aa r X i v : . [ m a t h . R T ] N ov THE STRUCTURE OF YOUNG MODULES ANDSCHUR ALGEBRAS IN CHARACTERISTIC 2
MORIAH ELKIN AND PETER WEBB
Abstract.
We compute explicitly the submodule structure of theYoung modules for symmetric groups S n over fields of character-istic 2, when n ≤
7. We use this information to compute thesubmodule structure of indecomposable projectives for the corre-sponding Schur algebras when n ≤
5, and we give give partialinformation when n = 6 ,
7, including the Gabriel quiver and thestructure of the Weyl modules. We resolve all Morita equivalencesbetween blocks of these algebras. Introduction
The Young modules Y λ play a fundamental role in the representa-tion theory of the symmetric groups S n . They are the indecompos-able summands of the Young permutation modules M λ , which are thepermutation modules on the cosets of the Young subgroups S λ . Theisomorphism types of the Y λ , as well as the M λ and S λ , are indexed bypartitions λ of n . Over fields of characteristic 0 the Young modules co-incide with the Specht modules, and form a complete set of irreduciblemodules. The situation is different over fields of positive characteristic p . Now the Young modules are no longer irreducible, in general, andthey provide a transition between the irreducible modules, the Spechtmodules, and the larger projective indecomposable modules. Beingtypically more tractable than the Specht modules or the irreduciblemodules, they are useful in constructing larger representations. Theyalso play a key role in the study of the Schur algebras, via the Schur-Weyl correspondence.Explicit computations of the structure of Young modules are onlyknown for symmetric groups of small degree, or for partitions with few Date : November 2020.2020
Mathematics Subject Classification.
Primary: 20C30; Secondary: 20G43,20-08.
Key words and phrases.
Young module, symmetric group, Schur algebra.The first author was supported by a University of Minnesota UndergraduateResearch Scholarship. parts, or of a special kind. In this work we extend such computations toall Young modules as far as the symmetric group S in characteristic 2.We also describe explicitly the structure of the Schur algebras S K ( n, n )with n ≤ K of characteristic 2, giving partial information when n = 6 ,
7. Our descriptions are by means of diagrams. Included in ourdescription of the Young modules are the filtrations by Specht modulesthat correspond to Weyl filtrations of the projective modules for theSchur algebra. Our methods depend heavily on computer calculation.Some of our descriptions of Young modules Y λ already appear inthe literature, and we include the known cases here for completeness,so as to give a comprehensive picture. Descriptions of Young modulesthat are already known can be found as follows. When λ = [ n ] and λ = [ n − ,
1] the Young modules are well-known and their structurefollows directly from [9]. When λ = [ n − ,
2] and λ = [ n − , ] they aredescribed in [4] and [16], for instance. In [13] the structure is given when λ = [ n − , λ is p -regular the Young module Y λ is projective,and when p = 2 the projective modules are known for S n when n ≤ n = 6 and are shown in [2], but a full description of the projectiveswhen n = 6, in terms of diagrams showing non-split sections, was notgiven there. We note that several of the references just mentionedprovide good examples of how precise descriptions of Young modulescan be used to resolve other questions in representation theory.From our calculations, we can make some observations about thestructure of the blocks that appear. We recall that each block of repre-sentations of a symmetric group is assigned a weight, that is the numberof rim p -hooks that must be removed from any partition parametrizinga representation such as Y λ so as to leave a p -core. The representationswe consider all lie in blocks of weight 0, 1, 2 or 3. The blocks of weight0 are simple projective modules (blocks of defect 0) and the blocks ofweight 1 were shown to be Morita equivalent by Scopes [19]. From ourcalculations with the remaining blocks of weights 2 and 3 we obtainthe following. Corollary 1.1.
The principal block of F S and the non-principal blockof F S of weight 2 are Morita equivalent, as are the correspondingblocks of the Schur algebras S F (5 , and S F (7 , . By contrast, F S (which consists of a single block, of weight 2) is notMorita equivalent to the principal block of F S or the non-principalblock of F S . At the same time, the Schur algebra S F (4 ,
4) is notMorita equivalent to the corresponding blocks of the Schur algebras S F (5 ,
5) and S F (7 , F S and OUNG MODULES AND SCHUR ALGEBRAS 3 F S (of weight 3), as well as the corresponding blocks of S F (6 , S F (7 ,
7) are not Morita equivalent. These inequivalences can beseen because the Cartan matrices are different, and this information isreadily available in the literature.
Proof.
The proofs depend on calculations done in later sections. The di-agrams for the projective modules in the blocks mentioned of F S and F S have the same structure, with different simple modules appear-ing. This shows that these group algebra blocks are Morita equivalent.The fact that the diagrams for all the Young modules in these blockshave the same structure shows that the blocks of the Schur algebrasare Morita equivalent. (cid:3) For standard notation and facts about group representations, suchas terminology for radical and socle layers etc, we refer to [20]. Forbackground on the representation theory of the symmetric groups andSchur algebras, see [9], [12], [18], [7] and [14]. For an overview of thistheory, the exposition [5] is very useful.2.
Young modules and Schur algebras
In this section we summarize the part of the theory of representa-tions of the symmetric groups necessary for our calculations, therebyestablishing our notation.For each partition λ = ( λ , λ , . . . , λ t ) of n the Young subgroup S λ of S n is defined as S λ = S λ × S λ × · · · × S λ t . and the corresponding permutation RS n -module on the left cosets of S λ is M λ = R [ S n /S λ ] = RS n ⊗ RS λ R, which we refer to as a Young permutation module . It has the
Spechtmodule S λ as a submodule.When R is a field of characteristic zero, these Specht modules areirreducible, and they provide a complete set of non-isomorphic irre-ducible representations of S n . When R is a field of characteristic p > λ is p -regular , meaning that λ has at most p − S λ has a unique simple quotient D λ , and these D λ form a complete list of isomorphism types of irreducible modules [9].The indecomposable direct summands of the Young permuatationmodules M λ were considered by James in [11], and a little later Grab-meier [6] and Green [8] referred to them as Young modules . Jamesshowed (at least over an infinite field, but this restriction was removedby Green [8]) that
MORIAH ELKIN AND PETER WEBB (1) the Young modules Y λ are parametrized up to isomorphism bythe partitions λ of n , in such a way that(2) the summands of M λ all have the form Y µ for some µ ≥ λ inthe dominance ordering, and that(3) Y λ occurs as a direct summand of M λ with multiplicity 1.These conditions imply(4) The Y λ are self-dual.We use these properties in constructing the Y λ and in analyzing theirstructure. Unlike the Specht modules, we are not able to identify Y λ as a specific subset of M λ , and so in the non-projective cases we con-struct Y λ by decomposing M λ into direct summands and identifyingthe unique new summand that has not appeared in earlier M µ . Theself-duality is a strong constraint that is very useful computationally.The Young modules have a close connection with the Schur algebras ,for which we refer to [7] and [14] as background. We are interested inthe submodule structure of the indecomposable projective modules ofSchur algebras, and for this it is sufficient to work with basic algebras,Morita equivalent to the Schur algebras. By abuse of a usual notationfor Schur algebras, we define S R ( n, r ) = End S r ( L λ Y λ ) when n ≥ r .For each r , the algebras defined in this way are all the same provided n ≥ r , and their common value is S R ( r, r ). Conventionally, r is thesubscript of the symmetric group in this context, but because we wantto use n as this symbol we prefer to write the algebra associated to S n as S R ( n, n ). This algebra is Morita equivalent to the usual Schuralgebra with these parameters.Our definition of Schur algebras is unusual in another respect, inthat we allow R to be any field (or complete local ring). Often theSchur algebras are only defined over infinite fields. When R is a primefield, it is already a splitting field for S R ( n, n ), by work of Green [8],so that in determining the submodule structure of the projectives wemight as well work over a prime field.It was shown by Donkin that the Young modules have Specht filtra-tions, that is, chains of submodules whose factors are Specht modules(see, for instance, [14]). Such filtrations may be obtained as the imagesunder the Schur functor of Weyl filtrations of the indecomposable pro-jective modules for S R ( n, n ). We indicate the Specht filtrations thatarise in this way in our results on the Young modules.3. Diagrammatic descriptions of modules
The diagrams we use to describe the structure of modules have along history, and within the context of representations of algebras go
OUNG MODULES AND SCHUR ALGEBRAS 5 back at least to the 1960s, if not before. We will use diagrams inthe sense of Benson and Carlson [3] and review what it means for adiagram to describe a module. An earlier approach to this kind oftheory, applicable in slightly more restrictive circumstances, was givenby Alperin [1]. A brief introduction to the kind of diagrams we shalluse can be found on pages 1556–7 of [15].Briefly, our module diagrams are directed graphs X without loopsor multiple edges, together with a labeling of the vertices by a fixed setof irreducible modules. In drawing these diagrams, the edges alwayspoint down. Each edge is labeled by a non-zero Ext class so that if theedge x α −→ y has source x labeled by the irreducible S and the target y is labeled by the irreducible T , then α ∈ Ext ( S, T ).To say what it means for such a diagram X to describe a certainmodule M , we consider subsets of the set of vertices of X that are closedunder following along directed edges. For each such ‘arrow closed’subset U of vertices there is a submodule M U associated to it, withthe property if U ⊇ U then M U ⊇ M U , and so that the compositionfactors of M U /M U are the labels of the vertices in U that are not in U . We require that M X = M and M ∅ = 0, from which it follows thatthe labels of the vertices of X are exactly the composition factors of M , taken with multiplicity and up to isomorphism. If U ⊇ U differby two vertices joined by an edge x α −→ y with x labeled by S and y labeled by T then the extension class of the short exact sequence0 → M U ∪{ T } /M U → M U /M U → M U /M U ∪{ T } → α . If there is no edge between x and y , the above extension must split.In our diagrams we do not label edges when dim Ext( S, T ) = 1, becauseover F there is only one non-zero extension. Thus the existence of anedge simply means a non-split extension.The only case of an Ext group of dimension larger than 1 that weshall encounter is with the group S , where dim Ext( D [5 , , D [5 , ) = 2,and in this case we give the edges labels. We point out that for theYoung module Y [3 , ] for S , the situation arises where three copies of D [5 , appear underneath a single copy of D [5 , , each edge labeled by adifferent non-zero element of the Ext group. It seems not to be possiblein this case to have a diagram where the labels on edges emanating froma single vertex are linearly independent.Part of the information conveyed by a diagram for a module M isa description of its radical and socle layers, Rad i M/ Rad i + r M andSoc i + r M/ Soc i M . To obtain this description, note that the radical of M has a diagram obtained by removing all vertices that are not targetsof arrows, and the socle of M has as a diagram the vertices that are not MORIAH ELKIN AND PETER WEBB sources of arrows (a discrete set of points). The powers of the radicaland socle are obtained by iterating these combinatorial procedures.As a special case of this, a module is semisimple if and only if it has adiagram with no edges. More generally, a module with a diagram thatis a disjoint union of subdiagrams is the direct sum of the submoduleswith those diagrams.At the other extreme, a diagram where the vertices are arranged ina single vertical line of edges represents a uniserial module , namely,one with a unique composition series or, equivalently, where the sub-modules are linearly ordered by inclusion. This is so because a moduleis uniserial if and only if all its radical layers Rad i M/ Rad i +1 M aresimple, and this happens if and only if the diagram has a single lineof edges. When drawing uniserial diagrams we often omit the directedarrows between the composition factors where this does not cause con-fusion. 4. Computational methods
Our methods are heavily computational, using routines developedby the authors and others in GAP, and available from from Webb’s in-ternet site [21]. At a basic level these routines provide functionality tocompute with submodules and quotient modules of given representa-tions. Further routines allow more elaborate operations. We highlightsome of the key approaches. Specific further detail is given in notes foreach module we consider.Our initial step is to construct the Young modules. These are sum-mands of permutation modules and, when they are small enough, canbe constructed by decomposing the permutation module into directsummands. The basic algorithm used to do this depends on Fitting’slemma, and finds a random endomorphism of the module that is notnilpotent and not an isomorphism. Raising this endomorphism to ahigh enough power, its kernel and image provide a direct sum decom-position of the module. This general approach is demanding on com-putational resources, but succeeds with modules whose dimensions arein the low hundreds.With larger modules we use a different approach to direct sum de-composition that applies when the isomorphism types of some sum-mands are already known. It works with the permutation modules M λ when the λ are taken in dominance order, because each M λ hasa unique new direct summand Y λ and all the other direct summandshave already been constructed. The largest module M λ that we de-composed in this way has dimension 840. This approach may be new OUNG MODULES AND SCHUR ALGEBRAS 7 in the computational context, although the idea is very simple. Given amodule and a candidate summand, random homomorphisms betweenthese modules in both directions are tested until they happen to befound so that the composite from the summand to the bigger module,back to the summand is an isomorphism. These maps are then splitmono and split epi, and express the candidate summand as a directsummand of the larger module. As a special case, this approach alsoprovides a way to test for isomorphism of two modules, and find anisomorphism when they are isomorphic.Many of the larger projective Young modules are constructed assummands of tensor products of suitably chosen smaller modules. Wecan test whether a summand of such a tensor product is projectiveby a routine that restricts to a Sylow p -subgroup and checks the rankof the sum of the group elements. If the module then has a uniquesimple quotient, it is the indecomposable projective associated to thatmodule.Once the Young modules have been constructed, a first approachto analyzing their structure is to compute the factors that appear inthe Zassenhaus Butterfly Lemma, associated to two chains of submod-ules. We use the radical and socle series as input, as computed by theMeat-axe [17]. This gives a list of composition factors of the module,positioned according to the radical and socle layers in which they ap-pear. The approach immediately identifies uniserial modules, and is astart for more complicated modules.After this, a very useful routine allows us to remove from the bottomof a module all homomorphic images of another module, meaning thatwe quotient out the sum of the images of all homomorphisms from onemodule to another. A dual routine computes the intersection of thekernels of all homomorphisms from one module to another, allowing usto remove composition factors from the top of a module in a controlledway. These routines allow us to remove parts of a module with someprecision so that we can examine what remains. By judiciously remov-ing submodules and factor modules in this way, we can test for splitextensions by direct sum decomposition and by examining radical andsocle layers.Certain kinds of modules appear repeatedly in our investigation, andare easily identified. The structure of uniserial modules is identified asalready noted. It is often useful to work with the heart of a module,which may be defined to be the quotient of its radical by its socle (insituations where the socle is contained in the radical). This is calculatedby removing the socle and the top as just noted, and the structure ofmodules whose hearts are the direct sum of two uniserial modules can MORIAH ELKIN AND PETER WEBB then be immediately verified. Such modules are called biserial . Wewill use the term string module to indicate a module with a diagramthat is combinatorially a subdivision of a single line, folded so that theedges may go up or down. Such modules are identified by removingcomposition factors from the socle and the top in a judicious way andthen showing that splitting occurs. Details are given of the preciseprocedure when this occurs.Other computational strategies proceeded on an ad hoc basis, andare detailed in notes after each diagram.5.
The module structure of Young modules incharacteristic 2
At the start of the tables for each symmetric group we give thetable of multiplicities of the Young modules as summands of the Youngpermutation modules. These multiplicities are known as the p -Kostkanumbers (here p = 2). They are already known, because they maybe computed from the decomposition numbers for the Schur algebrasgiven in [10] and [6]. We present the table for the convenience of thereader, because the multiplicities are important in our construction ofthe Young modules, and we do not know of a reference where they arestated explicitly.The factors in a Specht module filtration of the Young modules areindicated either by enclosing the factors for a single Specht module ina box, or by joining those factors by thick lines. We have chosen to usemore than one way to indicate the Specht factors because of the diffi-culty of incorporating this information in diagrams that may alreadybe quite complicated. Using thick lines is often more straightforward,but it is not always possible. When a Specht module is simple, thereare no lines to thicken, and in these cases we enclose the Specht modulein a box. The Specht modules S [5 , ] and S [3 , ] decompose as the directsum of two simple modules, and so we enclose those modules in a box,again because there is no edge to thicken. In each situation we havechosen thick lines or boxes to produce the diagram that conveys theinformation most clearly, in our view.The Specht module factors in our Specht filtrations are computed byapplying the Schur functor to the Weyl modules of the Schur algebra,which gives their composition factors and also the order in which theymust appear, using the quasi-hereditary property of the Schur algebra.Under this construction, the Schur module S λ appears as the bottomfactor in the Specht filtration of Y λ , so that our diagrams include thestructure of the Specht modules. OUNG MODULES AND SCHUR ALGEBRAS 9
Young modules for S . Young module summand multiplicities:Table of Permutation module2-Kostka numbers [1]Young module [1] 1Young module structure: λ [1] Y λ D [1] Young modules for S . Young module summand multiplicities:Table of Permutation module2-Kostka numbers [2] [1 ]Youngmodule [2] 1 0[1 ] 0 1Young module structure: λ [2] [1 ] Y λ D [2] D [2] D [2] Young modules for S . Young module summand multiplicities:Table of Permutation module2-Kostka numbers [3] [2 ,
1] [1 ]Youngmodule [3] 1 1 0[2 ,
1] 0 1 2[1 ] 0 0 1Young module structure: λ [3] [2 ,
1] [1 ] Y λ D [3] D [2 , D [3] D [3] Young modules for S . Young module summand multiplicities:
Table of Permutation module2-Kostka numbers [4] [3 ,
1] [2 ] [2 , ] [1 ]Youngmodule [4] 1 0 0 0 0[3 ,
1] 0 1 0 1 0[2 ] 0 0 1 0 0[2 , ] 0 0 0 1 2[1 ] 0 0 0 0 1Young module structure: λ [4] [3 ,
1] [2 ] [2 , ] [1 ] Y λ D [4] D [4] D [3 , D [4] D [4] D [3 , D [3 , D [4] D [3 , D [4] D [3 , D [4] D [3 , D [4] D [4] D [3 , D [3 , D [4] D [4] ⊕ Notes on Y [2 ] . We compute that the socle has the form D [3 , ⊕ D [4] . Quotienting out the socle D [4] leaves an indecomposable module,while quotienting out instead D [3 , leaves two uniserial summands. Thestructure of these summands implies the diagram shown.5.5. Young modules for S . Young module summand multiplicities:Table of Permutation module2-Kostka numbers [5] [4 ,
1] [3 ,
2] [3 , ] [2 ,
1] [2 , ] [1 ]Youngmodule [5] 1 1 0 0 0 0 0[4 ,
1] 0 1 1 2 2 2 0[3 ,
2] 0 0 1 0 1 0 0[3 , ] 0 0 0 1 0 1 0[2 ,
1] 0 0 0 0 1 2 4[2 , ] 0 0 0 0 0 1 4[1 ] 0 0 0 0 0 0 1Young module structure: OUNG MODULES AND SCHUR ALGEBRAS 11 λ [5] [4 ,
1] [3 ,
2] [3 , ] Y λ D [5] D [4 , D [5] D [3 , D [5] D [5] D [3 , D [5] D [5] D [3 , D [5] λ [2 ,
1] [2 , ] [1 ] Y λ D [3 , D [5] D [5] D [3 , D [5] D [5] D [3 , D [4 , D [4 , D [5] D [5] ⊕ D [5] D [3 , D [5] D [5] D [3 , D [3 , D [5] D [5] D [3 , D [5] Young modules for S . Young module summand multiplicities:
Table of Permutation module2-Kostka numbers [6] [5 ,
1] [4 ,
2] [4 , ] [3 ] [3 , ,
1] [3 , ] [2 ] [2 , ] [2 , ] [1 ]Youngmodule [6] 1 0 1 0 0 0 0 0 0 0 0[5 ,
1] 0 1 0 1 0 0 0 0 0 0 0[4 ,
2] 0 0 1 0 0 0 0 2 0 0 0[4 , ] 0 0 0 1 0 1 2 0 2 2 0[3 ] 0 0 0 0 1 1 0 0 1 0 0[3 , ,
1] 0 0 0 0 0 1 2 2 4 8 16[3 , ] 0 0 0 0 0 0 1 0 0 1 0[2 ] 0 0 0 0 0 0 0 1 0 0 0[2 , ] 0 0 0 0 0 0 0 0 1 2 4[2 , ] 0 0 0 0 0 0 0 0 0 1 4[1 ] 0 0 0 0 0 0 0 0 0 0 1 Young module structure: λ [6] [5 ,
1] [4 ,
2] [4 , ] [3 ] Y λ D [6] D [6] D [5 , D [6] D [5 , D [6] D [4 , D [6] D [5 , D [5 , D [6] D [4 , D [6] D [5 , D [6] D [4 , D [6] D [5 , D [6] D [5 , D [6] D [4 , D [6] D [5 , D [6] D [4 , Notes on Y [3 ] . The socle of the heart of Y [3 ] is D [5 , ⊕ D [4 , .Quotienting out D [4 , from the heart leaves two uniserial summandswith composition factors shown, while the result of quotienting outsuccessively D [5 , and D [6] remains indecomposable. Now, the factthat the heart is self dual implies that the heart is a string module inthe manner shown. λ [3 , ,
1] [3 , ] [2 ] Y λ D [3 , , D [6] D [5 , D [6] D [4 , D [6] D [5 , D [6] D [4 , D [6] D [4 , D [6] D [5 , D [6] D [4 , D [6] D [5 , D [4 , D [6] D [5 , D [6] D [4 , D [6] D [5 , D [6] D [4 , D [4 , D [6] D [6] Notes on Y [3 , ] . We first verify that each section of the formRad i Y [3 , ] / Rad i +3 Y [3 , ] for 1 ≤ i ≤ i = 3, quotienting out D [4 , and then D [6] yields an indecomposable module, while quotientingout instead D [5 , causes the module to decompose as the direct sum of OUNG MODULES AND SCHUR ALGEBRAS 13 two uniserial modules. The cases for i = 1 , i = 2 , D [6] modules correctly: where i = 2 forexample, we may work with the larger module Rad Y [3 , ] / Rad Y [3 , ] ,and remove D [5 , and then D [6] .Additionally, there is a homomorphism Y [4 , ] → Y [3 , ] whose imageis uniserial of length 8, and whose cokernel is also uniserial of length 8,with composition factors as shown. Therefore the only edges betweenthe two uniserial strands of the heart go from left to right as shown. Notes on Y [2 ] . The module remains indecomposable after remov-ing the top and bottom copies of D [6] on the right, so some edges mustconnect to the D [4 , on the right. Additionally, the quotient of theradical of Y [2 ] by its socle decomposes into two uniserial summands.This implies the existence of the edges between copies of D [4 , .Now, removing only the top and bottom copies of D [4 , from Y [2 ] leaves an indecomposable module, so some edges must connect to thecopies of D [6] on the right. However, if we further quotient out bothcopies of D [6] from the socle of this module, the new module doesdecompose. Therefore the edges between copies of D [6] must be asshown. λ [2 , ] [2 , ] Y λ D [4 , D [4 , D [6] D [5 , D [6] D [4 , D [6] D [5 , D [6] D [4 , D [4 , D [6] D [5 , D [6] D [4 , D [6] D [5 , D [6] D [5 , D [5 , D [6] D [4 , D [6] D [5 , D [6] D [4 , D [6] D [5 , D [5 , D [6] D [4 , D [6] D [5 , D [6] D [4 , D [6] Notes on Y [2 , ] . This module is the projective cover of D [4 , because [2 , ] and [4 ,
2] are conjugate partitions. There is a homo-morphism from Y [2 ] to Y [2 , ] having kernel the socle D [6] of Y [2 ] .The quotient of Y [2 , ] by the image of this homomorphism is unise-rial. This implies the right side of the diagram and the diagonal line,and that the only remaining edges are in the left uniserial quotient, andfrom that quotient down to the right. We now calculate with the sec-tions Rad i Y [2 , ] / Rad i +2 Y [2 , ] where 1 ≤ i ≤
7, and show that theyare all string modules by testing indecomposability of the quotients bythe simple modules in their socles.
Notes on Y [2 , ] . This module is constructed as a tensor product Y [4 , ] ⊗ U where U is a 2-dimensional uniserial module whose compo-sition factors are two copies of the trivial module D [6] . Such a module U may be constructed as a section of Y [3 , ] , or of Y [2 ] . The tensorproduct is verified to be projective and indecomposable with top D [5 , .We deduce that it is the projective cover of D [5 , . This is also a de-scription of Y [2 , ] because the partition [2 , ] is column 2-regular, andso the Young module is the projective cover of the simple indexed bythe conjugate partition (see [5]). Because the tensor product of theshort exact sequence 0 → D [6] → U → D [6] → Y [4 , ] is exact, itfollows that Y [2 , ] has a submodule isomorphic to the uniserial module Y [4 , ] , with quotient also isomorphic to this module. This establishesthe outer edges in the diagram, and that the only remaining edges gofrom the quotient to the submodule. Finally, examining 2-step lay-ers Rad i Y [2 , ] / Rad i +2 Y [2 , ] confirms that they are string modules asshown (see notes on Y [2 ] , Y [2 , ] , and Y [2 , ] for methods). OUNG MODULES AND SCHUR ALGEBRAS 15 λ [1 ] Y λ D [6] D [6] D [4 , D [5 , D [6] D [6] D [5 , D [4 , D [5 , D [4 , D [6] D [6] D [4 , D [5 , D [6] D [4 , D [5 , D [6] D [6] D [6] D [4 , D [5 , D [6] D [6] D [5 , D [4 , D [6] D [6] D [5 , D [4 , D [6] D [6] Notes on Y [1 ] . This module is the projective cover of D [6] (because[1 ] and [6] are conjugate partitions). It may be constructed as thetensor product Y [3 , ] ⊗ U where U is the 2-dimensional uniserial modulewith two copies of D [6] as composition factors, used in the constructonof Y [2 , ] . This tensor product is verified to be projective, it has a copyof D [6] in its top layer, and it has the correct dimension, known fromthe Cartan matrix.The quotient Rad Y [1 ] / Rad Y [1 ] decomposes as indicated by thediagram. We consider the 2-step layers of each of its two direct sum-mands, and conclude that those sections are string modules as shown(see notes on Y [2 ] , Y [2 , ] , and Y [2 , ] for methods). There can be nomore cross-diagonal edges within these summands because of the struc-tures of Y [2 , ] and Y [2 , ] , which are the projective covers of D [4 , and D [5 , . Finally, examination of the section Rad Y [1 ] / Rad Y [1 ] con-firms that it is a string as shown. The remaining edges in the diagramare established by duality.5.7. Young modules for S . For this group we find in the upcomingcalculations that there is a simple module admitting more than onenon-split self-extension: dim Ext F S ( D [5 , , D [5 , ) = 2. Because we areworking over F there are exactly three non-split extensions, and we are able to distinguish between them as follows. One non-split extensionoccurs in the middle of the Young module Y [5 , ] , and we denote thisby a dashed line. Another arises in the module D [5 , ⊗ U , where U is the 2-dimensional uniserial module with trivial composition factors,and we denote this with a dotted line. The third non-split extension,which is the sum of the other two, we denote by a dash-dotted line.Young module summand multiplicities: Table of Permutation module2-Kostka numbers [7] [6 ,
1] [5 ,
2] [5 , ] [4 ,
3] [4 , ,
1] [4 , ] [3 ,
1] [3 , ] [3 , , ] [3 , ] [2 ,
1] [2 , ] [2 , ] [1 ]Youngmodule [7] 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0[6 ,
1] 0 1 1 2 0 1 2 0 0 0 0 0 0 0 0[5 ,
2] 0 0 1 0 1 1 0 0 2 0 0 2 0 0 0[5 , ] 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0[4 ,
3] 0 0 0 0 1 1 0 2 2 2 0 3 2 0 0[4 , ,
1] 0 0 0 0 0 1 2 0 1 2 4 3 4 4 0[4 , ] 0 0 0 0 0 0 1 0 0 1 4 0 2 4 0[3 ,
1] 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0[3 , ] 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0[3 , , ] 0 0 0 0 0 0 0 0 0 1 2 2 5 10 20[3 , ] 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0[2 ,
1] 0 0 0 0 0 0 0 0 0 0 0 1 2 4 8[2 , ] 0 0 0 0 0 0 0 0 0 0 0 0 1 4 14[2 , ] 0 0 0 0 0 0 0 0 0 0 0 0 0 1 6[1 ] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 Young module structure: λ [7] [6 ,
1] [5 ,
2] [5 , ] [4 , Y λ D [7] D [6 , D [5 , D [7] D [5 , D [5 , D [7] D [6 , D [4 , D [6 , Notes on Y [5 , ] . Quotienting out the socle D [7] yields an indecom-posable module; instead quotienting out D [5 , also yields an indecom-posable module. Therefore the diagonal edges are as shown, and theradical and socle series require the diagram to be as shown. λ [4 , ,
1] [4 , ] [3 , Y λ D [5 , D [7] D [4 , , D [7] D [5 , D [5 , D [6 , D [4 , D [6 , D [6 , D [4 , D [6 , D [7] D [4 , , D [7] D [5 , D [5 , D [5 , D [5 , D [7] D [4 , , D [7] Notes on Y [4 , , . The module decomposes once the top and soclecopies of D [5 , are removed. OUNG MODULES AND SCHUR ALGEBRAS 17
Notes on Y [4 , ] . This module is the projective cover of D [6 , . It isa direct summand of the tensor product Y [5 , ] ⊗ D [6 , . Notes on Y [3 , . The module does not decompose when both so-cle copies of D [5 , are removed, but Y [3 , / Soc Y [3 , has the one re-maining middle copy of D [5 , as a direct summand. Therefore thatcopy of D [5 , must connect to the socle D [7] of Y [3 , . The edgesbetween copies of D [4 , , and D [7] , as well as all vertical edges inthe diagram, may be established by examining quotients of the formRad i Y [3 , / Rad i +2 Y [3 , for 1 ≤ i ≤
3, which are all strings as shownin the diagram.To demonstrate the absence of an edge between center copies of D [5 , ,as well as the existence of the cross-diagonal edges between copies of D [5 , , we remove copies of D [7] from the top and socle of Y [3 , . Theremaining module is indecomposable, but quotienting out the soclecopy of D [4 , , causes it to decompose as indicated by the diagram. λ [3 , ] [3 , , ] Y λ D [4 , , D [7] D [5 , D [7] D [4 , , D [7] D [4 , , D [7] D [4 , , D [7] D [5 , D [7] D [4 , , D [4 , , D [7] D [5 , D [7] D [4 , , Notes on Y [3 , ] . When D [4 , , , both copies of D [7] , and D [5 , aresequentially removed from the top of Y [3 , ] , the remaining moduleis a string as shown. Instead removing D [7] and then both copies of D [4 , , from the top of Y [3 , ] yields an indecomposable module; furtherremoving the top-left D [7] from this new module causes it to decompose,implying the existence of the edges between copies of D [7] . Notes on Y [3 , , ] . This module is the projective cover of D [4 , , . Toconfirm its structure, we first examine quotients of the form Rad i Y [3 , , ] / Rad i +2 Y [3 , , ] , which are all strings as shown in the diagram. Wethen isolate the second-highest and second-lowest copies of D [4 , , byremoving all other simple modules from the top and bottom of Y [3 , , ] ;the radical series of this new module demonstrates that the two copiesof D [4 , , are connected by the cross-diagonal edge shown. λ [3 , ] Y λ D [5 , D [7] D [5 , D [7] D [4 , , D [5 , D [7] D [5 , D [4 , , D [7] D [7] D [4 , , D [5 , D [7] D [5 , D [4 , , D [7] D [5 , D [7] D [5 , Notes on Y [3 , ] . Quotients of the form Rad i Y [3 , ] / Rad i +2 Y [3 , ] for i = 1 , Y [3 , ] / Rad Y [3 , ] follows from the way it decom-poses.We next examine Y [3 , ] / Soc Y [3 , ] , which is indecomposable. Re-moving both copies of D [7] from the bottom of this module yields thedirect sum indicated by the diagram, and demonstrates that the bot-tom copy of D [5 , is generated by one top copy of D [5 , and one topcopy of D [7] . Instead, removing that bottom copy of D [5 , causes themodule to decompose into two direct summands as shown; thereforethat D [5 , must be generated by one simple module from each of thosesummands. These manipulations imply that the entire quotient modulehas the structure shown.Finally, we investigate the edges that connect to the outer copies of D [5 , on the left and right. We first construct a module U with fourcomposition factors D [5 , : the top two copies of D [5 , and the two outercopies of D [5 , . (This can be done by first constructing a uniserial mod-ule with composition factors D [5 , , D [7] , D [4 , , , D [7] from Y [4 , , . Wequotient this uniserial module out of Y [3 , ] , and then remove excesssimple modules from the module that results.) The module U is inde-composable; this implies the existence of the outer edges from the topcopies of D [5 , to the outer copies, as well as either one or two cross-edges. Examination of Y [3 , ] / Soc Y [3 , ] , which decomposes after D [7] is removed from its top, confirms that there can be no other edges tothe outer copies of D [5 , .To examine the cross-edges, we next construct from Y [5 , ] a unise-rial module with two composition factors D [5 , , quotient this module OUNG MODULES AND SCHUR ALGEBRAS 19 out from Y [3 , ] , and remove excess simple modules until we are leftwith a module V that has four composition factors D [5 , . These mustbe the top two copies of D [5 , , the middle copy of D [5 , , and one ofthe outer copies of D [5 , . To determine which outer copy, we notethat V decomposes as ( D [5 , / ( D [5 , ⊕ D [5 , )) ⊕ D [5 , . If V containedthe left outer D [5 , without a cross-edge, it would have decomposedas ( D [5 , /D [5 , ) ⊕ ( D [5 , /D [5 , ); and if V contained either outer D [5 , with a cross-edge, it would have been a string and would not have de-composed. The given decomposition therefore implies that V containsthe right outer copy of D [5 , with no cross-edge. Since U is indecom-posable, the other cross-edge must exist (between the outer left andtop right copies of D [5 , ), so U is a string as shown. λ [2 ,
1] [2 , ] [2 , ] Y λ D [4 , D [6 , D [6 , D [4 , D [6 , D [6 , D [4 , D [5 , D [7] D [4 , , D [7] D [5 , D [5 , D [7] D [4 , , D [7] D [5 , D [5 , D [5 , D [6 , D [4 , D [6 , D [6 , D [4 , D [6 , D [6 , D [4 , D [6 , D [6 , D [4 , D [6 , ⊕ Notes on Y [2 , . This module is the projective cover of D [4 , . It isa direct summand of the tensor product Y [3 , ] ⊗ D [6 , . Notes on Y [2 , ] . This module is the projective cover of D [5 , . Itis a direct summand of the tensor product Y [3 , , ] ⊗ D [6 , . To examineits structure, we first construct from Y [5 , ] a uniserial module withtwo composition factors D [5 , . Removing this module from the topand bottom of Y [2 , ] isolates the four leftmost and four rightmostsimple modules; the structure of this outer ladder can be confirmed bypreviously explained methods (e.g. see notes on Y [3 , , ] and Y [3 , ] ).Additionally, isolating the four bottommost copies of D [5 , from thethird socle allows the edges between them to be confirmed. We finallycheck that there are no additional edges to the center copies of D [5 , byremoving the three bottommost copies of D [5 , and the topmost copyof D [5 , ; this new module has the remaining center copy of D [5 , as adirect summand. Notes on Y [2 , ] . This module is the projective cover of D [6 , . Itmay be constructed as the tensor product Y [4 , ] ⊗ U where U is the2-dimensional uniserial module with two copies of D [7] as compositionfactors. Such a module U may be constructed by removing copies of D [5 , from the top and bottom of Y [5 , ] . This tensor product is verifiedto be projective and indecomposable, and it has a copy of D [6 , in itstop layer. λ [1 ] Y λ D [7] D [4 , , D [7] D [5 , D [7] D [4 , , D [7] D [5 , D [5 , D [7] D [4 , , D [7] D [5 , D [7] D [4 , , D [7] Notes on Y [1 ] . This module is the projective cover of D [7] . It is adirect summand of the tensor product Y [2 , ] ⊗ D [6 , . To confirm thestructure of its heart, we first examine 2-step radical layers, findingthem to be strings and sums of strings as shown. We now performadditional checks to confirm that no other edges exist.Removing copies of D [7] from the top and bottom of the heart yieldsthe direct sum indicated by the diagram, which rules out the exis-tence of any additional edges that do not connect to one of thosecopies of D [7] . We eliminate the possibility of cross-diagonals in theladders by taking the top three-step radical layer of the heart, re-moving D [4 , , from its top and D [7] from its bottom, and noticingthat the leftmost copies of D [7] and D [5 , are a (uniserial) direct sum-mand of the resulting module. Finally, we construct two uniserialmodules from Y [4 , , . These are U := D [5 , /D [7] /D [4 , , /D [7] and V := D [7] /D [4 , , /D [7] /D [5 , . Removing sequentially U from the bot-tom and V from the top of Y [1 ] yields a module that decomposes as( D [7] /D [4 , , /D [7] ) ⊕ ( D [7] /D [4 , , /D [7] ); this decomposition rules outthe possibility of a connection between the two center columns of Y [1 ] . OUNG MODULES AND SCHUR ALGEBRAS 21 The module structure of projective modules forSchur algebras in characteristic 2
As noted previously, we compute with a basic algebra that is Moritaequivalent to the usual Schur algebra, and this makes no difference tothe structural properties that we consider. By abuse of notation wedenote the Morita equivalent algebra by the same symbols as are oftenused for the actual Schur algebra. Our algebra is S F ( n, n ) = End F S n ( M λ ⊢ n Y λ )and we store it on the computer as a matrix with rows and columnsindexed by partitions λ , where the ( λ, µ )-entry is a list of matricesthat is a basis for Hom F S n ( Y λ , Y µ ). The powers of the radical of S F ( n, n ) are stored in a similar way. Because the algebra is basic, theradical consists of those matrices that differ from S F ( n, n ) only on thediagonal, where the entries span the maximal ideal of End F S n ( Y µ ).This ideal can in turn be computed as the span of all composites Y µ → Y ν → Y µ with µ = ν , using the fact that S F ( n, n ) is quasi-hereditary,so that no simple module extends itself. Thus this endomorphism ringand its radical powers are immediately computable from the Youngmodules that have been constructed. From this we obtain the Gabrielquiver of the Schur algebra.The indecomposable projective P ( µ ) ∼ = Hom F S n ( Y µ , L λ ⊢ n Y λ ) isstored as row µ of the matrix that stores S F ( n, n ), and in this waywe parametrize the indecomposable projectives P ( λ ) and their uniquesimple quotients L λ by partitions of n . Its radical series arises as part ofthe computation of the radical series of S F ( n, n ). From the radical se-ries we use the quasi-hereditary structure given by the dominance orderon the partitions to deduce the standard (or Weyl) modules ∆ λ . Thediagrams for the projective modules are then built up from the stan-dard modules, and we exploit the information coming from the Gabrielquiver, comparison of structures in different projective modules, andthe fact that the projective modules are also injective precisely whenthe corresponding Young module Y λ is projective, and hence has asimple socle. This happens when λ is column p -regular, and it meansthat the ∆ µ appearing as a factor in such P ( λ ) with µ earliest in thedominance order has a simple socle.This latter result about injectivity is well known, but it is hard tofind an exact reference consistent with our conventions, and for theconvenience of the reader we prove the implication we need here. Forany F S n -module Y we write Y ♮ := Hom F S n ( Y, L λ ⊢ n Y λ ) and we will use the fact that the functor Y Y ♮ is a contravariant equivalencebetween the full subcategory of F S n -modules that are direct sums ofthe Y λ , and the full subcategory of S F ( n, n )-modules whose objectsare projective. Proposition 6.1.
Let Y µ be a projective Young module for F S n . Then Y µ♮ is injective (and also projective) as a S F ( n, n ) -module.Proof. We have already been assuming the projectivity of Y µ♮ and,indeed, it is a direct summand of S F ( n, n ). To show injectivity when Y µ is projective, we show that Ext S F ( n,n ) ( X, Y µ♮ ) = 0 for all S F ( n, n )-modules X . Take a S F ( n, n )-projective resolution · · · → Y ♮ → Y ♮ → Y ♮ → X → Y i are direct sums of Young modules. All projectives maybe written this way, by the equivalence of categories already referredto, and it arises by applying ♮ to a complex Y → Y → Y → · · · This latter complex is acyclic, except at 0, because L λ ⊢ n Y λ has all in-decomposable injective F S n modules as summands and the resolutionwas acyclic. Consider the cochain complexHom S F ( n,n ) ( Y ♮ , Y µ♮ ) → Hom S F ( n,n ) ( Y ♮ , Y µ♮ ) → · · · whose cohomology computes the Ext iS F ( n,n ) ( X, Y µ♮ ). By the equiva-lence of categories, it is isomorphic toHom F S n ( Y µ , Y ) → Hom F S n ( Y µ , Y ) → Hom F S n ( Y µ , Y ) → · · · and this is acyclic, except at 0, because Y µ is projective. This showsthat Ext S F ( n,n ) ( X, Y µ♮ ) = 0 so that Y µ♮ is injective. (cid:3) We display the standard or Weyl factors ∆ in a quasi-hereditaryfiltration of the projective modules using similar conventions to theones we used for Specht factors of Young modules: much of the time a∆ factor is indicated by joining its composition factors with thick lines,but sometimes we put the ∆ factor in a box.6.1.
Projective modules for S F (1 , . There is one projective mod-ule P [1], and it is simple. λ [1] P ( λ ) L [1] OUNG MODULES AND SCHUR ALGEBRAS 23
Projective modules for S F (2 , . λ [2] [1 ] P ( λ ) L [2] L [1 ] L [1 ] L [2] L [1 ] Projective modules for S F (3 , . λ [3] [2 ,
1] [1 ] P ( λ ) L [3] L [1 ] L [2 , L [1 ] L [3] L [1 ] Projective modules for S F (4 , . λ [4] [3 ,
1] [2 ] P ( λ ) L [4] L [2 ] L [3 , L [1 ] L [3 , L [2 ] L [4] L [2 ] L [3 , L [1 ] L [2 , ] L [1 ] L [2 ] L [4] L [2 ] L [3 , L [1 ] L [2 , ] L [2 , ] L [2 ] L [3 , L [1 ] λ [2 , ] [1 ] P ( λ ) L [2 , ] L [2 ] L [1 ] L [2 , ] L [3 , L [2 ] L [1 ] L [2 , ] L [1 ] L [2 , ] L [1 ] L [3 , L [1 ] L [2 ] L [4] L [2 ] L [3 , L [1 ] L [2 , ] Projective modules for S F (5 , . λ [5] [4 ,
1] [3 ,
2] [3 , ] P ( λ ) L [5] L [3 , L [3 , ] L [1 ] L [4 , L [2 , ] L [3 , L [5] L [3 , L [3 , ] L [1 ] L [2 , L [1 ] L [3 , ] L [3 , ] L [3 , L [5] L [3 , L [3 , ] L [1 ] L [2 , ] L [1 ] L [3 , ] L [1 ] L [2 , L [1 ] λ [2 ,
1] [2 , ] [1 ] P ( λ ) L [2 , L [1 ] L [3 , ] L [3 , L [1 ] L [2 , ] L [1 ] L [3 , ] L [1 ] L [2 , L [2 , ] L [4 , L [2 , ] L [1 ] L [3 , ] L [1 ] L [2 , L [1 ] L [3 , ] L [1 ] L [2 , L [1 ] L [3 , L [5] L [3 , L [2 , L [1 ] L [3 , ] L [1 ] L [2 , L [1 ] L [3 , ] The Schur algebra S F (6 , . We do not give the full structureof the projective modules in this case as the larger diagrams are com-plicated, with several tens of composition factors and elaborate Extstructure, so that the point of the diagram as a way to understand the
OUNG MODULES AND SCHUR ALGEBRAS 25 structure of the modules is defeated. As partial information we give di-agrams for the standard modules ∆ λ that arise in its quasi-hereditarystructure, also known as the Weyl modules. The composition factormultiplicities in these modules are the decomposition numbers and thisinformation is given in [6] and [10]. We note that the decompositionmatrix D satisfies D T D = C , the Cartan matrix.Weyl module structure: λ [6] [5 ,
1] [4 , λ L [6] L [5 , L [2 ] L [3 ] L [3 , ] L [1 ] L [5 , L [3 ] L [3 , ] L [4 , L [1 ] L [4 , ] L [2 , ] L [4 , L [4 , ] L [3 ] L [2 , ] L [3 , ] L [2 ] L [1 ] L [2 , ] λ [4 , ] [3 ] [3 , ,
1] [3 , ]∆ λ L [4 , ] L [2 , ] L [3 , ] L [2 ] L [1 ] L [2 , ] L [1 ] L [3 ] L [3 , ] L [2 ] L [1 ] L [2 , ] L [3 , , L [3 , ] L [1 ] L [2 , ] L [1 ] L [2 , ] λ [2 ] [2 , ] [2 , ] [1 ]∆ λ L [2 ] L [2 , ] L [1 ] L [2 , ] L [1 ] L [2 , ] L [2 , ] L [1 ] L [1 ] The Gabriel quiver of S F (6 ,
6) is the directed graph whose verticesare the isomorphism types of the simple modules, and where the num-ber of edges from a simple S to a simple T is dim Ext S F (6 , ( S, T ). ForSchur algebras this relation on S and T is symmetric because there isa duality on modules under which simple modules are self dual. Thismeans that for each edge S → T there is also an edge T → S .Gabriel quiver: L [6] L [5 , L [4 , L [4 , ] L [3 ] L [3 , , L [3 , ] L [2 ] L [2 , ] L [2 , ] L [1 ] The Schur algebra S F (7 , . Because S has two blocks over F ,so does S F (7 , F S block 1, and the other one block 2. We have seen thatthe non-principal block of F S is Morita equivalent to the principalblock of F S in such a way that the diagrams for the Young modulesin these blocks are the same, under a correspondence of partitions[6 , , , , ] ↔ [5][3 , , ] . Accordingly, block 2 of S F (7 ,
7) is Morita equivalent to block 1 of S F (5 , S F (7 ,
7) is harder to describe and we give less complete in-formation. As with S F (6 ,
6) we give diagrams for the standard modulesand the Gabriel quiver. The comments made about these structures inthe context of S F (6 ,
6) also apply here.Weyl module structure for block 1:
OUNG MODULES AND SCHUR ALGEBRAS 27 λ [7] [5 ,
2] [5 , ] [4 , , λ L [7] L [5 , ] L [3 , ] L [3 , L [3 , ] L [1 ] L [5 , L [3 , L [5 , ] L [3 , L [3 , ] L [4 , , L [3 , ] L [2 , ] L [5 , ] L [3 , L [3 , ] L [4 , , L [3 , ] L [2 , ] L [1 ] L [4 , , L [3 , ] L [3 , L [2 , ] L [3 , ] L [3 , ] L [1 ] L [3 , , ] λ [3 ,
1] [3 , ] [3 , , ] [3 , ] [2 , ] [1 ]∆ λ L [3 , L [3 , ] L [3 , ] L [1 ] L [3 , , ] L [3 , ] L [3 , , ] L [1 ] L [3 , , ] L [1 ] L [3 , ] L [2 , ] L [3 , ] L [2 , ] L [1 ] L [2 , ] L [1 ] Gabriel quiver for block 1: L [7] L [5 , L [5 , ] L [4 , , L [3 , L [3 , ] L [3 , , ] L [3 , ] L [2 , ] L [1 ] References [1] J.L. Alperin,
Diagrams for modules,
J. Pure Appl. Algebra 16 (1980), 111–119.[2] D.J. Benson,
Modular representation theory: new trends and methods,
LectureNotes in Mathematics 1081, Springer (1984).[3] D.J. Benson and J.F Carlson
Diagrammatic methods for modular representa-tions and cohomology,
Comm. Algebra 15 (1987), 53–121.[4] C. Bessenrodt and A.S. Kleshchev,
On tensor products of modular representa-tions of symmetric groups,
Bull. London Math. Soc. 32 (2000), 292–296.[5] K. Erdmann,
Symmetric groups and quasi-hereditary algebras,
Finite-dimensional algebras and related topics (Ottawa, ON, 1992), 123–161, NATOAdv. Sci. Inst. Ser. C Math. Phys. Sci., 424, Kluwer Acad. Publ., Dordrecht,1994. [6] J. Grabmeier,
Unerlegbare Moduln mit trivialer Youngquelle und Darstellungs-theorie der Schuralgebra,
Beyreuther Math. Schriften 20 (1985), 9–152.[7] J.A. Green,
Polynomial representations of GL n , Lecture Notes in Mathematics830, 2nd edition, Springer (2007).[8] J.A. Green, Functor categories and group representations,
Portugaliae Mathe-matica 43 (1985-86), 3–16.[9] G.D. James,
The representation theory of the symmetric groups,
Lecture Notesin Mathematics 682, Springer-Verlag (Berlin, Heidelberg, New York) (1978).[10] G.D. James,
The decomposition of tensors over fields of prime characteristic,
Math. Z. 172 (1980), 161–178.[11] G.D. James,
Trivial source modules for symmetric groups,
Arch. Math. (Basel)41 (1983), 294-?300.[12] G.D. James and A. Kerber,
The representation theory of the symmetric group,
Encyclopedia of Mathematics and its Applications, Vol. 16 (Addison-WesleyPublishing, Reading, MA, 1981).[13] A. Kleshchev, L. Morotti, P.H. Tiep,
Irreducible restrictions of representationsof symmetric groups in small characteristics: reduction theorems,
Math. Z. 293(2019), 677–723.[14] S. Martin,
Schur algebras and representation theory,
Cambridge Tracts inMathematics 112, Cambridge University Press (1993).[15] L. Morotti,
Irreducible tensor products for symmetric groups in characteristic2,
Proc. London Math. Soc. (3) 116 (2018), 1553–1598.[16] J. M¨uller and J. Orlob,
On the structure of the tensor square of the naturalmodule of the symmetric group,
Algebra Colloquium 18 (2011), 589–610.[17] R.A. Parker,
The computer calculation of modular characters (the meat-axe), in ed. M.D. Atkinson,Computational group theory (Durham, 1982), AcademicPress, London (1984), 267–274.[18] B.E. Sagan,
The symmetric group,
Springer-Verlag, New York (2001).[19] J.C. Scopes,
Cartan matrices and Morita equivalences for blocks of the sym-metric groups,
J. Algebra 142 (1991), 441–455.[20] P.J. Webb,
A course in finite group representation theory,
Cambridge studiesin advanced mathematics 161, Cambridge University Press (2016).[21] P.J. Webb et al,
The GAP package reps ∼ webb/GAPfiles/. Email address : [email protected] Email address : [email protected]@math.umn.edu