ , . . . , λ k ) is a partition of n .The problem of finding decomposition numbers for symmetric groups in primecharacteristic has been studied extensively. The decomposition matrix of the sym-metric group S n in prime characteristic p has rows labelled by the partitions of n ,and columns by the so called p -regular partitions of n , namely, partitions of n withless than p parts of any given size. The decomposition number d µν is the entry of thedecomposition matrix that records the number of composition factors of the Spechtmodule S µ , defined over a field of characteristic p , that are isomorphic to the simplemodule D ν , defined by James in [13] as the unique top composition factor of S ν .The main purpose of this paper is to begin the study, over a field of prime char-acteristic, of a new family of permutation modules known as Foulkes modules. The1ew information obtained will allows us to draw corollaries on decomposition num-bers. In particular we will give a new combinatorial description of certain columnsof the decomposition matrices of symmetric groups.In characteristic zero the study of Foulkes modules was mainly connected to thelong standing Foulkes’ Conjecture, stated by H.O. Foulkes in [6]. For some recentadvances on the ordinary character of the Foulkes modules see [7] and [15]. Forany a and n natural numbers the Foulkes module H ( a n ) is obtained by inducingthe trivial representation of the wreath product S a ≀ S n up to the full symmetricgroup S an . If the prime p is the characteristic of the underlying field F , we observethat whenever n < p then the indecomposable summands of the Foulkes moduleare Young modules, since H ( a n ) is isomorphic to a direct summand of the Youngpermutation module M ( a n ) in this case. On the other hand for n ≥ p we do nothave any precise information on the non-projective indecomposable summands of H ( a n ) . We will focus on the study of these new indecomposable F S an -modules. Inparticular we will give the following description of the possible vertices for all a and n such that a < p ≤ n . Theorem 1.1.
Let p be an odd prime and let a and n be natural numbers such that a < p ≤ n . Let U be an indecomposable non-projective summand of the F S an -module H ( a n ) and let Q be a vertex of U . Then there exists s ∈ { , , . . . , ⌊ np ⌋} ∩ N such that Q is conjugate to a Sylow p -subgroup of S a ≀ S sp . Moreover the Green correspondentof U admits a tensor factorization V ⊠ Z as a module for F (( N S asp ( Q ) /Q ) × S a ( n − sp ) ) ,where V is isomorphic to the projective cover of the trivial F ( N S asp ( Q ) /Q ) -moduleand Z is an indecomposable projective summand of H ( a n − sp ) . This result should be compared with [8, Theorem 1.2] which studies summands ofcertain twists by the sign character of the permutation module H (2 n ) . The main toolused to study the vertices of H ( a n ) is the Brauer correspondence for p -permutationmodules as developed by Brou´e in [4].It is clear by Theorem 1.1 that the non-projective indecomposable summands ofthe permutation module H ( a n ) are not Young modules for any a < p ≤ n , since theirvertices are not Sylow p -subgroups of Young subgroups of S an . This observation willbe sufficient to disprove the modular version of Foulkes’ Conjecture.In order to present our new result on decomposition numbers we need to in-troduce the following definition. Let γ be a p -core partition and let φ ( a n ) be theordinary character afforded by the Foulkes module H ( a n ) . Denote by F ( γ ) the setcontaining all the partitions µ of an such that the p -core of µ is equal to γ and suchthat the irreducible ordinary character of S an labelled by µ has non-zero multiplicityin the decomposition of φ ( a n ) as a sum of irreducible characters. In symbols F ( γ ) = { µ ⊢ an : γ ( µ ) = γ and (cid:10) χ µ , φ ( a n ) (cid:11) = 0 } . The new results obtained in Theorem 1.1 will lead us to prove the followingtheorem.
Theorem 1.2.
Let a, n be natural numbers and let p be a prime such that a < p ≤ n .Let λ be a p -regular partition of na such that λ has weight w < a . Denote by γ the p -core of λ . If λ is maximal in F ( γ ) , then the only non-zero entries in the column abelled by λ of the decomposition matrix of S an are in the rows labelled by partitions µ ∈ F ( γ ) . Moreover [ S µ : D λ ] ≤ (cid:10) φ ( a n ) , χ µ (cid:11) . Theorem 1.2 allows us to detect new information on decomposition numbersfrom the study of the ordinary structure of the Foulkes character. In particular thestudy of the zero-multiplicity characters in the decomposition of φ ( a n ) leads to somenew non-obvious zeros in certain columns of the decomposition matrix of S an (seeCorollary 4.3 and Example 4.4).The paper is structured as follows. The necessary background on the Brauer cor-respondence and a detailed description of the combinatorial structure of the Foulkesmodules are given in Section 2. In Section 3 we will prove Theorem 1.1. The proofis split into a series of lemmas and propositions describing the structural propertiesof the Foulkes modules. Finally in Section 4 we will prove Theorem 1.2 and givesome applications. In this section we briefly present the theoretical tools that we will extensively use todeduce our new results. A partition λ of a natural number n is a weakly decreasingsequence of positive numbers λ = ( λ , λ , . . . , λ k ) such that P ki =1 λ i = n . Given anatural number n and a partition λ of n we will denote by S λ the Specht modulelabelled by λ and by χ λ the ordinary character afforded by S λ . This notation agreeswith the notation of [10] and we refer the reader to it for a comprehensive expositionof the general theory of the symmetric group representations. As mentioned in the introduction we will study, in the next section, the verticesof a family of p -permutation modules. One of the most important techniques usedwill be the Brauer construction applied to p -permutation modules. Here we willsummarize the main results of Brou´e’s paper [4].Let F be a field of prime characteristic p and G a finite group. An F G -module V is called a p -permutation module if for every P ∈ Syl p ( G ) there exists a linear basis B P of V that is permuted by P . It is quite easy to derive from the definition thatthe direct sum and the tensor product of p -permutation modules are p -permutationmodules; if H ≤ G then the restriction to H of a p -permutation F G -module is a p -permutation module as well as the induction from H to G of a p -permutation F H -module; in conclusion every direct summand of a p -permutation module is a p -permutation module. A complete characterization of such modules is contained inthe following theorem. Theorem 2.1.
An indecomposable F G -module V is a p -permutation module if andonly if there exists P ≤ p G such that V | F ↑ GP .
3e recall now the definition and the first basic properties of the Brauer con-struction for F G -modules. Given an F G -module V and Q ≤ p G we denote by V Q the set of fixed elements { v ∈ V | vg = v for all g ∈ Q } . It is easy to see that V Q is an F N G ( Q )-module on which Q acts trivially. For P a proper subgroup of Q , therelative trace map Tr QP : V P → V Q is the linear map defined byTr QP ( v ) = X g ∈ Z vg, where Z is a set of right coset representatives for P in Q . It is easy to notice thatthe definition of the map does not depend on the choice of the set of representatives.We observe that also Tr Q ( V ) := X P Let V be an indecomposable p -permutation module and Q be a vertexof V . Let P be a p -subgroup of G , then V ( P ) = 0 if and only if P ≤ Q g for some g ∈ G . If V is an F G -module with p -permutation basis B with respect to a Sylow p -subgroup P of G and R ≤ P , then taking for each orbit of R on B the sum of thevectors in that orbit, we obtain a basis for V R . The sums over vectors lying in orbitsof size p or more are relative traces from proper subgroups of R , and so V ( R ) isisomorphic to the F -span of B ( R ) := { v ∈ B : vg = v for all g ∈ R } . Thus Theorem 2.2 has the following corollary, Corollary 2.3. Let V be a p -permutation F G -module with p -permutation basis B with respect to a Sylow p -subgroup P of G . Let R ≤ P . Then the F N G ( R ) -module V ( R ) is equal to hB ( R ) i and V has an indecomposable summand with a vertex con-taining R if and only if B ( R ) = ∅ . The next result [4, 3.4] explains what is now known as Brou´e’s correspondence. Theorem 2.4. An indecomposable p -permutation module V has vertex Q if and onlyif V ( Q ) is a projective F ( N G ( Q ) /Q ) -module. Furthermore • The Brauer map sending V to V ( Q ) is a bijection between the set of indecom-posable p -permutation F G -modules with vertex Q and the set of indecomposableprojective F ( N G ( Q ) /Q ) -modules. Regarded as an F N G ( Q ) -module, V ( Q ) is theGreen correspondent of V . Let V be a p -permutation F G -module and E an indecomposable projective F ( N G ( Q ) /Q ) -module. Then E is a direct summand of V ( Q ) if and only ifits correspondent U ( i.e the F G -module U such that U ( Q ) ∼ = E ) is a directsummand of V . Some important consequences that we will use extensively in the next sectionsare stated below. Corollary 2.5. Let U be a p -permutation F G -module and let Q be a p -subgroup of G . The Brauer correspondent U ( Q ) of U is a p -permutation F N G ( Q ) -module.Proof. Let P ′ be a Sylow p -subgroup of N G ( Q ) and let P be a Sylow p -subgroup of G containing P ′ . Denote by B P a p -permutation basis of U with respect to P . ByCorollary 2.3 we have that the F N G ( Q )-module U ( Q ) has linear basis B P ( Q ). It iseasy to observe that B P ( Q ) is a p -permutation basis with respect to P ′ . Let P ′′ beanother Sylow p -subgroup of N G ( Q ) and let g ∈ N G ( Q ) such that P ′′ = P ′ g . Thenwe have that B ′′ := { xg | x ∈ B P ( Q ) } is a p -permutation basis of U ( Q ) with respect to P ′′ . This completes the proof. Corollary 2.6. Let G and H be two finite groups and let C be a subgroup of G .Let U be an indecomposable p -permutation F G -module and V be a p -permutation F H -module. Then • If U ↓ C = W ⊕· · ·⊕ W k , then there exist a vertex R of U and vertices Q , . . . , Q k of W , . . . , W k respectively, such that Q i ≤ R for all i ∈ { , , . . . , k } . • The indecomposable F ( G × H ) -module U ⊠ V has a vertex containing both Q and P , vertices of U and V respectively. Lemma 2.7. Let U and V be p -permutation F G -modules and let P be a p -subgroupof G , then ( U ⊕ V )( P ) ∼ = U ( P ) ⊕ V ( P ) as F p ( N G ( P )) -modules. The following lemma is stated in [8, Lemma 4.7]; we include the proof here tomake this article more self contained. Lemma 2.8. Let Q and R be p -subgroups of a finite group G and let U be a p -permutation F G -module. Let K = N G ( R ) . If R is normal in Q then the Brauercorrespondents U ( Q ) and (cid:0) U ( R ) (cid:1) ( Q ) are isomorphic as F N K ( Q ) -modules.Proof. Let P be a Sylow p -subgroup of N G ( R ) containing Q and let B be a p -permutation basis for U with respect to P . By Corollary 2.3 we have U ( Q ) = hB ( Q ) i as an F N G ( Q )-module. In particular U ( Q ) y N K ( Q ) = hB ( Q ) i as an F N K ( Q )-module. On the other hand U ( R ) = hB ( R ) i as an F N G ( R )-module.Now B ( R ) is a p -permutation basis for U ( R ) with respect to K ∩ P . Since thissubgroup contains Q we have (cid:0) U ( R ) (cid:1) ( Q ) = hB ( R ) i ( Q ) = h ( B ( R ))( Q ) i = hB ( Q ) i , as F N K ( Q )-modules, as required. 5 emma 2.9. Let G and G ′ be finite groups and let U and U ′ be p -permutationmodules for F G and F G ′ , respectively. If Q ≤ G is a p -subgroup then ( U ⊠ U ′ )( Q ) = U ( Q ) ⊠ U ′ , where on the left-hand side Q is regarded as a subgroup of G × G ′ in the obviousway.Proof. This follows easily from Corollary 2.3 by taking p -permutation bases for U and U ′ . Proposition 2.10. Let M be a p -permutation F G -module and let P be a p -subgroupof G . If M ( P ) is an indecomposable F ( N G ( P )) -module then M has a unique inde-composable summand U such that P is contained in a vertex of U .Proof. Suppose by contradiction that exist V and V indecomposable summands of M with vertices Q and Q respectively, such that P ≤ Q , Q . Then by Lemma 2.7we have that V ( P ) ⊕ V ( P ) | M ( P ) . This contradicts the indecomposability of M ( P ) since by Theorem 2.4 we have that V i ( P ) = 0 for i ∈ { , } .An argument that we shall use several times is stated in the lemma below: Lemma 2.11. If P is a p -group and Q is a subgroup of P then the permutationmodule F ↑ PQ is indecomposable, with vertex Q . We conclude the section by recalling the definition and the basic properties ofScott modules. We refer the reader to [4, Section 2] for a more detailed account.Given a subgroup H of G there exists a unique indecomposable summand U of thepermutation module F ↑ GH such that the trivial F G -module is a submodule of U . Wesay that U is the Scott module of G associated to H and we denote it by Sc( G, H ).The following theorem summarizes the main properties of Scott modules. Theorem 2.12. Let G be a finite group, H a subgroup of G and P a Sylow p -subgroup of H . Then the Scott module Sc( G, P ) is isomorphic to Sc( G, H ) and isuniquely determined up to isomorphism by either of the following properties: • The trivial F G -module is isomorphic to a submodule of Sc( G, P ) . • The trivial F G -module is isomorphic to a quotient of Sc( G, P ) .Moreover, Sc( G, P ) has vertex P and the Brou´e correspondent (Sc( G, P ))( P ) isisomorphic to the projective cover of the trivial F ( N G ( P ) /P ) -module .2 Blocks of symmetric groups The blocks of symmetric groups are combinatorially described by Nakayama’s Con-jecture, first proved by Brauer and Robinson in [3] and [16]. In order to state thisresult, we must recall some definitions.Let λ be a partition. A p -hook in λ is a connected part of the rim of the Youngdiagram of λ consisting of exactly p boxes, whose removal leaves the diagram of apartition. By repeatedly stripping off p -hooks from λ we obtain the p -core of λ ; thenumber of hooks we remove is the weight of λ . Theorem 2.13 (Nakayama’s Conjecture) . Let p be prime. The p -blocks of S n arelabelled by pairs ( γ, w ) , where γ is a p -core and w ∈ N is the associated weight,such that | γ | + wp = n . Thus the Specht module S λ lies in the block labelled by ( γ, w ) if and only if λ has p -core γ and weight w . (cid:3) Often it is best to consider partitions using James’ abacus : for a detailed accountof this tool see [12, pages 76-78]. For example given a partition λ , it is very easy tounderstand its p -core γ ( λ ) by using the abacus.The following result on the defect group of Blocks of the symmetric group (see[12, Theorem 6.2.45]) will be important in the proof of our main Theorem 1.2. Theorem 2.14. The defect group of a symmetric group block of weight w is conju-gate to a Sylow p -subgroup of S wp . Let K be a field and a , n two non-zero natural numbers. Let Ω ( a n ) be the collectionof all set partitions of { , , . . . , an } into n sets of size a . We will denote an arbitraryelement ω ∈ Ω ( a n ) by ω = { ω , ω , · · · , ω n } , where ω j ⊆ { , , · · · , an } , | ω j | = a and ω i ∩ ω j = ∅ for all 1 ≤ i < j ≤ n . We willcall ω j a set of ω . The symmetric group S an acts transitively in a natural way onΩ ( a n ) by permuting the numbers in each set of every set partition. Let H ( a n ) be the K S an -permutation module generated as a K -vector space by the elements of Ω ( a n ) ,with the action of S an defined as the natural linear extension of the action on Ω ( a n ) .The module H ( a n ) is called a Foulkes module . Since the action of S an is transitiveon Ω ( a n ) and the stabilizer of any set partition ω ∈ Ω ( a n ) is isomorphic to S a ≀ S n itis finally easy to deduce that H ( a n ) ∼ = K x S an S a ≀ S n . When the field K is the field of complex numbers C , the study of the decompo-sition into irreducible direct summands of the Foulkes module is closely related tothe problem known as Foulkes’ Conjecture as stated firstly in [6] by H.O. Foulkes in1950. Conjecture. Let K = C and let a and n be natural numbers such that a < n . Then H ( n a ) is a direct summand of H ( a n ) . 7f we replace C with F (a field of prime characteristic p ) in the statement abovewe obtain a modular version of Foulkes’ Conjecture. This version is known to befalse but we are not aware of any explicit reference in the literature. We will give ashort proof of this fact as a corollary of Theorem 1.1. H ( a n )This section is devoted to the proof of Theorem 1.1. As explained in the introduction,the proof is split into a series of lemmas and propositions. The structure of thesection is similar to Section 4 of [8] but different ideas and further ad hoc argumentsare needed here. We start by fixing some notation. Let F be a field of odd primecharacteristic p and let a and n be natural numbers such that a < p ≤ n . Let S a ≀ S n be the subgroup of S an acting transitively and imprimitively on { , , . . . , an } andhaving as blocks of imprimitivity the sets T j = { j, n + j, n + j, . . . , ( a − n + j } for j ∈ { , . . . , n } . In this setting we have that for any Sylow p -subgroup P of S a ≀ S n there exists a Sylow p -subgroup Q of S { ,...,n } , such that P is conjugate to Q = { x | x ∈ Q } , where ( j + kn ) x = ( j ) x + kn for all j ∈ { , . . . , n } and all k ∈ { , , . . . , a − } .Let ρ be an element of order p in S a ≀ S n . By the above remarks there exists s ∈ { , , . . . , ⌊ np ⌋} ∩ N such that ρ has sa orbits of order p and a ( n − sp ) fixed pointsin its natural action on { , , . . . , an } .For all j ∈ N such that pj ≤ an let z j be the p -cycle of S an defined by z j = ( p ( j − 1) + 1 , p ( j − 1) + 2 , . . . , pj ) . Denote by R ℓ the cyclic subgroup of S an of order p generated by z z · · · z ℓ . We willcall O , . . . , O ℓ the p -orbits of R ℓ . Note that O j = supp( z j ) for all j ∈ { , , . . . , ℓ } .In the following lemma we will study the Brou´e correspondence for H ( a n ) withrespect to R ℓ . Lemma 3.1. Let a and n be natural numbers and p an odd prime such that a < p ≤ n . Let ℓ be a natural number such that ℓp ≤ an . If ℓ = as for some natural number s , then H ( a n ) ( R as ) ∼ = H ( a sp ) ( R as ) ⊠ H ( a n − sp ) as F N S an ( R as ) -modules. If ℓ is not an integer multiple of a then H ( a n ) ( R ℓ ) = 0 .Proof. We already noticed that the number of p -orbits of an element of order p in S a ≀ S n must be a multiple of a . Therefore if ℓ is not an integer multiple of a then R ℓ is not conjugate to any subgroup of S a ≀ S n . This implies that H ( a n ) ( R ℓ ) = 0 byTheorem 2.2. 8uppose now that ℓ = as for some s ∈ N . Let ω = { ω , ω , . . . , ω n } ∈ Ω ( a n ) befixed by R as . Then there exist ω j , . . . , ω j sp sets of ω such that sp [ i =1 ω j i = supp( R as )since no set of a fixed set partition can contain two numbers x and y such that x ∈ supp( R as ) and y / ∈ supp( R as ). So we can write each ω ∈ Ω ( a n ) ( R as ) as ω = u ω ∪ v ω where u ω = (cid:8) ω j , . . . , ω j sp (cid:9) ∈ Ω ( a sp ) ( R as ) , and v ω is a set partition in Ω ( a n − sp )+ , that is the collection of all the set partitionsof { asp + 1 , . . . , an } into n − sp sets of size a . We will also denote by H ( a n − sp )+ the F S { asp +1 ,...,an } -permutation module generated by Ω ( a n − sp )+ as a vector space. The map ψ : Ω ( a n ) ( R as ) −→ Ω ( a sp ) ( R as ) × Ω ( a n − sp )+ that associates to each ω ∈ Ω ( a n ) ( R as ) the element u ω × v ω ∈ Ω ( a sp ) ( R as ) × Ω ( a n − sp )+ isa well defined bijection. This factorization of the linear basis of H ( a n ) ( R as ) inducesan isomorphism of vector spaces between H ( a n ) ( R as ) and H ( a sp ) ( R as ) ⊠ H ( a n − sp )+ thatis compatible with the action of N S an ( R as ) ∼ = N S asp ( R as ) × S a ( n − sp ) . Therefore wehave that H ( a n ) ( R as ) ∼ = H ( a sp ) ( R as ) ⊠ H ( a n − sp )+ as F ( N S an ( R as ))-modules. The proposition now follows after identifying S { asp +1 ,...,an } with S a ( n − sp ) and H ( a n − sp )+ with H ( a n − sp ) .Lemma 3.1 allows us to restrict for the moment our attention to the study of theBrou´e correspondent H ( a sp ) ( R as ) of H ( a sp ) . In particular we will now give a precisedescription of its canonical basis Ω ( a sp ) ( R as ) constitued by the set partitions fixedunder the action of R as . In order to do this we need to introduce a new importantconcept.Let δ = { δ , δ , . . . , δ s } be a set partition of { , , . . . , as } into s sets of size a (namely δ ∈ Ω ( a s ) ). Let A , A , . . . , A s be subsets of { , , . . . , asp } of size a suchthat for each i ∈ { , . . . , s } and j ∈ { , . . . , as } we have | A i ∩ O j | = ( j ∈ δ i j / ∈ δ i . In particular each set A i contains at most one element of a given orbit of R as .Consider now ω to be the element of Ω ( a sp ) ( R as ) of the form ω = { A , A σ, A σ , . . . , A σ p − , A , . . . A σ p − , . . . . . . , A s , . . . , A s σ p − } , where σ = z z · · · z as . We will say that the set partition ω has type δ . Notice thatfrom the type we can read how the orbits of R as are relatively distributed in the setsof the set partition ω .In the following lemma we will show that the set partitions of Ω ( a sp ) that arefixed by the action of R as are precisely the ones of the form described above.9 emma 3.2. Let the set partition ω = { ω , . . . , ω sp } be an element of Ω ( a sp ) . Then ω is fixed by R as if and only if there exists a corresponding set partition δ = { δ , . . . , δ s } ∈ Ω ( a s ) and s sets A , . . . , A s of ω such that | A i ∩ O j | = ( if j ∈ δ i if j / ∈ δ i and ω = { A , A σ, A σ , . . . , A σ p − , A , . . . A σ p − , . . . . . . , A s , . . . , A s σ p − } , where σ = z z · · · z as .Proof. Suppose that ω is fixed by R as = h σ i . Let O j be an orbit of R as for some j ∈ { , , . . . , as } and let ω j , ω j , . . . , ω j l be the sets of ω such that ω j i ∩ O j = ∅ .Clearly l ≤ p because |O j | = p . Since ωσ = ω we have that for all x ∈ { j , . . . , j l } there exists y ∈ { j , . . . , j l } such that x = y and ω j x σ = ω j y and no ω j i is fixed by σ because a < p . In particular we have that R as acts without fixed points on the set { ω j , ω j , . . . , ω j l } . Therefore there exists a number k ≥ p k = |{ ω j , ω j , . . . , ω j l }| = l ≤ p. This immediately implies that l = p and therefore that | ω j i ∩ O j | = 1 for all i ∈{ , , . . . , l } . This argument holds for all the R as orbits O , O , . . . , O as . Hence for all x ∈ { , , . . . , sp } the set ω x of ω contains a numbers no two of which are in the same R as -orbit. Consider one of those sets, say A , of ω . Define the correspondent set δ of size a as follows: for all i ∈ { , , . . . , as } , let i ∈ δ if and only if | A ∩ O i | = 1.Observe that since ωσ = ω we have that A , A σ, . . . , A σ p − are p distinct sets of ω such that for all k ∈ { , , . . . , p − } we have that | A σ k ∩ O j | = ( j ∈ δ j / ∈ δ . We now repeat the above construction by considering a set A of ω such that A = A σ k for any k ∈ { , , . . . , p − } and defining the corresponding set δ exactly asabove. After s iterations of the process we obtain the claim, where the set partition δ ∈ Ω ( a s ) corresponding to ω is δ = { δ , δ , . . . , δ s } .The converse is trivial since a set partition ω of the form described in the hy-pothesis is clearly fixed by the action of σ .From Lemma 3.2 we obtain that every ω ∈ Ω ( a sp ) ( R as ) is of a well defined type δ ∈ Ω ( a s ) . In the next lemma we will fix a δ ∈ Ω ( a s ) and we will calculate explicitlythe number of set partitions of type δ in Ω ( a sp ) ( R as ). Lemma 3.3. For every given δ ∈ Ω ( a s ) there are p ( a − s distinct set partitions in Ω ( a sp ) ( R as ) of type δ . roof. Define δ ⋆ ∈ Ω ( a s ) by δ ⋆ = (cid:8) { , s, . . . , a − s } , { , s, . . . , a − s } , · · · , { s, s, . . . , as } (cid:9) . By Lemma 3.2 we have that given any set partition ω = { ω , . . . , ω sp } ∈ Ω ( a sp ) ( R s )of type δ ⋆ , then each set ω j contains exactly one element lying in { , , . . . , sp } , theunion of the first s orbits O , O , . . . , O s of R as . Therefore, without loss of generality,we can relabel the indices of the sets of ω in order to have for all j ∈ { , , . . . , sp } ω j = { j, x j , x j , . . . , x a − j } , where x ij is a number lying in the R as -orbit of j + isp for each i ∈ { , , . . . , a − } .Notice that this implies that there are p possible different choices for each x ij . Ifwe fix j ∈ { , , . . . , sp } such that j is not divisible by p then there exist uniquenatural numbers t and k in { , , . . . , s − } and { , , . . . , p − } respectively, suchthat j = tp + k . Moreover, by definition of σ it follows that (( t + 1) p ) σ k = j . Since ωσ k = ω , we must have ω ( t +1) p σ k = ω j . Therefore for all i ∈ { , , . . . , a − } wehave that x ij = x i ( t +1) p σ k Hence the set partition ω is uniquely determined by its sets ω p , ω p , . . . , ω sp . Thisimplies that there are exactly p ( a − s different set partitions of type δ ⋆ in Ω ( a sp ) ( R as ).It is an easy exercise to verify that, changing the labels, the argument above worksfor any other type δ in Ω ( a s ) .Consider now the subgroup of N S asp ( R as ) defined by C := h z i × h z i × . . . × h z as i . Notice that C preserves the type of set partitions in its action on Ω ( a sp ) ( R as ). There-fore we have that the sub-vectorspace K δ of H ( a sp ) ( R as ) generated by all the fixed setpartitions of type δ is an F C -submodule of H ( a sp ) ( R as ) for any given δ . Moreover,we deduce the following result: Proposition 3.4. The following isomorphism of F C -modules holds: H ( a sp ) ( R as ) y C ∼ = M δ ∈ Ω ( as ) K δ . Proof. For any given δ ∈ Ω ( a s ) denote by B δ the subset of Ω ( a sp ) ( R as ) containing allthe set partitions of type δ . Clearly H ( a sp ) ( R as ) decomposes as a vector space intothe direct sum of all the K δ for δ ∈ Ω ( a s ) . Moreover we observe that the orbits of C on { , , . . . , asp } are exactly the same as the orbits of R as and therefore if ω ∈ B δ then ωc ∈ B δ for any c ∈ C . This implies that H ( a sp ) ( R as ) ↓ C ∼ = M δ ∈ Ω ( as ) K δ as F C -modules, as desired. 11e will now define three p -subgroups of S asp that will play a central role in thenext part of the section. For all j ∈ { , , . . . , s } denote by π j the p -element of C given by π j = z j = z j z j + s z j +2 s · · · z j +( a − s . Let E s be the elementary abelian subgroup of C of order p s defined by E s = h π i × · · · × h π s i . Let P s be a Sylow p -subgroup of S { ,...,sp } with base group h z , . . . , z s i , chosenso that z z · · · z s is in its centre. (The existence of such Sylow p -subgroups followsfrom the construction of Sylow p -subgroups of symmetric groups as iterated wreathproducts in [12, 4.1.19 and 4.1.20]).Let Q s be the group consisting of all permutations g where g lies in P s . Forthe reader’s convenience we recall that for all k ∈ { , , . . . , a − } and all j ∈{ , , . . . , sp } , we have that ( j + ksp ) g = ( j ) g + ksp. In particular we observe that this implies that g = g g · · · g a − , where for all k ∈ { , , . . . a − } , g k is the element of S asp that fixes all the numbers outside { ksp + 1 , ksp + 2 , . . . , ( k + 1) sp } and such that ( j + ksp ) g k = ( j ) g + ksp for all j ∈ { , , . . . , sp } . Notice that Q s has E s as normal base group by construction andclearly R as ⊳ E s ⊳ C and R as ⊳ Q s .We are now very close to deducing the indecomposability of H ( a sp ) ( R as ) as an F N S asp ( R as )-module. In order to prove this we need to observe an important struc-tural property of the F C -modules K δ for all δ ∈ Ω ( a s ) . Proposition 3.5. For any δ ∈ Ω ( a s ) there exists g ∈ N S asp ( R as ) such that K δ ∼ = F ↑ CE gs Proof. As usual, define δ ⋆ ∈ Ω ( a s ) by δ ⋆ = { δ , δ , . . . , δ s } , where δ i = { i, i + s, i + 2 s, . . . , i + ( a − s } and let ω ⋆ be any fixed element of B δ ⋆ .Then by Lemma 3.2 we have that ω ⋆ = { A , A σ, . . . , A σ p − , A , . . . A σ p − , . . . . . . , A s , . . . , A s σ p − } , for some sets A , A , . . . , A s such that | A i ∩ O j | = ( j ∈ δ ⋆i j / ∈ δ ⋆i . This implies that we can equivalently rewrite ω ⋆ as ω ⋆ = { A , A π , . . . , A π p − , A , A π . . . A π p − , . . . . . . , A s , . . . , A s π p − s } . ω ⋆ is fixed by the action of E s . Moreover if we denote by L the stabilizerin C of ω ⋆ we have that as F C -modules K δ ⋆ ∼ = F ↑ CL since C acts transitively on the elements of B δ ⋆ .Lemma 3.3 implies that dim F ( K δ ⋆ ) = p s ( a − , therefore by [1, page 56] we havethat p s ( a − = | C : L | ≤ | C : E s | = p s ( a − . Hence E s = L and K δ ⋆ ∼ = F ↑ CE s as F C -modules. Since N S asp ( R as ) acts as the fullsymmetric group on the set {O , O , . . . , O as } , we obtain that for any δ ∈ Ω ( a s ) thereexists g ∈ N S asp ( R as ) such that any set partition of type δ in Ω ( a sp ) ( R as ) is fixed by E gs . With an argument completely similar to the one used above we deduce that K δ ∼ = F ↑ CE gs . The following corollary of Proposition 3.5 will be extremely useful in the lastpart of the section. Corollary 3.6. Every indecomposable summand of H ( a sp ) ( R as ) has vertex contain-ing E s .Proof. Let U be an indecomposable summand of H ( a sp ) ( R as ). By Lemma 2.11 andProposition 3.5 we observe that the restriction of U to C is isomorphic to a direct sumof indecomposable p -permutation F C -modules with vertices conjugate in N S asp ( R as )to E s . Therefore by the first part of Corollary 2.6 we obtain that E s is contained ina vertex of U .It is now possible to determine a vertex of H ( a sp ) ( R as ) as an F N S asp ( R as )-module. Proposition 3.7. The F N S asp ( R as ) -module H ( a sp ) ( R as ) is indecomposable and hasvertex Q s ∈ Syl p ( S a ≀ S sp ) Proof. Let δ ⋆ be the set partition of Ω ( a s ) defined at the beginning of the proof ofProposition 3.5. Since ω ∈ Ω ( a sp ) ( R as ) is fixed by E s if and only if ω ∈ B δ ⋆ and since E s ⊳ C , we have that( H ( a sp ) ( R as ))( E s ) ↓ C = H ( a sp ) ( R as ) ↓ C ( E s ) = K δ ⋆ ( E s ) ∼ = F ↑ CE s , as F C -modules. By Lemma 2.11 we have that ( H ( a sp ) ( R as ))( E s ) ↓ C is indecompos-able, hence also ( H ( a sp ) ( R as ))( E s ) is indecomposable. Therefore by Proposition 2.10there exists a unique summand of H ( a sp ) ( R as ) with vertex containing E s , but thisimplies that H ( a sp ) ( R as ) is indecomposable by Corollary 3.6.Let Q ≤ p N S asp ( R as ) be a vertex of H ( a sp ) ( R as ). Consider ω ⋆ to be the setpartition in Ω ( a sp ) ( R as ) defined by ω ⋆ = { ω , ω , . . . , ω sp } , ω j = { j, j + sp, j + 2 sp, . . . , j + ( a − sp } , for all j ∈ { , , . . . , sp } . Byconstruction we have that Q s fixes ω ⋆ . Therefore a conjugate of Q s is a subgroupof Q . On the other hand by Corollary 2.3 there exists ω ∈ Ω ( a sp ) ( R as ) such that Q fixes ω . Since the stabilizer of ω in S asp is isomorphic to S a ≀ S sp , we deduce that Q isisomorphic to a subgroup of a Sylow p -subgroup of S a ≀ S sp . In particular this impliesthat | Q | ≤ | Q s | and therefore we obtain that Q s is a vertex of H ( a sp ) ( R as ). Corollary 3.8. The Foulkes module H ( a sp ) has a unique indecomposable summand U with vertex Q s ∈ Syl p ( S a ≀ S sp ) . In particular U is the Scott module Sc( S asp , S a ≀ S sp ) and we have that H ( a sp ) ( Q s ) = U ( Q s ) ∼ = P F as F ( N S asp ( Q s ) /Q s ) -modules. Moreover, any other indecomposable summand of H ( a sp ) has vertex conjugate to a subgroup of Q s .Proof. Since H ( a sp ) is isomorphic to the permutation module induced by the action of S asp on the cosets of S a ≀ S sp , it is clear that any vertex of an indecomposable summandis contained in a Sylow p -subgroup of S a ≀ S sp and therefore is conjugate to a subgroupof Q s . From Proposition 3.7 we have that H ( a sp ) ( R as ) is indecomposable. Thereforeby Proposition 2.10 we deduce that exists a unique indecomposable summand U of H ( a sp ) such that R as is contained in a vertex of U . We know that R as ≤ Q s andcertainly the Scott module Sc( S asp , S a ≀ S sp ) = Sc( S asp , Q s ) is an indecomposablesummand of H ( a sp ) , with vertex Q s . Therefore U = Sc( S asp , Q s ) and by Theorem2.12 module we have that H ( a sp ) ( Q s ) = U ( Q s ) ∼ = P F , as F ( N S asp ( Q s ) /Q s )-modules.In order to prove Theorem 1.1 we need the following technical lemma, which isthe analogous of [8, Lemma 4.6]. Denote by D s the group C ∩ N S asp ( Q s ). Lemma 3.9. Let p be a prime and let a and s be natural numbers such that a < p . Then the unique Sylow p -subgroup of N S asp ( Q s ) is the subgroup h D s , Q s i of N S asp ( R as ) .Proof. Keeping the notation introduced after Proposition 3.4, for j ∈ { , , . . . , as } let O j = { ( j − p + 1 , . . . , jp } and for k ∈ { , . . . , s } let X k = a − [ l =0 O ls + k . Since Q s normalizes E s , it permutes the sets X , . . . , X s as blocks for its action.Moreover given x ∈ N S asp ( Q s ) we have that π xj ∈ h π , . . . , π s i for all j ∈ { , . . . , s } .Therefore also N S asp ( Q s ) permutes as blocks for its action the sets X , . . . , X s .Let g be a p -element of N S asp ( Q s ). The group h Q s , g i permutes the sets in X := { X , . . . , X s } as blocks for its action. Let π : h Q s , g i → S X 14e the corresponding group homomorphism. By construction Q s acts on the sets X , . . . , X s as a Sylow p -subgroup of S { X ,...,X s } ; hence Q s π is a Sylow p -subgroupof S X . Therefore, since h Q s , g i is a p -group, there exists ˜ g ∈ Q s such that gπ = ˜ gπ .Let y = g ˜ g − . Since y acts trivially on the sets in X , we may write y = g . . . g s where g j ∈ S X j for each j . The p -group h Q s , y i has as a subgroup h π j , y i . The per-mutation group induced by the subgroup on X j , namely h π j , g j i , is a p -group actingon a set of size ap . Since p > a , the unique Sylow p -subgroup of S X j containing π j is h z j , z j + s , . . . , z j +( a − s i . Hence g j ∈ h z j , z j + s , . . . , z j +( a − s i for each j ∈ { , . . . , s } .Therefore y ∈ h z , z . . . , z as i = C . We also know that y ∈ h Q s , g i ≤ N S asp ( Q s ).Therefore y ∈ D s , and since ˜ g ∈ Q s , it follows that g ∈ h D s , Q s i , as required. Con-versely, the subgroup h D s , Q s i is contained in N S asp ( Q s ) because both D s and Q s are. It follows that h D s , Q s i is the unique Sylow p -subgroup of N S asp ( Q s ).We are now ready to prove Theorem 1.1 Proof of Theorem 1.1. To simplify the notation we denote N S asp ( R as ) by K s . Let U be an indecomposable summand of H ( a n ) with vertex Q . Let ℓ ∈ { , . . . , ⌊ anp ⌋} ∩ N be maximal with respect to the property that R ℓ is a subgroup of (a conjugate of)the vertex Q . The Brou´e correspondent U ( R ℓ ) is a non-zero direct summand of H ( a n ) ( R ℓ ) by Theorem 2.2. Therefore we deduce by Lemma 3.1 that there exist anatural number s such that ℓ = as and Z a non-zero summand of H ( a n − sp ) such that U ( R as ) ∼ = H ( a sp ) ( R as ) ⊠ Z as F ( K s × S a ( n − sp ) )-modules. Since R as is normal in Q s , it follows from Lemmas 2.8and 2.9 that there is an isomorphism of F ( N K s ( Q s ) × S a ( n − sp ) )-modules U ( Q s ) ∼ = ( U ( R as ))( Q s ) ∼ = ( H ( a sp ) ( R as ))( Q s ) ⊠ Z. By Proposition 3.7 we deduce that H ( a sp ) ( Q s ) ⊠ Z = 0. Hence we have that Q s ≤ Q .Let B be a p -permutation basis for the F K s -module H ( a sp ) ( R as ) with respect to aSylow p -subgroup of K s containing Q s . It follows from Corollary 2.3 and Lemma 3.9that C = B Q s is a p -permutation basis for the F N K s ( Q s )-module ( H ( a sp ) ( R as ))( Q s )with respect to the unique Sylow p -subgroup P := h D s , Q s i of N K s ( Q s ). Let C ′ be a p -permutation basis for Z with respect to P ′ , a Sylow p -subgroup of S { asp +1 ,...,an } ∼ = S a ( n − sp ) . Hence C ⊠ C ′ = { v ⊗ v ′ : v ∈ C , v ′ ∈ C ′ } is a p -permutation basis for ( H ( a sp ) ( R as ))( Q s ) ⊠ Z with respect to the Sylow p -subgroup P × P ′ of N K s ( Q s ) × S a ( n − sp ) .Suppose, for a contradiction, that Q strictly contains Q s . Since Q is a p -groupthere exists a p -element g ∈ N Q ( Q s ) such that g Q s . Notice that Q s has orbits oflength at least p on { , . . . , asp } and fixes { asp + 1 , . . . , an } . Since g permutes theseorbits as blocks for its action, we may factorize g as g = hh + where h ∈ N S asp ( Q s )and h + ∈ S a ( n − sp ) . By Lemma 3.9 we have that h Q s , h i ≤ N K ( Q s ).15orollary 2.3 now implies that ( C ⊠ C ′ ) h Q s ,g i = ∅ . Let v ⊗ v ′ ∈ C ⊠ C ′ be suchthat ( v ⊗ v ′ ) g = v ⊗ v ′ . Then v ∈ B h Q s ,h i . But Q s is a vertex of H ( a sp ) ( R as ), so itfollows from Corollary 2.3 that h ∈ Q s . Hence h ′ is a non-identity element of Q .By taking an appropriate power of h ′ we find that Q contains a product of one ormore p -cycles with support contained in { asp + 1 , . . . , an } . This contradicts ourassumption that l = as was maximal such that R as is contained in a vertex of U .Therefore U has vertex Q s .We saw above that there is an isomorphism U ( Q s ) ∼ = ( H ( a sp ) ( R as ))( Q s ) ⊠ Z of F ( N K s ( Q s ) × S a ( n − sp ) )-modules. This identifies U ( Q s ) as a vector space on which N S an ( Q s ) = N S asp ( Q s ) × S a ( n − sp ) acts. It is clear from the isomorphism in Lemma3.1 that N S asp ( Q s ) acts on the first tensor factor and S a ( n − sp ) acts on the second.Hence the action of N K s ( Q s ) on ( H ( a sp ) ( R as ))( Q s ) extends to an action of N S asp ( Q s )on ( H ( a sp ) ( R as ))( Q s ) and we obtain a tensor factorization V ⊠ Z of U ( Q s ) as a N S asp ( Q s ) × S a ( n − sp ) -module. An outer tensor product of modules is projective ifand only if both factors are projective, so by Theorem 2.4 and Corollary 3.8, V is isomorphic to the projective cover of the trivial F ( N S asp ( Q s ) /Q s )-module, Z is aprojective F S a ( n − sp ) -module, and U ( Q s ) is the Green correspondent of U .Notice that Theorem 1.1 implies that the only indecomposable summands of theFoulkes module H ( a n ) that are Young modules are the projective ones, because nonon-projective Young module can possibly have vertex of the form Q s ∈ Syl p ( S a ≀ S sp ).Another consequence of Theorem 1.1 is that the modular version of Foulkes’Conjecture is false. In fact if we consider a < p ≤ n then H ( n a ) cannot be a directsummand of H ( a n ) since there exists a non-projective Young module Y in the directsum decomposition of H ( n a ) , namely the Scott module Sc( S an , S n ≀ S a ). As explainedabove such Y cannot appear as a direct summand of H ( a n ) .It is important to underline that for a given natural number a < p , Theorem1.1 tells us that an indecomposable summand U of H ( a n ) has vertex Q s for some s ∈ { , . . . , ⌊ np ⌋} ∩ N but it does not guarantee that for every s ∈ { , . . . , ⌊ np ⌋} ∩ N there exists an indecomposable summand with vertex Q s . We believe that suchsituation occurs. It will be material of another article to prove this property for themodules H (2 n ) for any odd prime p and any n ∈ N . In this section we will give upper bounds to the entries of some columns of thedecomposition matrix of F p S an when a < p . In particular we will prove Theorem1.2.For the rest of the section let F p be the finite field of size p and let a, w benatural numbers such that w < a < p . Let B := B ( γ, w ) be a block of the groupalgebra F p S an such that F ( γ ) = ∅ , where F ( γ ) is the set defined in the introduction,before the statement of Theorem 1.2. Denote by D a defect group of B . For every p -regular partition ν of an , we will denote by P ν the projective cover of the simple F p S an -module D ν . 16 roposition 4.1. The block component of H ( a n ) for the block B is projective.Proof. Let U be an indecomposable summand of H ( a n ) lying in the block B . Supposethat U is non-projective. Then by Theorem 1.1 there exists t ∈ N non-zero suchthat Q t ∈ Syl p ( S a ≀ S tp ) is a vertex of U . By [1, Theorem 5, page 97] we deduce thata conjugate of Q t is contained in D . Moreover, by Theorem 2.14 we have that D isconjugate to a Sylow p -subgroup of S wp . This leads to a contradiction, because | supp( Q t ) | = atp ≥ ap > wp = | supp( D ) | . In order to prove Theorem 1.2, we will need Scott’s lifting theorem [2, Theorem3.11.3]. For the reader convenience we state it below. We will denote by Z p the ringof p -adic integers. Theorem 4.2. If U is a direct summand of a permutation F p G -module M then thereis a Z p G -module U Z p , unique up to isomorphism, such that U Z p is a direct summandof M Z p and U Z p ⊗ Z p F p ∼ = U . Let P ν Z p be the Z p -free Z p S an -module whose reduction modulo p is P ν . By Brauerreciprocity (see for instance [17, § P ν Z p is ψ ν = X µ d µν χ µ . ( ⋆ )It is well known that if d µν = 0 then ν dominates µ . Hence the sum may be takenover those partitions µ dominated by ν .We are now ready to prove Theorem 1.2. Proof of Theorem 1.2. Since F ( γ ) = ∅ the block component W of H ( a n ) for B isnon zero. Let ζ , . . . , ζ s be the p -regular partitions of an such that W = P ζ ⊕ P ζ ⊕ · · · ⊕ P ζ s . From the definition of F ( γ ) and by Theorem 4.2, it follows that the ordinary char-acter of W is ψ ζ + · · · + ψ ζ s = X µ ∈F ( γ ) (cid:0) s X i =1 d µζ i (cid:1) χ µ . By hypothesis λ is a maximal partition in the dominance order on F ( γ ), and by ( ⋆ )each ψ ζ j is a sum of ordinary irreducible characters χ µ for partitions µ dominatedby ζ j . Therefore one of the partitions ζ j must equal λ , as required.Therefore P λ is a direct summand of H ( a n ) and ψ λ is a summand of the Foulkescharacter φ ( a n ) . Hence d µλ = (cid:10) ψ λ , χ µ (cid:11) ≤ (cid:10) φ ( a n ) , χ µ (cid:11) , for all µ ⊢ an . In particular if µ / ∈ F ( γ ) then d µλ = 0.As already mentioned in the introduction, Theorem 1.2 allows us to recover newinformation on the decomposition numbers via the study of the ordinary Foulkescharacter φ ( a n ) . An example of this possibility is the following result.17 orollary 4.3. Let λ be a p -regular partition of na . Denote by γ the p -core of λ . If λ is maximal in F ( γ ) , then [ S µ : D λ ] = 0 for all µ ⊢ na such that µ has more than n parts.Proof. It is a well known fact (see for instance [7, Proposition 2.7]) that if µ hasmore than n parts then (cid:10) φ ( a n ) , χ µ (cid:11) = 0 . The statement now follows from Theorem 1.2.We conclude with an explicit example. Example 4.4. Let a = 4 , n = p = 5 and let λ = (18 , be a weight- partitionof . The -core of λ is γ = (3 , and the multiplicity of χ λ as an irreducibleconstituent of φ (4 ) is , by [12, Theorem 5.4.34]. Therefore λ ∈ F ( γ ) and it isclearly maximal under the dominance order. By Corollary 4.3 we obtain a numberof non-trivial zeros in the column labelled by λ of the decomposition matrix of S incharacteristic . For instance, all the partitions µ obtained from (3 , , ) by addingtwo -hooks lie in F ( γ ) and are such that [ S µ : D λ ] = 0 . Acknowledgments I would like to thank my PhD supervisor Dr. Mark Wildon for his helpful advicewhich guided me throughout this work. References [1] Alperin, J. L. Local representation theory , Cambridge studies in advancedmathematics, vol. 11, CUP, 1986.[2] Benson, D. J. Representations and cohomology I , Cambridge studies inadvanced mathematics, vol. 30, CUP, 1995.[3] Brauer, R. On a conjecture by Nakayama , Trans. Roy. Soc. Canada. Sect.III. (3) (1947), 11–19.[4] Brou´e, M. On Scott modules and p -permutation modules: an approach throughthe Brauer homomorphism. Proc. Amer. Math. Soc. 93, no. 3 (1985), 401–408.[5] Erdmann, K. Young modules for symmetric groups . J. Aust. Math. Soc. 71,number 2 (2001), 201–210.[6] Foulkes, H. O. Concomitants of the quintic and sextic up to degree four inthe coefficients of the ground form , J. London Math. Soc. (1950), 205–209.[7] Giannelli, E. On the decomposition of the Foulkes module , Arch. Math.(Basel) (2013), 201–214. 188] Giannelli, E. and Wildon, M. Foulkes modules and decomposition numbersof the symmetric group , arXiv:1310.2661 [math.RT][9] Grabmeier, J. Unzerlegbare Moduln mit trivialer Younquelle und Darstel-lungstheorie der Schuralgebra , Bayreuth. Math. Schr. (1985), 9–152.[10] James, G. D. The representation theory of the symmetric groups , vol. 682 ofLecture Notes in Mathematics. Springer-Verlag, 1978.[11] James, G. D. Trivial source modules for symmetric groups . Arch. Math. 41,(1983), 294–300.[12] James, G. D., and Kerber, A. The representation theory of the symmetricgroup . Addison-Wesley, 1981.[13] G. D. James, The irreducible representations of the symmetric groups , Bull.London Math. Soc. (1976), 229–232.[14] Klyachko, A.A. Direct summands of permutation modules. Selecta Math.Soviet. (I) 3, (1983–1984), 45–55.[15] Paget, R. and Wildon, M. Set families and Foulkes modules , J. AlgebraicCombin. (2011),[16] Robinson, G. de B. On a conjecture by Nakayama , Trans. Roy. Soc. Canada.Sect. III. (1947), 20–25.[17] J. P. Serre,