Decomposable Specht modules indexed by bihooks II
DDecomposable Specht modules indexed by bihooks II
Robert Muth
Washington & Jefferson CollegeWashington, PA 15301 [email protected]
Liron Speyer
Okinawa Institute of Science and TechnologyOkinawa, Japan 904-0495 [email protected]
Louise Sutton
University of ManchesterManchester, UK M13 9PL [email protected]
Abstract
Previously, the last two authors found large families of decomposable Specht modules labelledby bihooks, over the Iwahori–Hecke algebra of type B . In most cases we conjectured that thesewere the only decomposable Specht modules labelled by bihooks, proving it in some instances.Inspired by a recent semisimplicity result of Bowman, Bessenrodt and the third author, we lookback at our decomposable Specht modules and show that they are often either semisimple, orvery close to being so. We obtain their exact structure and composition factors in these cases. Inthe process, we determine the graded decomposition numbers for almost all of the decomposableSpecht modules indexed by bihooks. The Specht modules are a natural family of modules for the symmetric group, indexed by partitions,whose combinatorial construction has natural generalisations to Hecke algebras, and more generally tocyclotomic Hecke algebras, where multipartitions play the corresponding indexing role. In all of thesecases, these modules are the ordinary irreducible modules for these algebras, and it is an open anddifficult problem to determine their structure outside of the semisimple situation. One natural questionone can ask is when these important modules are decomposable . For the symmetric groups, an oldresult of James [Jam78, Corollary 13.18] tells us that the Specht modules are always indecomposableover fields of characteristic not equal to 2; this has an analogue for their Hecke algebras too – theSpecht modules are indecomposable outside of quantum characteristic e = 2. Already in this setting,it is an unresolved, hard problem to determine which Specht modules are decomposable when e = 2.Some progress has been made by Murphy [Mur80] and the second author [Spe14], where decomposableSpecht modules indexed by hook partitions were classified in the symmetric group and Hecke algebracases, respectively. Some more decomposable Specht modules for symmetric groups and their Heckealgebras can be found in [DF12, DG20, BBS19].The analogous question for cyclotomic Hecke algebras have not yet been studied. For (integral)Hecke algebras of type B , which may be seen as level 2 cyclotomic Hecke algebras, this study wasinitiated by the second and third authors [SS20]. These algebras have Specht modules S λ that arenow indexed by bipartitions λ , and it is known (for example by [FS16, Corollary 3.12]) that all Specht1 a r X i v : . [ m a t h . R T ] J a n Robert Muth, Liron Speyer & Louise Sutton modules are indecomposable unless e = 2 or the bicharge is of the form κ = ( i, i ) for some i . (Upto isomorphism we may assume that i = 0 in this case.) For κ = (0 , p denotes the characterstic of the ground field F . Theorem 1.1 [SS20, Theorems 4.1 and 5.4].
Let κ = (0 , , and λ = (( ke + a, b ) , ( je + a, b )) or (( b + 1 , je + a − ) , ( b + 1 , ke + a − )) , for some j, k (cid:62) , < a (cid:54) e and (cid:54) b < e with a + b (cid:54) = e , orfor a = b = 0 .(i) For j, k > , if j + k is even and p (cid:54) = 2 , or if j + k is odd, then S λ is decomposable.(ii) If j = 1 or k = 1 , then S λ is decomposable if and only if p (cid:45) j + k . Note that if e = 2 then we also have that S (( ke ) , ( je )) ∼ = S (( ke ) , (1 je )) , which allows the above results toalso cover λ = ((2 k + a, b ) , ( a, j + b )) or (( a, k + b ) , (2 j + a, b )). We conjectured [SS20, Conjecture4.2] that if e (cid:54) = 2 and p (cid:54) = 2, these are all of the decomposable Specht modules indexed by bihooks.As in the previous paper, our approach here uses the cyclotomic KLR algebras introduced byKhovanov, Lauda and Rouquier ([KL09, Rou08]), via the now-famous isomorphism of Brundan andKleshchev [BK09a]. This perspective allows for the study of the graded representation theory ofcyclotomic Hecke algebras.We observed – using the LLT algorithm – that, over a field of characteristic 0, the above Spechtmodules seemed to have all composition factors focussed in a single degree. A recent result of Bowman,Bessenrodt and the third author [BBS19, Theorem 3.2] then implies that the Specht modules must besemisimple. The purpose of the current paper is to study the structure of these decomposable Spechtmodules we found previously, and in particular determine when they are semisimple.Our first main result determines precisely when the Specht modules are semisimple, and determinesthe summands (the composition factors, with grading shift). Theorem 5.2.
Suppose k (cid:62) j (cid:62) . Then S (( ke ) , ( je )) is semisimple if and only if one of the followingholds: (cid:5) p (cid:54) = 2 and p does not divide any of the integers k + j, k + j − , . . . , k − j + 2 ; (cid:5) p = 2 , j = 1 , and k is even; (cid:5) p = 2 , j = 2 , and k ≡ .When semisimple, S (( ke ) , ( je )) is isomorphic to j (cid:77) r =1 D ((( k + j − r ) e, ( r − e +1) , ( e − (cid:104) j (cid:105) ⊕ D (( ke + je ) , ∅ ) (cid:104) j (cid:105) . From this, Theorem 5.4 and Corollary 5.7 give the corresponding result for the more generalbihooks of Theorem 1.1, by arguments using graded i -induction functors, which were also key in theproof of Theorem 1.1.Our methods for this use a Morita equivalence of Kleshchev and the first author [KM17b], thatallows us to pass the Specht modules S (( ke ) , ( je )) to the tensor product of Weyl modules ∆(1 k ) ⊗ ∆(1 j )over the usual Schur algebra S ( k + j, k + j ) – see Corollary 3.11. We are then able to perform a largepart of our analysis in the Schur algebra setting, pulling results back through the Morita equivalence.This analysis on the Schur algebra side is supplemented on the KLR algebra side by some structurethat we are able to extract from Specht module homomorphisms that we define along the way.Recasting our problem in the Schur algebra setting allows us to also gain traction in some ofthe non-semisimple cases. In particular we are able to give the complete structure of the Spechtmodules appearing in Theorem 1.1 whenever j = 1 or 2, or p divides precisely one of the integers k + j, k + j − , . . . , k − j + 2 (see Corollaries 5.8 and 5.9, Proposition 5.13 and Corollary 5.14, andTheorem 5.20, respectively). In most of these cases, our analysis hinges on piecing together Weyl ecomposable Specht modules indexed by bihooks II j, k > p = 2, filling a gap in Theorem 1.1.In ongoing work, we are utilising the methods and results from this paper to yield presentationsand graded dimension formulae for some simple modules in level 2. In fact, we are studying the simplemodules appearing in Theorems 5.2 and 5.4, under the same conditions.Our paper is organised as follows. In Section 2, we lay the foundations for our work, recalling thenecessary Lie theory setup, tableau-combinatorics, as well as definitions and some properties of KLRalgebras and their Specht modules. In Section 3, we recall work of Kleshchev and the first author[KM17a, KM17b] studying imaginary Schur algebras, and apply their results to our situation. Themain outcome of this section is a Morita equivalence that allows us to largely transport our problemto the classical Schur algebra, where we only need to work with Weyl modules indexed by two-columnpartitions. Section 4 is devoted to proving Propositions 4.5 and 4.7, which provide homomorphismsbetween certain Specht modules. These will provide the driving force for our proofs from one side ofthe Morita equivalence of Section 3. Our main results come in Section 5, where we use our Moritaequivalence to combine information from our Specht module homomorphisms with some known resultsfor Weyl modules (over the Schur algebra) indexed by two-column partitions and obtain our maintheorem on semisimple Specht modules, Theorem 5.2, as well as determine the structures of somenon-semisimple Specht modules. Acknowledgements.
The authors thank Kay Jin Lim for useful comments about [DL17], and MatthewFayers whose GAP packages were used in LLT computations as well as homomorphism computations.The authors are also grateful for the support from Singapore MOE Tier 2 AcRF MOE2015-T2-2-003, which funded a research visit of the first two authors to the third at the National University ofSingapore. The second author is partially supported by JSPS Kakenhi grant number 20K22316.
In this section we give an overview of KLR algebras, Specht modules labelled by bihooks, and theassociated combinatorics, as in our previous paper. For our methods here, we will also need to recallsome information about Weyl modules over Schur algebras, and their connection to KLR algebras viaImaginary Schur–Weyl duality. Throughout, F will denote an arbitrary field of characteristic p (cid:62) Let e ∈ { , , . . . } , which we call the quantum characteristic . We set I := Z /e Z , which we identifywith the set { , , . . . , e − } . We let Γ be the (type A (1) e − ) quiver with vertex set I and an arrow i → i − i ∈ I .Following Kac’s book [Kac90], we recall standard notation for the Kac–Moody algebra associatedto the generalised Cartan matrix ( a ij ) i,j ∈ I . We have simple roots { α i | i ∈ I } , fundamental dominantweights { Λ i | i ∈ I } , and the invariant symmetric bilinear form ( , ) such that ( α i , α j ) = a i,j and(Λ i , α j ) = δ ij , for all i, j ∈ I . Let Φ + be the set of positive roots, and let Q + := (cid:76) i ∈ I Z (cid:62) α i bethe positive cone of the root lattice. We write δ := α + · · · + α e − ∈ Φ + for the null root . If α = (cid:80) i ∈ I c i α i ∈ Q + , then we define the height of α to be ht( α ) = (cid:80) i ∈ I c i .An e -bicharge is an ordered pair κ = ( κ , κ ) ∈ I . We define its associated dominant weight Λ oflevel two to be Λ = Λ κ := Λ κ + Λ κ . Let S n be the symmetric group on n letters. We let s , . . . , s n − denote the standard Coxeter gener-ators, where s i is the simple transposition ( i, i + 1) for 1 (cid:54) i < n . We define a reduced expression for Robert Muth, Liron Speyer & Louise Sutton a permutation w ∈ S n to be an expression s i . . . s i m such that m is minimal, and call m the lengthof w , denoted (cid:96) ( w ).For h, n ∈ N , we will let S F ( h, n ) denote the classical Schur algebra over F ; see for instance [BDK01, §
1] for definitions and representation theoretic details.We define the
Bruhat order (cid:54) on S n as follows. If x, w ∈ S n , then we write x (cid:54) w if there is areduced expression for x which is a subexpression of a reduced expression for w . A composition λ of n is a sequence of non-negative integers λ = ( λ , λ , . . . ) such that | λ | := (cid:80) λ i = n .We say λ is a partition of n provided that the sequence is weakly decreasing, i.e. λ i (cid:62) λ j for all i (cid:54) j .We write ∅ for the empty partition (0 , , . . . ). We will denote the set of all compositions of n by C n ,and the set of all partitions of n by P n . We also define: C n ( a ) = { λ a composition of n | λ i = 0 for all i > a } P n ( a ) = { λ a partition of n | λ i = 0 for all i > a } for the sets of compositions and partitions of n , respectively, with a or fewer parts .A bipartition λ of n is a pair λ = ( λ (1) , λ (2) ) of partitions such that | λ | = | λ (1) | + | λ (2) | = n . Werefer to λ (1) and λ (2) as the 1 st and nd component , respectively, of λ . We abuse notation and alsowrite ∅ for the empty bipartition ( ∅ , ∅ ). We denote the set of all bipartitions of n by P n . Definition 2.1.
We call a bipartition λ a bihook if it is of the form λ = (( a, b ) , ( c, d )) for someintegers a, c (cid:62) b, d (cid:62) λ, µ ∈ P n , we say that λ dominates µ , and write λ (cid:81) µ , if for all k (cid:62) k (cid:88) j =1 λ (1) j (cid:62) k (cid:88) j =1 µ (1) j and | λ (1) | + k (cid:88) j =1 λ (2) j (cid:62) | µ (1) | + k (cid:88) j =1 µ (2) j . The
Young diagram of λ = ( λ (1) , λ (2) ) ∈ P n is defined to be[ λ ] := { ( i, j, m ) ∈ N × N × { , } | (cid:54) j (cid:54) λ ( m ) i } . We refer to elements of [ λ ] as nodes of λ . We draw the Young diagram of a bipartition as a columnvector of Young diagrams [ λ (1) ] , [ λ (2) ] from top to bottom. We say that a node A ∈ [ λ ] is removable if[ λ ] \ { A } is a Young diagram of a bipartition, while a node A (cid:54)∈ [ λ ] is addable if [ λ ] ∪ { A } is a Youngdiagram of a bipartition.If λ is a partition, the conjugate partition , denoted λ (cid:48) , is defined by λ (cid:48) i = | { j (cid:62) | λ j (cid:62) i } | . If λ ∈ P n , then we define the conjugate bipartition, also denoted λ (cid:48) , to be λ (cid:48) = ( λ (2) (cid:48) , λ (1) (cid:48) ). Let λ ∈ P n . Then a λ -tableau is a bijection T : [ λ ] → { , . . . , n } . We depict a λ -tableau T by insertingentries 1 , . . . , n into the Young diagram [ λ ] with no repeats; we let T ( i, j, m ) denote the entry lyingin node ( i, j, m ) ∈ [ λ ]. We say that T is standard if its entries increase down each column and alongeach row, within each component, and denote the set of all standard λ -tableaux by Std( λ ).The column-initial tableau T λ is the λ -tableau where the entries 1 , . . . , n appear in order downconsecutive columns, working from left to right, first in component 2, then in component 1.The symmetric group S n acts naturally on the left on the set of all λ -tableaux. For T a λ -tableau,we define the permutation w T ∈ S n by w T T λ = T .Suppose λ ∈ P n . Let S and T be λ -tableaux with corresponding reduced expressions w S and w T ,respectively. Then we say that T dominates S , written as T (cid:81) S , if and only if w T (cid:62) w S . ecomposable Specht modules indexed by bihooks II e -bicharge κ = ( κ , κ ). The e -residue of a node A = ( i, j, m ) ∈ N × N × { , } is definedto be res A := κ m + j − i (mod e ) . We call a node of residue r an r - node .Let T be a λ -tableau. If T ( i, j, m ) = r , we set res T ( r ) = res( i, j, m ). The residue sequence of T isdefined to be i T = (res T (1) , . . . , res T ( n )) . We denote the residue sequence of the column-initial tableau T λ by i λ := i T λ .We now define the degree and codegree of a standard tableau, as in [BKW11, § λ ∈ P n and an i -node A of λ , we define d A ( λ ) : = { addable i -nodes of λ strictly below A }− { removable i -nodes of λ strictly below A } ; d A ( λ ) : = { addable i -nodes of λ strictly above A }− { removable i -nodes of λ strictly above A } . Let T ∈ Std( λ ) with T − ( n ) = A . We define the degree and codegree of T , denoted deg( T ) andcodeg( T ), recursively, by setting deg( ∅ ) := 0 =: codeg( ∅ ), anddeg( T ) := d A ( λ ) + deg( T Khovanov–Lauda–Rouquier (KLR) algebra or quiver Hecke algebra R α to be the unitalassociative F -algebra with generating set { e ( i ) | i ∈ I α } ∪ { y , . . . , y n } ∪ { ψ , . . . , ψ n − } and relations e ( i ) e ( j ) = δ i , j e ( i ); (cid:88) i ∈ I α e ( i ) = 1; y r e ( i ) = e ( i ) y r ; ψ r e ( i ) = e ( s r i ) ψ r ; y r y s = y s y r ; ψ r y s = y s ψ r if s (cid:54) = r, r + 1; ψ r ψ s = ψ s ψ r if | r − s | > y r ψ r e ( i ) = ( ψ r y r +1 − δ i r ,i r +1 ) e ( i ); y r +1 ψ r e ( i ) = ( ψ r y r + δ i r ,i r +1 ) e ( i ); ψ r e ( i ) = i r = i r +1 ,e ( i ) if i r +1 (cid:54) = i r , i r ± , ( y r +1 − y r ) e ( i ) if i r = i r +1 + 1 , ( y r − y r +1 ) e ( i ) if i r = i r +1 − Robert Muth, Liron Speyer & Louise Sutton ψ r ψ r +1 ψ r e ( i ) = ( ψ r +1 ψ r ψ r +1 + 1) e ( i ) if i r +2 = i r = i r +1 + 1 , ( ψ r +1 ψ r ψ r +1 − e ( i ) if i r +2 = i r = i r +1 − , ( ψ r +1 ψ r ψ r +1 ) e ( i ) otherwise;for all admissible r, s, i , j . When e = 2, we actually have slightly different ‘quadratic’ and ‘braid’relations, which may be found, for example, in [KMR12, § e = 2.Recalling that Λ = Λ κ , we define R Λ α by imposing one additional relation on R α : y (Λ ,α i )1 e ( i ) = 0 , for all i ∈ I α . Lemma 2.2 [BK09b, Corollary 1]. There is a unique Z -grading on R Λ α such that, for all admissible r and i , deg( e ( i )) = 0 , deg( y r ) = 2 , deg ψ r ( e ( i )) = − a i r ,r r +1 . The cyclotomic KLR algebra or cyclotomic quiver Hecke algebra R Λ n is defined to be the directsum (cid:76) α R Λ α , where the sum is taken over all α ∈ Q + of height n .These Z -graded algebras are connected to the Hecke algebras of type B via (a special case of)Brundan and Kleshchev’s Graded Isomorphism Theorem . Theorem 2.3 [BK09a, Main Theorem]. If e = char( F ) or char( F ) (cid:45) e , then R Λ n is isomorphicto the integral Hecke algebra H n ( ξ, Q , Q ) of type B with parameters ξ ∈ F a primitive e th root ofunity, Q = ξ κ , and Q = ξ κ . That is, H n ( ξ, Q , Q ) has generators T , . . . , T n − satisfying type B Coxeter relations, with the quadratic relations replaced with ( T − ξ κ )( T − ξ κ ) = 0 and ( T i − ξ )( T i + 1) = 0 ∀ (cid:54) i (cid:54) n − . Finally, we end the subsection by briefly recalling the sign isomorphism sgn : R κn → R κ (cid:48) n from [KMR12, § κ (cid:48) = ( − κ , − κ ). It is the homogeneous algebra isomorphism defined bysgn : e ( i , i , . . . , i n ) (cid:55)→ e ( − i , − i , . . . , − i n ) , y r (cid:55)→ − y r , ψ s (cid:55)→ − ψ s , for all ( i , i , . . . , i n ) , r, s. We begin by recalling some necessary definitions pertaining to graded modules. If M is any graded R Λ n -module, we let M (cid:104) r (cid:105) denote the graded module obtained by shifting the grading on M up by r ;that is, M (cid:104) r (cid:105) d = M d − r . For q an indeterminate, the Grothendieck groups of R n and R Λ n are Z [ q, q − ]-modules by letting q m act by a degree shift by m . We let M denote the (ungraded) module obtainedby forgetting the grading on M . The (graded) dual of M is M (cid:126) := Hom F ( M, F ) with R Λ n -action givenby ( h · f )( m ) = f ( τ ( h ) m ) for f ∈ M (cid:126) , m ∈ M , h ∈ R Λ n . Here, τ denotes the homogeneous algebraanti-involution of R Λ n that sends each generator to themselves.For any graded R α -module, and i ∈ I α , we write M i := e ( i ) M , giving a vector space decomposition M = (cid:76) i ∈ I α M i . If M i (cid:54) = 0, we say that i is a word of M . The formal q -character of M isch q M := (cid:88) i ∈ I α (qdim M i ) · i , where qdim M ∈ Z [ q, q − ] is the graded dimension of M i . We have ch q M (cid:126) = ch q M , where the barindicates the bar involution , i.e. the automorphism of Z [ q, q − ] which swaps q and q − , extendedlinearly to (cid:76) i ∈ I α Z [ q, q − ] i .There is an induction functor Ind α,β , which associates to an R α -module M and an R β -module N the R α + β -module M ◦ N := Ind α,β M (cid:2) N for α, β ∈ Q + . ecomposable Specht modules indexed by bihooks II The graded R Λ n -modules of prime importance to us are the Specht modules. We use the followingpresentation of Specht modules from [KMR12, Definition 7.11]; we give only the presentations for thetwo families of bipartitions that we will explicitly need. Definition 2.4. Let λ = (( ke, je − e + 1) , ( e − ∈ P n for some j, k ∈ N . The (column) Spechtmodule S λ is the cyclic R Λ n -module generated by z λ of degree deg( z λ ) := codeg( T λ ) subject to therelations: (cid:5) e ( i λ ) z λ = z λ ; (cid:5) y r z λ = 0 for all r ∈ { , . . . , n } ; (cid:5) ψ r z λ = 0 for all r ∈ { , . . . , e − } ∪ { e, e + 2 , e + 4 , . . . , je − e } ∪ { je − e + 2 , . . . , ke + je − } ; (cid:5) ψ r ψ r +1 z λ = 0 for all r ∈ { e, e + 2 , e + 4 , . . . , je − e } ; (cid:5) ψ r ψ r − z λ = 0 for all r ∈ { e + 2 , e + 4 , . . . , je − e } .Let µ = (( ke ) , ( je )) ∈ P n for some j, k ∈ N . The (column) Specht module S µ is the cyclic R Λ n -module generated by z µ of degree deg( z µ ) := codeg( T µ ) subject to the relations: (cid:5) e ( i µ ) z µ = z µ ; (cid:5) y r z µ = 0 for all r ∈ { , . . . , n } ; (cid:5) ψ r z µ = 0 for all r ∈ { , . . . , n − } \ { je } .For each w ∈ S n , we fix a reduced expression w = s i . . . s i m throughout. We define the associatedelement of R Λ n to be ψ w := ψ i . . . ψ i m , which, in general, depends on the choice of reduced expressionfor w . For λ ∈ P n and a λ -tableau T , we define v T := ψ w T z λ . Whilst these vectors v T of S λ alsodepend on the choice of reduced expression in general, the following result does not. Theorem 2.5. [BKW11, Corollary 4.6], [KMR12, Proposition 7.14 and Corollary 7.20] For λ ∈ P n , the set of vectors { v T | T ∈ Std( λ ) } is a homogeneous F -basis of S λ , with deg( v T ) = codeg( T ) . Assumption 2.6. In light of Theorem 1.1, whose Specht modules are those of interest in the presentpaper, we will assume that κ = (0 , 0) throughout the remainder of the paper.We will compute homomorphisms of Specht modules, for which the following lemma will be useful. Lemma 2.7 [BKW11, Lemma 4.4]. Let λ ∈ P n , and T ∈ Std( λ ) . Then e ( i ) v T = δ i , i T v T . The next lemma serves as a reduction result for analysing our Specht modules. Lemma 2.8. Suppose that k (cid:62) j . Then S (( je ) , ( ke )) ∼ = S (cid:126) (( ke ) , ( je )) (cid:104) j + k (cid:105) .Proof. By examining the presentations, and noting that codeg( T (( je ) , ( ke )) ) = k while deg( T (( ke ) , ( je )) ) = j + 2 k , we see that S (( je ) , ( ke )) (cid:104) j + k (cid:105) ∼ = S (( ke ) , ( je )) . Next, [KMR12, Theorem 7.25] and an easy degree calculation gives us thatS (( ke ) , ( je )) ∼ = S (cid:126) (( ke ) , ( je )) (cid:104) j + 2 k (cid:105) , and hence that S (( je ) , ( ke )) ∼ = S (cid:126) (( ke ) , ( je )) (cid:104) j + k (cid:105) . Robert Muth, Liron Speyer & Louise Sutton Let λ ∈ P n . We define the i -signature of λ by reading the Young digram [ λ ] from the top of thefirst component down to the bottom of the last component, writing a + for each addable i -node anda − for each removable i -node. We obtain the reduced i -signature of λ by successively deleting alladjacent pairs + − from the i -signature of λ , always of the form − · · · − + · · · +.The removable i -nodes corresponding to the − signs in the reduced i -signature of λ are called the normal i -nodes of λ , while the addable i -nodes corresponding to the + signs in the reduced i -signatureof λ are called the conormal i -nodes of λ . The lowest normal i -node of [ λ ], if there is one, is calledthe good i -node of λ , which corresponds to the last − sign in the i -signature of λ . Analogously, thehighest conormal i -node of [ λ ], if there is one, is called the cogood i -node of λ , which corresponds tothe first + sign in the i -signature of λ .We say that a bipartition λ ∈ P n is regular , or conjugate-Kleshchev , if [ λ ] can be obtained bysuccessively adding cogood nodes to ∅ . That is, we have a sequence ∅ = λ (0) , λ (1) , . . . , λ ( n ) = λ such that [ λ ( i )] ∪ { A } = [ λ ( i + 1)], where A is a cogood node of λ ( i ). Equivalently, λ is regular if andonly if ∅ can be obtained by successively removing good nodes from [ λ ]. Observe in level one thatthe set of all regular partitions coincides with the set of all e -regular partitions.For each regular bipartition λ ∈ P n , the Specht module S λ has a simple head, denoted by D λ . Theorem 2.9 [BK09b, Theorem 5.10]. The modules { D λ | λ ∈ P n , λ regular } give a completeset of graded simple R Λ n -modules up to isomorphism and grading shift. Moreover, each D λ is self-dualas a graded module. There is a bijection m e,κ : P n → P n such that (D λ ) sgn ∼ = D m e,κ ( λ ) . Strictly speaking, the signmap actually takes a simple R Λ κ n -module to a R Λ − κ n -module, but we may ignore this here as we onlyconsider κ = (0 , R Λ κ n -modules. This higher levelanalogue of the Mullineux involution was introduced by Fayers in [Fay08, § 2] (see also [JL09]). Moreprecisely, if λ ∈ P κn is obtained from ∅ by adding cogood nodes with residues i , i , . . . , i n , then m e,κ ( λ ) is the multipartition obtained from ∅ by adding cogood nodes of residues − i , − i , . . . , − i n . Let ϕ i ( λ ) denote the number of conormal i -nodes of a multipartition λ , and ˜ f i λ the partition obtainedfrom λ by adding the cogood i -node. In [BK09b, Sections 4.4 and 4.6], graded induction functors andtheir divided powers are introduced. Here, we denote these by f i and f ( n ) i , respectively. We refer to[BK09b] for further details, and here only mention some key facts that we will require later.For a non-negative integer n , we define the quantum integer [ n ] := 1 + q + · · · + q n − and the quantum factorial [ n ]! := [1][2] . . . [ n ]. Lemma 2.10 [BK09b, Lemma 4.8, Theorem 4.12]. There is an isomorphism f ni ∼ = [ n ]! f ( n ) i .Moreover, f i D λ is non-zero if and only if ϕ i ( λ ) (cid:54) = 0 , in which case f i D λ has irreducible socle isomor-phic to D ˜ f i λ (cid:104) ϕ i ( λ ) − (cid:105) and head isomorphic to D ˜ f i λ (cid:104) − ϕ i ( λ ) (cid:105) . It will be useful for us to know which Weyl modules (over the usual Schur algebra – i.e. level 1, with e = p ) ∆( λ ) indexed by two-column partitions are irreducible. In such cases, ∆( λ ) is equal to itssimple head, L ( λ ). Here, by Weyl modules, we mean the Weyl modules used in [Mat99]. The Schurfunctor maps these modules to ‘row Specht modules’. Because of the distinction between level 1 andlevel 2 cases, this will not be problematic for us. For ( a, b ) ∈ [ λ ], we denote by h λ ( a, b ) its hook length,i.e. h λ ( a, b ) = λ a − b + λ (cid:48) b − a + 1. For a prime p , define ν p ( h ) to be the largest power of p dividing h . Theorem 2.11 [Mat99, Proposition 5.39]. Let λ (cid:96) n , and let F be a field of characteristic p . TheWeyl module ∆( λ ) over the Schur algebra S ( n, n ) is irreducible if and only if ν p ( h λ ( a, b )) = ν p ( h λ ( a, c )) for every ( a, b ) , ( a, c ) ∈ [ λ ] . ecomposable Specht modules indexed by bihooks II Corollary 2.12. Let e = p .(i) If p (cid:54) = 2 and n (cid:62) j , the Weyl modules ∆(1 n ) , ∆(2 , n − ) , . . . , ∆(2 j , n − j ) are all simultaneouslyirreducible if and only if p does not divide any of the integers n, n − , . . . , n − j + 2 .(ii) If p = 2 , then the Weyl modules:(a) ∆(1 n ) and ∆(2 , n − ) are simultaneously irreducible if and only if n is odd;(b) ∆(1 n ) and ∆(2 , n − ) , and ∆(2 , n − ) are simultaneously irreducible if and only if n ≡ ;(c) ∆(1 n ) and ∆(2 , n − ) , ∆(2 , n − ) , and ∆(2 , n − ) are never simultaneously irreducible.Proof. We proceed by induction on j , with some special care needed for small j . If j = 1, we wantto know when ∆(1 n ) and ∆(2 , n − ) are simultaneously irreducible. The former module is alwaysirreducible, and the latter is irreducible if and only if p (cid:45) n . If p | n , then ∆(2 , n − ) is comprised of asimple head L (cid:0) , n − (cid:1) , and a simple socle L (1 n ).If j = 2, we use the fact that ∆(1 n ) and ∆(2 , n − ) are simultaneously irreducible if and onlyif p (cid:45) n , and check when ∆(2 , n − ) is also irreducible. We apply Theorem 2.11, noting that weonly need to check the valuations of hook lengths in the first two rows. The nodes (1 , 2) and (2 , , n − )] have hook lengths 1 and 2, respectively. In characteristic p (cid:54) = 2, both give p-adicvaluations 0. The nodes (1 , 2) and (2 , 2) have hook lengths n − n − 2, respectively. Thus if p (cid:54) = 2, ∆(2 , n − ) is irreducible if and only if p (cid:45) n − , n − 2. If p = 2, then ν p ( h λ (1 , , n − ) is irreducible if and only if n ≡ n ≡ , n − ) is reducible, completing the proof when p = 2.We now assume that p (cid:54) = 2, j > 2, and that the Weyl modules ∆(1 n ) , ∆(2 , n − ) , . . . , ∆(2 j − , n − j +2 )are all simultaneously irreducible if and only if p does not divide any of the integers n, n − , . . . , n − j + 4. In particular, since these are 2 j − p > j − 3. We need to check when ∆(2 j , n − j )is irreducible. Since p > j − 3, the p-adic valuations in the second column of [(2 j , n − j )] are all 0.The hook lengths in nodes (1 , , (2 , , . . . , ( j, 1) are n − j + 1 , n − j, . . . , n − j + 2, respectively, andthe result follows from Theorem 2.11.We end this subsection with a result computing the composition factors of Weyl modules indexedby two-column partitions. This result will be useful to us in Section 5. First we must introduce somenotation. Suppose that a and b are positive integers with p -adic expansions a = a + pa + p a + · · · and b = b + pb + p b + · · · with 0 (cid:54) a i , b i < p for all i . Write a (cid:52) p b if, for all i either a i = 0 or a i = b i . Theorem 2.13 [Mat99, Section 6.4, Rule 15]. Let λ = (2 m , n − m ) and µ = (2 j , n − j ) . Then [∆( λ ) : L ( µ )] = (cid:40) if (cid:98) m − jp (cid:99) (cid:52) p (cid:98) n − j +1 p (cid:99) and either p | m − j or p | n − m − j +1 , otherwise. Here we import some useful results from [KM17b]. Let (cid:60) be a balanced convex preorder on Φ + (see for instance [Kle14, § R δ -module L δ,e − such that for all words j = ( j , . . . , j e ) in L δ,e − , we have j = 0 and j e = e − ( e ) is a simple one-dimensional R δ -module with character i = (0 , , , . . . , e − ( e ) ∼ = L δ,e − .For ν = ( n , . . . , n a ) ∈ C n ( a ) define the associated Gelfand–Graev words in I nδ via g ( n ) = 0 n n . . . ( e − n , g ν = g ( n ) . . . g ( n a ) . Robert Muth, Liron Speyer & Louise Sutton We also define c ( ν ) = ([ n ]! . . . [ n a ]!) e . Following [KM17b], for n ∈ N , denote the R nδ -module M n := L ◦ nδ,e − , the imaginary tensor spacemodule (of colour e − ) and the finite-dimensional quotient algebra S n := R nδ / Ann R nδ M n the imag-inary Schur algebra . This quotient algebra earns its name via a Morita equivalence with the classicalSchur algebra S F ( n, n ). Theorem 3.1 below is a special case of [KM17b, Theorems 4 & 5]. Theorem 3.1 [KM17b, Theorems 4 & 5]. Let h (cid:62) n ∈ N . There exists a projective generator Z for S n such that End S n ( Z ) ∼ = S F ( h, n ) , and considering Z as a right S F ( h, n ) -module, we have End S F ( h,n ) ( Z ) ∼ = S n . This gives mutually inverse equivalences of categories: M h,n : S n → S F ( h, n ) , V (cid:55)→ Hom S n ( Z, V ) (cid:99) M h,n : S F ( h, n ) → S n , W (cid:55)→ Z ⊗ S F ( h,n ) W. This equivalence intertwines induction in the imaginary Schur algebra and the tensor product in theclassical Schur algebra, in the following sense. Let ν = ( n , . . . , n a ) ∈ C n ( a ) . We have an isomorphismof functors: Ind nδn δ,...,n a δ ( (cid:99) M h,n (?) (cid:2) · · · (cid:2) (cid:99) M h,n a (?)) ∼ = (cid:99) M h,n (? ⊗ · · · ⊗ ?) from S F ( h, n )-mod × · · · × S F ( h, n a )-mod to S n -mod . In this paper we use the Morita equivalence above as a black box, and will not need to be concernedwith the specifics of the projective generator Z . We will write M n for M n,n , and M for M n when n is clear from context, and similarly for (cid:99) M .Let h (cid:62) n . Recall (see [Don98]) that S F ( h, n ) is a finite-dimensional quasi-hereditary algebra withirreducible, standard, and indecomposable tilting modulesL h ( λ ) , ∆ h ( λ ) , T h ( λ ) , ( λ ∈ P n ) . We will omit the superscript h in the situation that h = n .Via Theorem 3.1 we have that S n is itself a finite-dimensional quasi-hereditary algebra withirreducible, standard, and indecomposable tilting modules (cid:99) M (L( λ )) , (cid:99) M (∆( λ )) , (cid:99) M (T( λ )) , ( λ ∈ P n ) . Lemma 3.2. [KM17b, Lemma 6.1.3] Let h (cid:62) n , and λ ∈ P n . Then we have (cid:99) M h,n (L h ( λ )) ∼ = (cid:99) M (L ( λ )) and (cid:99) M h,n (∆ h ( λ )) ∼ = (cid:99) M (∆( λ )) . Lemma 3.3. [KM17b, Lemma 6.3.2] Let λ ∈ P n . Then we have that (cid:99) M (L( λ )) (cid:126) ∼ = (cid:99) M (L( λ )) . Since the characters of images of simple S F ( n, n )-modules under (cid:99) M are bar-invariant, we have thefollowing immediate result. Corollary 3.4. Let W ∈ S F ( n, n )-mod . Then ch q (cid:99) M ( W ) = ch q (cid:99) M ( W ) = ch q (cid:99) M ( W ) (cid:126) . Knowledge of the usual formal character ch W of a module W over the classical Schur algebra S F ( n, n ) (i.e., the dimension of the usual weight spaces W λ = e ( λ ) W for λ ∈ C n ( n ), provides partialinformation about the graded character of M ( W ): Theorem 3.5. [KM17b, Theorem 9] Let λ ∈ C n ( n ) and W ∈ S F ( n, n )-mod . Then dim q (cid:99) M ( W ) g λ = c ( λ ) dim W λ . ecomposable Specht modules indexed by bihooks II λ ∈ P n , µ ∈ C n ( n ), let K λ,µ denote the usual Kostka number; the dimension of the µ -weight space ∆( λ ) µ in the standard module ∆( λ ), given by the number of semistandard λ -tableauxof weight µ . Let k λ,µ denote the dimension of the µ -weight space L ( λ ) µ in the simple module L ( λ ).If char F = 0, then K λ,µ = k λ,µ . The following theorem, a special case of [KM17b, Theorem 10],provides important partial information about the character of these modules: Theorem 3.6. [KM17b] For λ ∈ P n , µ ∈ C n ( n ) , we have dim (cid:99) M (∆( λ )) g µ = c ( µ ) K λ,µ and dim (cid:99) M (L ( λ )) g µ = c ( µ ) k λ,µ . Corollary 3.7. If L is a simple S n -module, and there exists λ ∈ P n such that dim L g λ > and dim L g µ = 0 for all µ ∈ P n with µ (cid:66) λ , then L ∼ = (cid:99) M (L ( λ )) .Proof. Simple modules for S F ( n, n ) are distinguished by their highest weights (see for instance [Gre07,Theorem 3.5a]), so if L (cid:48) is a simple S F ( n, n )-module such that dim L (cid:48) λ > (cid:48) µ = 0 for all µ (cid:66) λ , it follows that L (cid:48) ∼ = L ( λ ). Then Theorem 3.6 implies the result.We now show that the self-duality of tilting modules is preserved by (cid:99) M : Proposition 3.8. Let λ ∈ P n . Then we have (cid:99) M (T( λ )) (cid:126) ∼ = (cid:99) M (T( λ )) .Proof. By [Don98, 3.3(1)], the tilting module T( λ ) arises as an indecomposable summand of somemodule W = L n (1 µ ) ⊗ · · · ⊗ L n (1 µ n ), where µ ∈ P n and moreover, all indecomposable summands of W are tilting modules T ( ν ), where ν ∈ P n . Now we note that (cid:99) M ( W ) (cid:126) = (cid:99) M (L n (1 µ ) ⊗ · · · ⊗ L n (1 µ n )) (cid:126) ∼ = ( (cid:99) M (L n (1 µ )) ◦ · · · ◦ (cid:99) M (L n (1 µ n ))) (cid:126) ∼ = (cid:99) M (L(1 µ n )) (cid:126) ◦ · · · ◦ (cid:99) M (L(1 µ )) (cid:126) ∼ = (cid:99) M (L(1 µ n )) ◦ · · · ◦ (cid:99) M (L(1 µ )) ∼ = (cid:99) M (L n (1 µ )) ◦ · · · ◦ (cid:99) M (L n (1 µ n )) ∼ = (cid:99) M (L n (1 µ ) ⊗ · · · ⊗ L n (1 µ n )) = (cid:99) M ( W ) , where the second isomorphism follows from Lemma 3.2 and [KM17b, Lemma 3.4.2], the third isomor-phism follows from Lemma 3.3, and the fourth isomorphism follows from Lemma 3.2 and [KM17b,Lemma 6.2.2]. Since (cid:99) M ( W ) (cid:126) ∼ = (cid:99) M ( W ), it follows that we must have (cid:99) M (T( λ )) ∼ = (cid:99) M (T( ν )) (cid:126) forsome tilting module T ( ν ) which is an indecomposable summand of W . But then by Corollary 3.4 wehave ch q (cid:99) M (T( λ )) = ch q (cid:99) M (T( ν )) (cid:126) = ch q (cid:99) M (T( ν )) . Then by Theorem 3.5 we have that ch T( λ ) = ch T( ν ). Then, as noted in [Don98, Remark 3.3(i)] wemust have λ = ν , so the result follows. Lemma 3.9. For k ∈ N , we have S ( ke ) ∼ = (cid:99) M (∆(1 k )) ∼ = (cid:99) M (L (cid:0) k (cid:1) ) .Proof. S ( ke ) is a simple one-dimensional module with character g (1 k ) = i k . Up to scalar, the onlyelement of R kδ that does not annihilate S ( ke ) is the idempotent e ( i k ). But i k is a word of M k , so e ( i k ) M k (cid:54) = 0. So we have that S ( ke ) factors through to a simple S k -module. It follows by Corollary 3.7that S ( ke ) ∼ = (cid:99) M (L (cid:0) k (cid:1) ), and (cid:99) M (∆(1 k )) ∼ = (cid:99) M (L (cid:0) k (cid:1) ) since ∆(1 k ) = L (cid:0) k (cid:1) as S F ( n, n )-modules. Lemma 3.10. Let k, j ∈ N , and write n = k + j . Then we have S (( ke ) , ( je )) ∼ = (cid:99) M (∆ n (1 j ) ⊗ ∆ n (1 k )) (cid:104) j (cid:105) ∼ = (cid:99) M (L n (1 j ) ⊗ L n (1 k )) (cid:104) j (cid:105) . Proof. We haveS (( ke ) , ( je )) ∼ = S ( je ) ◦ S ( ke ) (cid:104) j (cid:105) by [KMR12, Theorem 8.2]2 Robert Muth, Liron Speyer & Louise Sutton ∼ = (cid:99) M (∆(1 j )) ◦ (cid:99) M (∆(1 k )) (cid:104) j (cid:105) by Lemma 3.9 ∼ = (cid:99) M n,j (∆ n (1 j )) ◦ (cid:99) M n,k (∆ n (1 k )) (cid:104) j (cid:105) by Lemma 3.2 ∼ = (cid:99) M (∆ n (1 j ) ⊗ ∆ n (1 k )) (cid:104) j (cid:105) by Theorem 3.1 . As ∆(1 a ) ∼ = L (1 a ) for all a ∈ N , this completes the proof.Lemmas 3.9 and 3.10 imply the immediate Corollary 3.11. Let n = j + k . The Specht module S (( ke ) , ( je )) factors through the surjection from R nδ onto its quotient, so that it is naturally an S n -module. Moreover, M (S (( ke ) , ( je )) ) = ∆ n (1 j ) ⊗ ∆ n (1 k ) ∼ = ∆ n (1 k ) ⊗ ∆ n (1 j ) ∈ S F ( n, n )-mod . Next, it will also be useful to us to know a little about the indecomposable summands of Young per-mutation modules of the form M ( k, j ) over the symmetric group in characteristic p . These summandsare by definition Young modules, and the information we need is the following result of Henke.Suppose that a and b are positive integers with p -adic expansions a = a + pa + p a + · · · and b = b + pb + p b + · · · with 0 (cid:54) a i , b i < p for all i . Write a (cid:54) p b if, for all i , a i (cid:54) b i . It will sometimes be helpful to compactlywrite the p -adic expansions in the form [ a , a , a , . . . ]. Theorem 3.12 [Hen05, Theorem 3.3]. Let λ = ( n − j, j ) and µ = ( n − m, m ) . Then the Youngmodule Y ( µ ) appears as a summand of M ( λ ) exactly once if j − m (cid:54) p n − m , and does not appearotherwise. Proposition 3.13. For any j, k (cid:62) , any e (cid:62) , and over any field F , S (cid:126) (( ke ) , ( je )) ∼ = S (( ke ) , ( je )) (cid:104)− j (cid:105) . Moreover, if M is an indecomposable summand of S (( ke ) , ( je )) , then M (cid:126) ∼ = M (cid:104)− j (cid:105) .Proof. The indecomposable summands of L j + k (1 j ) ⊗ L j + k (1 k ) are tilting modules, so the result followsimmediately from Lemma 3.10 and Proposition 3.8. The main results of this section are the homomorphisms in Propositions 4.5 and 4.7. The reader maywant to skip ahead to the next section, where these homomorphisms are applied – the computationscarried out in order to prove that these maps are indeed homomorphisms are not so instructive.In light of Lemma 2.8, we can assume that k (cid:62) j as we determine the structure of S (( ke ) , ( je )) .In computing homomorphisms, we inherently make use of Lemma 2.7 throughout this section – itenables us to write the image of the cyclic generator z λ in terms of standard tableaux with residue i λ .We recall the necessary notation and results from [SS20], in preparation for computing Spechtmodule homomorphisms with domain S (( ke ) , ( je )) .For 1 (cid:54) i (cid:54) j (cid:54) n − 1, we define s j ↓ i := s j s j − . . . s i and s j ↑ i := s i s i +1 . . . s j . Similarly, we define ψ x ↓ y := ψ x ψ x − . . . ψ y and ψ x ↑ y := ψ y ψ y +1 . . . ψ x if x (cid:62) y , and set both equal to 1 F if x < y . ecomposable Specht modules indexed by bihooks II v T of S (( ke ) , ( je )) . Observe that a standard λ -tableau T is determined by the entries a r := T (1 , r, 2) lying in its second component, for all r ∈ { , . . . , je } .We can thus write T = w T T (( ke ) , ( je )) , where w T := s a − ↓ s a − ↓ . . . s aje − ↓ je ∈ S n . It follows that v T = ψ T z (( ke ) , ( je )) where ψ T := ψ a − ↓ ψ a − ↓ . . . ψ aje − ↓ je ∈ R Λ n . In order to distinguish our standard tableaux compactly, we will often write v ( a , a , . . . , a je ) for thestandard λ -tableau with entries a , a , . . . , a je in the second component.We recall from [SS20, § 4] the notions of e -bricks , e -brick tableaux and brick transpositions . Let T be a standard (( ke ) , ( je ))-tableau. We define an e -brick to be a sequence of e adjacent nodescontaining entries je + 1 , je + 2 , . . . , ( j + 1) e for j (cid:62) 0. We say that T is an e -brick tableau if all entriesof T lie in e -bricks. For any e -brick tableau T , we number the e -bricks in the order of their entries,i.e. T comprises of bricks 1 , , . . . , j + k . Then we have brick transpositions and their corresponding ψ expressions, which we will denote by Ψ r . In particular, the brick transposition that swaps the r thand ( r +1)th bricks corresponds toΨ r = ψ re ↓ ( r − e +1 re +1 ↓ ( r − e +2 . . . ( r +1) e − ↓ re . As with our ψ generators, we introduce the shorthand Ψ x ↓ y = Ψ x Ψ x − . . . Ψ y and Ψ x ↑ y = Ψ y Ψ y +1 . . . Ψ x .The following results will be crucial in our computations. Lemma 4.1 [SS20, Lemma 6.8]. Let e > , T ∈ Std( λ ) , v T = v ( a , . . . , a je ) , (cid:54) r < n , and (cid:54) s < je such that r (cid:54)≡ s (mod e ) .(i) If a s = r , a s +1 = r + 1 , then ψ r v ( a , . . . , a je ) = 0 .(ii) If s is maximal such that a s (cid:54) r − , and r, r + 1 / ∈ { a , . . . , a je } , then ψ r v ( a , . . . , a je ) = 0 . Lemma 4.2 [SS20, Lemma 6.14(i) & (ii)]. Let e > , (cid:54) s (cid:54) je − e and v T = v ( a , . . . , a je ) .(i) If a s + e = r for some (cid:54) r (cid:54) n such that r (cid:54)≡ s, s + 1 (mod e ) and r − , r + 1 , r + 2 , r +3 , . . . , r + e − (cid:54)∈ { a , . . . , a je } , then y r − v T = 0 .(ii) If a s + e = r for some (cid:54) r (cid:54) n such that r (cid:54)≡ s, s +1 (mod e ) and r +1 , r +2 , r +3 , . . . , r + e − (cid:54)∈{ a , . . . , a je } , then y r v T = 0 . Analogues for the previous two lemmas hold for e = 2, but were not proved in [SS20]. They canbe proved by similar computations, and are needed to prove that Proposition 4.7 also holds for e = 2.Next, we note properties of brick transpositions, KLR generators and basis vectors. Lemma 4.3 [SS20, Lemma 4.11]. (i) If | r − s | > , then Ψ r Ψ s = Ψ s Ψ r .(ii) If λ = (( ke ) , ( je )) , for some j, k (cid:62) and r (cid:54) = j , then Ψ r z λ = 0 . Proposition 4.4 [SS20, Lemmas 4.9 and 4.10 and Proposition 4.12]. Let λ = (( ke ) , ( je )) ,for some j, k (cid:62) . If v ∈ e ( i λ ) S λ , and (cid:54) r (cid:54) j + k − , then(i) y s v = 0 for all (cid:54) s (cid:54) ( k + j ) e ;(ii) ψ s v = 0 for all (cid:54) s (cid:54) ( k + j ) e − with s (cid:54)≡ e ) ; Robert Muth, Liron Speyer & Louise Sutton (iii) ψ re Ψ r v = − ψ re v ;(iv) for r < j + k − , ψ re Ψ r +1 Ψ r v = ψ re v ;(v) for r > , ψ re Ψ r − Ψ r v = ψ re v . Now we are ready to compute our first homomorphisms. The image of the cyclic generator underthe homomorphism below is seen to be a linear combination of basis vectors indexed by brick-tableaux. Proposition 4.5. Let k (cid:62) j (cid:62) . There exists a degree 1 Specht module homomorphism α k,j : S (( ke + e ) , ( je − e )) −→ S (( ke ) , ( je )) z (( ke + e ) , ( je − e )) (cid:55)−→ k (cid:88) i =0 ( k + 1 − i )Ψ j + i − ↓ j z (( ke ) , ( je )) . Proof. Let λ = (( ke + e ) , ( je − e )) and µ = (( ke ) , ( je )). We know from parts (i) and (ii) of Proposi-tion 4.4 that we need only show that ψ re α k,j ( z λ ) = 0 for each r ∈ { , , . . . , j − } ∪ { j, j + 1 , . . . , k + j − } . This is obvious if r ∈ { , . . . , j − } . We now suppose that r ∈ { j, j + 1 , . . . , k + j − } .Firstly, if s (cid:62) r + 2 then ψ se Ψ r ↓ j z µ = Ψ r ↓ j ( ψ se z µ ) = 0 . Secondly, if s (cid:54) r − ψ se Ψ r ↓ j z µ = Ψ r ↓ s +2 ( ψ se Ψ s +1 Ψ s ) Ψ s − ↓ j z µ = Ψ r ↓ s +2 ψ se Ψ s − ↓ j z µ = 0 , by Proposition 4.4.We thus have ψ re α k,j ( z λ ) = ψ re (cid:18) ( k − r + j − r +1 ↓ j +( k − r + j )Ψ r ↓ j +( k − r + j + 1)Ψ r − ↓ j (cid:19) z µ = (cid:18) ( k − r + j − ψ re Ψ r +1 Ψ r )Ψ r − ↓ j +( k − r + j )( ψ re Ψ r )Ψ r − ↓ j +( k − r + j + 1) ψ re Ψ r − ↓ j (cid:19) z µ = (( k − r + j − − k − r + j ) + ( k − r + j + 1)) Ψ r − ↓ j ψ re z µ = 0 , by Proposition 4.4.Finally, ψ ( k + j − e α k,j ( z λ ) = ψ ( k + j − e (cid:18) Ψ k + j − ↓ j +2Ψ k + j − ↓ j (cid:19) z µ = (cid:18) ( ψ ( k + j − e Ψ ( k + j − )Ψ k + j − ↓ j +2 ψ ( k + j − e Ψ k + j − ↓ j (cid:19) z µ = 0 . The degree of the homomorphism being 1 follows from an easy check that codeg T = j for any T ∈ Std(( ke ) , ( je )) with res T = i (( ke ) , ( je )) . Lemma 4.6. Let k (cid:62) j (cid:62) . Let r = 2 je − e − i for i (cid:62) . Then ψ r v (1 , . . . , r + e , r + 3 , r + 5 , r + 7 , . . . , je − e + 1) ∈ S (( ke ) , ( je )) . ecomposable Specht modules indexed by bihooks II Proof. We proceed by induction on i . Suppose that i = 1. By applying part (i) of [SS20, Corollary6.9], we have ψ je − e − v (1 , . . . , je − , je − e + 1) = ψ je − e ψ je − e − v (1 , . . . , je − , je − e )= ψ je − e v (1 , . . . , je − , je − e − ψ je − e ψ je − e − ↓ je z λ = ψ je − e − ↓ je ψ je − e z λ = 0 . Now suppose that the statement holds for some i > 1. Then by applying part (i) of Corollary6.9 [SS20], we have ψ r v (1 , . . . , r + e , r + 3 , r + 5 , r + 7 , . . . , je − e + 1)= ψ r +2 ψ r v (1 , . . . , r + e , r + 2 , r + 5 , r + 7 , . . . , je − e + 1)= ψ r +2 v (1 , . . . , r + e , r, r + 5 , r + 7 , . . . , je − e + 1)= ψ r − ↓ r + e +22 ψ r +2 v (1 , . . . , r + e +22 , r + 5 , r + 7 , . . . , je − e + 1)= 0 by the above inductive hypothesis.We now come to our second family of homomorphisms. Unlike α k,j in Proposition 4.5, the imageof the cyclic generator under a homomorphism γ k,j below is not a linear combination of basis vectorsindexed by brick-tableaux, as the generator has a different residue sequence. Proposition 4.7. There is a degree j Specht module homomorphism γ k,j : S (( ke,je − e +1) , ( e − −→ S (( ke ) , ( je )) z (( ke,je − e +1) , ( e − (cid:55)−→ v (1 , , . . . , e − , e + 1 , e + 3 , e + 5 , . . . , je − e + 1) . Proof. We will prove this for e (cid:54) = 2, and omit the proof for e = 2 – it is similar enough in spirit, butrather lengthy.Let λ = (( ke ) , ( je )). We show that v (1 , , . . . , e − , e + 1 , e + 3 , e + 5 , . . . , je − e + 1) satisfies therelations that z (( ke,je − e +1) , ( e − satisfies in Definition 2.4.(i) We first show that all y terms kill v (1 , , . . . , e − , e + 1 , e + 3 , e + 5 , . . . , je − e + 1). (cid:5) For r ∈ { , . . . , e − } ∪ { je − e + 2 , . . . , ke + je } , it is obvious that y r v (1 , , . . . , e − , e +1 , e + 3 , e + 5 , . . . , je − e + 1) = 0. (cid:5) Suppose that r ∈ { e, e + 2 , e + 4 , . . . , je − e } , and let r = e + 2 i for 0 (cid:54) i (cid:54) je − e . Observethat y r v (1 , , . . . , e − , e + 1 , e + 3 , e + 5 , . . . , je − e + 1) = (cid:15) · y r ψ r ↓ e + i ψ r +2 ↓ e + i +1 ψ r +4 ↓ e + i +2 . . . ψ je − e ↓ je z λ , where (cid:15) := ψ e ψ e +2 ↓ e +1 ψ e +4 ↓ e +2 . . . ψ r − ↓ e + i − . To prove that the above expression is zero, we now proceed to show that y r + ae ψ r + ae ↓ ( a +1) e + i ψ r + ae +2 ↓ ( a +1) e + i +1 ψ r + ae +4 ↓ ( a +1) e + i +2 . . . ψ je − ( a +1) e ↓ je z λ = 06 Robert Muth, Liron Speyer & Louise Sutton for all a ∈ { , . . . , j − } . We proceed by reverse induction on a . For a = j − 1, we musthave i = 0, in which case this expression becomes( y je ψ je ( − , z λ = ψ je y je +1 z λ = 0 . Now assuming the above statement holds for some a − < j − 1, we have( y r + ae ψ r + ae ( i − , i )) ψ r + ae − ↓ ( a +1) e + i ψ r + ae +2 ↓ ( a +1) e + i +1 ψ r + ae +4 ↓ ( a +1) e + i +2 . . . ψ je − ( a +1) e ↓ je z λ = ψ r + ae ↓ ( a +1) e + i ψ r + ae +2 ( y r + ae +1 ψ r + ae +1 ( i, i )) ψ r + ae ↓ ( a +1) e + i +1 ψ r + ae +4 ↓ ( a +1) e + i +2 ψ r + ae +6 ↓ ( a +1) e + i +3 . . . ψ je − ( a +1) e ↓ je z λ = ψ r + ae ↓ ( a +1) e + i ψ r + ae +2 ( ψ r + ae +1 y r + ae +2 − ψ r + ae ↓ ( a +1) e + i +1 ψ r + ae +4 ↓ ( a +1) e + i +2 ψ r + ae +6 ↓ ( a +1) e + i +3 . . . ψ je − ( a +1) e ↓ je z λ The first term is ψ r + ae ↓ ( a +1) e + i ψ r + ae +2 ↓ ( a +1) e + i +1 ψ r + ae +4 ↓ r + ae +3 ( y r + ae +2 ψ r + ae +2 ( i + 1 , i )) ψ r + ae +1 ↓ ( a +1) e + i +2 · ψ r + ae +6 ↓ ( a +1) e + i +3 ψ r + ae +8 ↓ ( a +1) e + i +4 . . . ψ je − ( a +1) e ↓ je z λ = ψ r + ae ↓ ( a +1) e + i ψ r + ae +2 ↓ ( a +1) e + i +1 ψ r + ae +4 ↓ ( a +1) e + i +2 ψ r + ae +6 ↓ r + ae +4 ( y r + ae +3 ψ r + ae +3 ( i + 2 , i )) ψ r + ae +2 ↓ ( a +1) e + i +3 · ψ r + ae +8 ↓ ( a +1) e + i +4 ψ r + ae +10 ↓ ( a +1) e + i +5 . . . ψ je − ( a +1) e ↓ je z λ ... = ψ r + ae ↓ ( a +1) e + i ψ r + ae +2 ↓ ( a +1) e + i +1 ψ r + ae +4 ↓ ( a +1) e + i +2 . . . ψ r + ae +2 e − ↓ ( a +2) e + i − y r + ae + e · ψ r + ae +2 e ↓ ( a +2) e + i ψ r + ae +2 e +2 ↓ ( a +2) e + i +1 ψ r + ae +2 e +4 ↓ ( a +2) e + i +2 . . . ψ je − ( a +1) e ↓ je z λ = (cid:18) ψ r + ae ↓ ( a +1) e + i ψ r + ae +2 ↓ ( a +1) e + i +1 . . . ψ r + ae +2 e − ↓ ( a +2) e + i − (cid:19) (cid:18) ψ r + ae +2 e ↓ r + ae + e +1 ψ r + ae +2 e +2 ↓ r + ae + e +3 . . . ψ je − ( a +1) e ↓ je − ( a +2) e +1 (cid:19) · y r + ae + e ψ r + ae + e ↓ ( a +2) e + i ψ r + ae + e +2 ↓ ( a +2) e + i +1 ψ r + ae + e +4 ↓ ( a +2) e + i +2 . . . ψ je − ( a +2) e ↓ je z λ = 0 by the inductive hypothesis.The second term is − ψ r + ae +2 ( ψ r + ae ψ r + ae − ψ r + ae ( i − , i, i − ψ r + ae − ↓ ( a +1) e + i r + ae − ↓ ( a +1) e + i +1 · ψ r + ae +4 ↓ ( a +1) e + i +2 ψ r + ae +6 ↓ ( a +1) e + i +3 . . . ψ je − ( a +1) e ↓ je z λ = ψ r + ae +2 ( ψ r + ae − ψ r + ae ψ r + ae − + 1) ψ r + ae − ↓ ( a +1) e + i r + ae − ↓ ( a +1) e + i +1 ψ r + ae +4 ↓ ( a +1) e + i +2 ψ r + ae +6 ↓ ( a +1) e + i +3 . . . ψ je − ( a +1) e ↓ je z λ . The first term of this expression is zero by Lemma 4.1(i), while the second term is zero bya computation analogous to Lemma 4.6. (cid:5) Suppose that r ∈ { e + 1 , e + 3 , . . . , je − e + 1 } , and let r = e + 2 i + 1 for 0 (cid:54) i (cid:54) je − e .Observe that y r v (1 , , . . . , e − , e + 1 , e + 3 , e + 5 , . . . , je − e + 1) = (cid:15) · y r ψ r − ↓ i + e ψ r +1 ↓ i + e +1 ψ r +3 ↓ i + e +2 . . . ψ je − e ↓ je z λ , ecomposable Specht modules indexed by bihooks II (cid:15) := ψ e ψ e +2 ↓ e +1 ψ e +4 ↓ e +2 . . . ψ r − ↓ i + e − . It thus suffices to show that y r ψ r − ↓ i + e ψ r +1 ↓ i + e +1 ψ r +3 ↓ i + e +2 . . . ψ je − e ↓ je z λ = 0. We have y r ψ r − ↓ i + e ψ r +1 ↓ i + e +1 ψ r +3 ↓ i + e +2 . . . ψ je − e ↓ je z λ = ( y r ψ r − ( i − , i )) ψ r − ↓ i + e ψ r +1 ↓ i + e +1 ψ r +3 ↓ i + e +2 . . . ψ je − e ↓ je z λ = ψ r − y r − ψ r − ↓ i + e ψ r +1 ↓ i + e +1 ψ r +3 ↓ i + e +2 . . . ψ je − e ↓ je z λ . If i = 0, then the above is ψ e y e ψ e +2 ↓ e +1 ψ e +4 ↓ e +2 . . . ψ je − e ↓ je z λ = ψ e ψ e +2 ↓ e +1 ψ e +4 ↓ e +2 . . . ψ je − e ↓ je y e z λ = 0 . If i > 0, then the above becomes ψ r − ( y r − ψ r − ( i − , i − ψ r − ↓ i + e ψ r +1 ↓ i + e +1 ψ r +3 ↓ i + e +2 . . . ψ je − e ↓ je z λ = ψ r − ( ψ r − y r − + 1) ψ r − ↓ i + e ψ r +1 ↓ i + e +1 ψ r +3 ↓ i + e +2 . . . ψ je − e ↓ je z λ . The second term of this expression becomes ψ r − ψ r − ↓ i + e ψ r +1 ↓ i + e +1 ψ r +3 ↓ i + e +2 . . . ψ je − e ↓ je z λ = ψ r − ↓ i + e ψ r − v (1 , . . . , i + e, r + 2 , r + 4 , . . . , je − e + 1)= 0 by Lemma 4.6.If i = 1, then the first term becomes ψ e +2 ↓ e +1 y e +1 ψ e +4 ↓ e +2 ψ e +5 ↓ e +3 . . . ψ je − e ↓ je z λ = ψ e +2 ↓ e +1 ψ e +4 ↓ e +2 ψ e +5 ↓ e +3 . . . ψ je − e ↓ je y e +1 z λ = 0 . If i > 1, then this term becomes ψ r − ↓ r − y r − v (1 , . . . , i + e − , r − , r + 2 , r + 4 , . . . , je − e + 1) = 0 , by Lemma 4.2(ii). If e > 5, we have ψ r − ψ r − ( y r − ψ r − ( i − , i − ψ r − ↓ i + e ψ r +1 ↓ i + e +1 ψ r +3 ↓ i + e +2 . . . ψ je − e ↓ je z λ = ψ r − ψ r − ψ r − y r − ψ r − ↓ i + e ψ r +1 ↓ i + e +1 ψ r +3 ↓ i + e +2 . . . ψ je − e ↓ je z λ . If i = 2, then this expression becomes ψ e +4 ↓ e +2 y e +2 ψ e +6 ↓ e +3 ψ e +8 ↓ e +4 . . . ψ je − e ↓ je z λ = ψ e +4 ↓ e +2 ψ e +6 ↓ e +3 ψ e +8 ↓ e +4 . . . ψ je − e ↓ je y e +2 z λ = 0 . Now assuming that i > 2, the expression is ψ r − ψ r − ψ r − y r − v (1 , . . . , i + e − , r − , r + 2 , r + 4 , . . . , je − e + 1) = 0by Lemma 4.2(ii) if e = 6. If e > 6, we continue in this way until we reach ψ r − ↓ i +4 y i +4 v (1 , . . . , i + e − , i + 4 , r + 2 , r + 4 , . . . , je − e + 1) = 0by Lemma 4.2(ii).8 Robert Muth, Liron Speyer & Louise Sutton (ii) We now show that v (1 , . . . , e − , e + 1 , e + 3 , . . . , je − e + 1) satisfies each of the ψ relations inDefinition 2.4. (cid:5) For all r ∈ { , . . . , e − } ∪ { je − e + 2 , . . . , ke + je − } , it is obvious that ψ r kills v (1 , . . . , e − , e + 1 , e + 3 , . . . , je − e + 1). (cid:5) Suppose that r ∈ { e, e + 2 , e + 4 , . . . , je − e } , and let r = e + 2 i for 0 (cid:54) i (cid:54) je − e . Then ψ r ψ r +1 v (1 , . . . , e − , e + 1 , e + 3 , . . . , je − e + 1)= ψ e ψ e +2 ↓ e +1 ψ e +4 ↓ e +2 . . . ψ r − ↓ e + i − ψ r v (1 , . . . , e + i − , r + 2 , r + 3 , r + 5 , r + 7 , . . . , je − e + 1)= 0 by Lemma 4.1(ii) . (cid:5) Suppose that r ∈ { e + 2 , e + 4 , . . . , je − e } , and let r = e + 2 i for 1 (cid:54) i (cid:54) je − e . Then ψ r ψ r − v (1 , . . . , e − , e + 1 , e + 3 , . . . , je − e + 1)= ψ e ψ e +2 ↓ e +1 ψ e +4 ↓ e +2 . . . ψ r − ↓ e + i − ψ r v (1 , . . . , e + i − , r, r + 1 , r + 3 , r + 5 , . . . , je − e + 1)= 0 by Lemma 4.1(i). (cid:5) Suppose that r ∈ { e, e + 2 , e + 4 , . . . , je − e } , and let r = e + 2 i for some 0 (cid:54) i (cid:54) je − e .Observe that ψ r v (1 , . . . , e − , e + 1 , e + 3 , . . . , je − e + 1)= ψ e ψ e +2 ↓ e +1 ψ e +4 ↓ e +2 . . . ψ r − ↓ e + i − (cid:0) ψ r e ( i − , i ) (cid:1) ψ r − ↓ e + i ψ r +2 ↓ e + i +1 ψ r +4 ↓ e + i +2 . . . ψ je − e ↓ je z λ = ψ e ψ e +2 ↓ e +1 ψ e +4 ↓ e +2 . . . ψ r − ↓ e + i − ( y r − y r +1 ) ψ r − ↓ e + i ψ r +2 ↓ e + i +1 ψ r +4 ↓ e + i +2 . . . ψ je − e ↓ je z λ . In the first term, we have ψ e ψ e +2 ↓ e +1 ψ e +4 ↓ e +2 . . . ψ r − ↓ e + i − ( y r ψ r − e ( i − , i − ψ r − ↓ e + i ψ r +2 ↓ e + i +1 ψ r +4 ↓ e + i +2 . . . ψ je − e ↓ je z λ = ψ e ψ e +2 ↓ e +1 ψ e +4 ↓ e +2 . . . ψ r − ↓ e + i − ( ψ r − y r − + 1) ψ r − ↓ e + i ψ r +2 ↓ e + i +1 ψ r +4 ↓ e + i +2 . . . ψ je − e ↓ je z λ , and the first term can be shown to be 0, as at the end of part (i) of this proof. We now let ζ := ψ r +2 ↓ e + i +1 ψ r +4 ↓ e + i +2 . . . ψ je − e ↓ je . By applying Lemma 4.1(i) throughout, we thus have ψ e ψ e +2 ↓ e +1 ψ e +4 ↓ e +2 . . . ψ r − ↓ e + i − ( ψ r − ψ r − ψ r − e ( i − , i − , i − ψ r − ↓ e + i − ψ r − ↓ e + i · ζz λ = ψ e ψ e +2 ↓ e +1 ψ e +4 ↓ e +2 . . . ψ r − ↓ e + i − ( (cid:40)(cid:40)(cid:40)(cid:40)(cid:40)(cid:40)(cid:40) ψ r − ψ r − ψ r − + 1) ψ r − ↓ e + i − ψ r − ↓ e + i · ζz λ = ψ e ψ e +2 ↓ e +1 ψ e +4 ↓ e +2 . . . ψ r − ↓ e + i − ( ψ r − ψ r − ψ r − e ( i − , i − , i − ψ r − ↓ e + i − ψ r − ↓ e + i − ψ r − ↓ e + i · ζz λ = ψ e ψ e +2 ↓ e +1 ψ e +4 ↓ e +2 . . . ψ r − ↓ e + i − ( (cid:40)(cid:40)(cid:40)(cid:40)(cid:40)(cid:40)(cid:40) ψ r − ψ r − ψ r − + 1) ψ r − ↓ e + i − ψ r − ↓ e + i − ψ r − ↓ e + i · ζz λ ... = ψ e ψ e +2 ↓ e +1 ( ψ e +4 ψ e +3 ψ e +4 e (1 , , ψ e +2 ψ e +3 ψ e +5 ↓ e +4 ψ e +7 ↓ e +5 . . . ψ r − ↓ e + i · ζz λ ecomposable Specht modules indexed by bihooks II ψ e ψ e +2 ↓ e +1 ( (cid:40)(cid:40)(cid:40)(cid:40)(cid:40)(cid:40)(cid:40) ψ e +3 ψ e +4 ψ e +3 + 1) ψ e +2 ψ e +3 ψ e +5 ↓ e +4 ψ e +7 ↓ e +5 . . . ψ r − ↓ e + i · ζz λ = ψ e ( ψ e +2 ψ e +1 ψ e +2 e (0 , , ψ e +3 ψ e +5 ↓ e +4 ψ e +7 ↓ e +5 . . . ψ r − ↓ e + i · ζz λ = ψ e ( ψ e +1 ψ e +2 ψ e +1 + 1) ψ e +3 ψ e +5 ↓ e +4 ψ e +7 ↓ e +5 . . . ψ r − ↓ e + i · ζz λ = ψ e ψ e +1 ψ e +2 ψ e +3 ψ e +5 ↓ e +4 ψ e +7 ↓ e +5 . . . ψ r − ↓ e + i · ζ ( ψ e +1 z λ ) + ψ e +3 ψ e +5 ↓ e +4 ψ e +7 ↓ e +5 . . . ψ r − ↓ e + i · ζ ( ψ e z λ )= 0 . Now let η := ψ e ψ e +2 ↓ e +1 ψ e +4 ↓ e +2 . . . ψ r − ↓ e + i − . Then the second y term of the expression above becomes η · ψ r − ↓ e + i ψ r +2 ( y r +1 ψ r +1 ( i, i )) ψ r ↓ e + i +1 ψ r +4 ↓ e + i +2 ψ r +6 ↓ e + i +3 . . . ψ je − e ↓ je z λ = η · ψ r − ↓ e + i ψ r +2 ( ψ r +1 y r +2 − ψ r ↓ e + i +1 ψ r +4 ↓ e + i +2 ψ r +6 ↓ e + i +3 . . . ψ je − e ↓ je z λ , where ψ r +2 ψ r +4 ↓ e + i +2 ψ r +6 ↓ e + i +3 . . . ψ je − e ↓ je z λ = 0 by Lemma 4.6. We thus have η · ψ r − ↓ e + i ψ r +2 ↓ e + i +1 y r +2 ψ r +4 ↓ e + i +2 ψ r +6 ↓ e + i +3 . . . ψ je − e ↓ je z λ = η · ψ r − ↓ e + i ψ r +2 ↓ e + i +1 ψ r +4 ↓ r +3 ψ r +6 ↓ r +5 . . . ψ je − e ↓ je − e − y r +2 ψ r +2 ↓ e + i +2 ψ r +4 ↓ e + i +3 . . . ψ je − e − ↓ je z λ , where y r +2 ψ r +2 ↓ e + i +2 ψ r +4 ↓ e + i +3 . . . ψ je − e − ↓ je z λ = 0 as in the proof of the y r relations above.Finally, we prove that γ k,j has degree j . If e = 2, then T (( ke,je − e +1) , ( e − has degree 2 j and the degreeof the tableau corresponding to v (1 , , . . . , e − , e + 1 , e + 3 , e + 5 , . . . , je − e + 1) is 3 j . If e > 2, theabove degrees are 1 and j + 1, respectively. To begin with, we introduce some notation. We will write M ∼ = L | L | . . . | L r to mean that the module M is uniserial with composition factors L , . . . , L r listed from socle to head.We will write M ∼ = L | . . . | L r ⊕ · · · ⊕ N | . . . | N r to mean that the module is isomorphic to the direct sum of uniserial modules L | . . . | L r , . . . , N | . . . | N r .We will also have occasion, in Proposition 5.13(vi) and Example 5.21, to indicate module structurevia Alperin diagrams, see [Alp80].We will assume that k, j (cid:62) (( ke ) , ( je )) is simple. First,we handle the case where j = 1. Proposition 5.1. Suppose k (cid:62) , and let n = ke + e . Then if p (cid:45) k + 1 , S (( ke ) , ( e )) is semisimple andis isomorphic to D (( ke, , ( e − (cid:104) (cid:105) ⊕ D (( n ) , ∅ ) (cid:104) (cid:105) . If p | k + 1 , then S (( ke ) , ( e )) ∼ = D (( n ) , ∅ ) (cid:104) (cid:105) | D (( ke, , ( e − (cid:104) (cid:105) | D (( n ) , ∅ ) (cid:104) (cid:105) . Robert Muth, Liron Speyer & Louise Sutton Proof. Using the Morita equivalence M of Section 3, we have that M (S (( ke ) , ( e )) ) ∼ = ∆(1 k ) ⊗ ∆(1), whichhas a filtration by the modules ∆(1 k +1 ) = L (cid:0) k +1 (cid:1) and ∆(2 , k − ). By Corollary 2.12, ∆(2 , k − )is irreducible if and only if p (cid:45) k + 1. Since ∆(1 k ) ⊗ ∆(1) must be self-dual by Proposition 3.13, itfollows that if p (cid:45) k + 1, ∆(1 k ) ⊗ ∆(1) ∼ = L (cid:0) k +1 (cid:1) ⊕ L (cid:0) , k − (cid:1) . This implies that M (S (( ke ) , ( e )) ) mustalso be a direct sum of two simple modules, since it is the Morita pre-image of ∆(1 k ) ⊗ ∆(1). Finally,by Propositions 4.5 and 4.7, we know that D (( ke, , ( e − (cid:104) (cid:105) and D (( n ) , ∅ ) (cid:104) (cid:105) are composition factors of M (S (( ke ) , ( e )) ), which completes the proof if p (cid:45) k + 1.If p | k + 1, then ∆(2 , k − ) ∼ = L (cid:0) k +1 (cid:1) | L (cid:0) , k − (cid:1) . Since ∆(1 k ) ⊗ ∆(1) must be self-dual, itfollows that ∆(1 k ) ⊗ ∆(1) ∼ = L (cid:0) k +1 (cid:1) | L (cid:0) , k − (cid:1) | L (cid:0) k +1 (cid:1) . Proposition 4.5 tells us that D (( n ) , ∅ ) (cid:104) (cid:105) is a submodule of S (( ke ) , ( e )) , while Proposition 4.7 implies that D (( ke, , ( e − (cid:104) (cid:105) is a composition factorof S (( ke ) , ( e )) . It follows that S (( ke ) , ( e )) ∼ = D (( n ) , ∅ ) (cid:104) (cid:105) | D (( ke, , ( e − (cid:104) (cid:105) | D (( n ) , ∅ ) (cid:104) r (cid:105) for some r ∈ Z . Tosee that r = 1 as claimed, we apply Proposition 3.13. In this section, we will handle all the semisimple Specht modules and determine their decompositions. Theorem 5.2. Suppose k (cid:62) j (cid:62) . Then S (( ke ) , ( je )) is semisimple if and only if one of the followingholds: (cid:5) p (cid:54) = 2 and p does not divide any of the integers k + j, k + j − , . . . , k − j + 2 ; (cid:5) p = 2 , j = 1 , and k is even; (cid:5) p = 2 , j = 2 , and k ≡ .When semisimple, S (( ke ) , ( je )) is isomorphic to D ((( k + j − e, , ( e − (cid:104) j (cid:105) ⊕ D ((( k + j − e,e +1) , ( e − (cid:104) j (cid:105) ⊕ · · · ⊕ D (( ke, ( j − e +1) , ( e − (cid:104) j (cid:105) ⊕ D (( ke + je ) , ∅ ) (cid:104) j (cid:105) = j (cid:77) r =1 D ((( k + j − r ) e, ( r − e +1) , ( e − (cid:104) j (cid:105) ⊕ D (( ke + je ) , ∅ ) (cid:104) j (cid:105) . Proof. For j = 1, the result is contained in Proposition 5.1. So assume now that j > M , our Specht module is mapped to a module that isfiltered by the Weyl modules ∆(1 k + j ) , ∆(2 , k + j − ) , . . . , ∆(2 j , k − j ). Since Weyl modules are always indecomposable, S (( ke ) , ( je )) cannot possibly be semisimple unless each of those Weyl modules is irre-ducible. By Corollary 2.12, this happens exactly in the cases of our theorem statement, which provesthe ‘only if’ part of the theorem.For the ‘if’ part, it suffices to note that under the stated conditions, M (S (( ke ) , ( je )) ) is a self-dualmodule with simple factors L (cid:0) k + j (cid:1) , L (cid:0) , k + j − (cid:1) , . . . , L (cid:0) j , k − j (cid:1) , each occurring exactly once.Finally, we prove the stated decomposition by induction on j , keeping k + j fixed. For j = 1, theresult is contained in Proposition 5.1, so we may assume thatS (( ke + e ) , ( je − e )) ∼ = j − (cid:77) r =1 D ((( k + j − r ) e, ( r − e +1) , ( e − ⊕ D (( ke + je ) , ∅ ) . We also know that M (S (( ke ) , ( je )) ) = L (cid:16) k + j (cid:17) ⊕ L (cid:16) , k + j − (cid:17) ⊕ · · · ⊕ L (cid:16) j − , k − j +2 (cid:17) , while M (S (( ke + e ) , ( je − e )) ) = L (cid:16) k + j (cid:17) ⊕ L (cid:16) , k + j − (cid:17) ⊕ · · · ⊕ L (cid:16) j − , k − j +2 (cid:17) ⊕ L (cid:16) j , k − j (cid:17) . It follows that M (D (( ke + je ) , ∅ ) ) ∼ = L (cid:0) k + j (cid:1) and M (D ((( k + j − r ) e, ( r − e +1) , ( e − ) ∼ = L (cid:0) r , k + j − r (cid:1) for r = 1 , . . . , j − 1. To see that M (D (( ke,je − e +1) , ( e − ) ∼ = L (cid:0) j , k − j (cid:1) , it suffices to note that S (( ke + e ) , ( je − e )) ⊆ ecomposable Specht modules indexed by bihooks II (( ke ) , ( je )) , and that these two Specht modules only differ by a single extra simple summand whose im-age under M is L (cid:0) j , k − j (cid:1) . Proposition 4.7 tells us that D (( ke,je − e +1) , ( e − (cid:104) j (cid:105) is a composition factorof S (( ke ) , ( je )) . Since this simple is not already accounted for, the result follows. To see the gradingshifts of the other simple factors, we may argue the same as in the proof of Proposition 5.1. Example. For e = 3, k = 7, and j = 5,S ((21) , (15)) (cid:104)− (cid:105) ∼ = D ((36) , ∅ ) ⊕ D ((33 , , (2)) ⊕ D ((30 , , (2)) ⊕ D ((27 , , (2)) ⊕ D ((24 , , (2)) ⊕ D ((21 , , (2)) . It will be useful to introduce some notation for the combinatorial map on labels matching an L ( λ )with D µ that is implicit in the above proof. Definition 5.3. We let T denote the combinatorial map that sends a two-column partition µ to abipartition λ precisely when M (D λ ) = L ( µ ). In particular, given a two-column partition µ of n , wedefine T ( µ ) := (cid:40) (( ne ) , ∅ ) if µ = (1 n );((( n − m ) e, ( m − e + 1) , ( e − µ = (2 m , n − m ) with m (cid:62) Example. For any e , T (1 ) = ((7 e ) , ∅ ), T (2 , ) = ((6 e, , ( e − T (2 , ) = ((5 e, e +1) , ( e − T (2 , 1) = ((4 e, e + 1) , ( e − i -induction functors yields: Theorem 5.4. Let λ = (( ke + a, b ) , ( je + a, b )) , for some k (cid:62) j (cid:62) , with < a (cid:54) e and (cid:54) b < e with a + b (cid:54) = e , or for a = b = 0 and set n := | λ | . Suppose that p does not divide any of the integers k + j, k + j − , . . . , k − j + 2 , or else that p = 2 = j and k ≡ . Then, S λ ∼ = (cid:76) jr =1 D (( n − re − a, ( r − e +1+ a ) , ( e − (cid:104) j (cid:105) ⊕ D (( n − a ) , ( a )) (cid:104) j (cid:105) if b = 0 , (cid:76) jr =1 D(( n − re − a − b, ( r − e +1+ a, b − ) , ( e, b ) ) (cid:104) j (cid:105) ⊕ D (( n − a − b, b ) , ( a, b )) (cid:104) j (cid:105) if b (cid:62) , Proof. The proof is in essence a similar argument to that of [SS20, Propositions 4.3 and 4.4]. Asthere, we apply the divided power functors of Subsection 2.9 to S (( ke ) , ( je )) , whose decomposition weknow by Theorem 5.2. In particular, if a + b < e , we examine f (2) e − b f (2) e − b +1 . . . f (2) e − · f (2) a − f (2) a − · · · f (2)0 S (( ke ) , ( je )) , while if a + b > e , we examine f (2) e − b f (2) e − b +1 · · · f (2) a − f (2) a − · f (4) a − · f (2) a f (2) a +1 . . . f (2) e − · f (2) a − f (2) a − · · · f (2)0 S (( ke ) , ( je )) . At each step, we apply a divided power f (2) i to a Specht module S µ , where µ has two addable i -nodes,and thus f (2) i S µ = S ν , where ν is obtained from µ by adding both of these i -nodes. The only exceptionis that if a + b > e , there is one step at which µ has four addable i -nodes, and f (4) i S µ = S ν , where ν isobtained from µ by adding all four of these i -nodes. So each case above yields the Specht module S λ .Now, Theorem 5.2 gives the decomposition of S (( ke ) , ( je )) , and so it suffices to check the result ofapplying our series of divided power functors (which are exact) to the simple modules appearing inthe decomposition. Indeed, we will show that for each simple summand and at each step, we have f (2) i D µ = D ν for some ν , eventually leading to the desired decomposition.First, we begin with the case b = 0, and examine the summand D (( n ) , ∅ ) (cid:104) j (cid:105) . At each step in f (2) a − f (2) a − · · · f (2)0 D (( n ) , ∅ ) , we have some f (2) i D µ , where µ has two addable i -nodes (one at the end ofthe first row of each component), and no removable i -nodes. Thus both addable i -nodes are conormaland by Lemma 2.10, f i D µ has socle D ˜ f i µ (cid:104) (cid:105) and head D ˜ f i µ (cid:104)− (cid:105) . Applying f i once more, we see that f i D µ has socle D ˜ f i µ (cid:104) (cid:105) and head D ˜ f i µ (cid:104)− (cid:105) . However, we know that in the ungraded setting we2 Robert Muth, Liron Speyer & Louise Sutton have f (2) i D µ = D ˜ f i µ ([BK03, Lemma 2.12]). It follows that f i D µ = D ˜ f i µ (cid:104)− (cid:105) ⊕ D ˜ f i µ (cid:104) (cid:105) , and thus f (2) i D µ = D ˜ f i µ .A similar argument applies to the other summands, noting that if i < e − 2, then the bipartition((( j + k − r ) e + i, ( r − e + 1 + i ) , ( e − i -nodes – in the first two rows of thefirst component – and no removable i -nodes. If i = e − 2, then the above bipartition has an addable i -node in each of the first three rows of the first component, and a removable i -node at the end of thefirst row of the second component. In both cases, there are two conormal i -nodes.The case b (cid:54) = 0 is almost identical, with conormal nodes now appearing in the first column ofeach component in many of the steps. The only non-trivial difference is when a + b > e , and we areapplying the functor f (4) a − to a simple module f (2) a f (2) a +1 . . . f (2) e − · f (2) a − f (2) a − · · · f (2)0 D µ . In each case, weare applying f (4) a − to a module D ν = f (2) a f (2) a +1 . . . f (2) e − · f (2) a − f (2) a − · · · f (2)0 D µ indexed by a bipartition ν that has four addable ( a − a − ν has four conormal( a − f a − to the head four times tells us that the desired module hashead D ˜ f (4) a − ν (cid:104)− (cid:105) and a similar application to the socle tells us that the desired module has socleD ˜ f (4) a − ν (cid:104) (cid:105) , both by Lemma 2.10. Since f i ∼ = [4]! f (4) i , it follows that f (4) a − D ν = D ˜ f a − ν . Finally, thedegree shifts are unchanged by application of these functors, since f ( r ) i (D µ (cid:104) k (cid:105) ) = f ( r ) i (D µ ) (cid:104) k (cid:105) .It will be useful to introduce some notation for the combinatorial map on simple labels takinglabels of composition factors of S (( ke ) , ( je )) to the corresponding labels of composition factors of S λ asin the above proof. Definition 5.5. We let F a,b denote the map on bipartitions defined by F a,b ( µ ) = ˜ f e − b ˜ f e − b +1 . . . ˜ f e − · ˜ f a − ˜ f a − · · · ˜ f µ if a + b < e ,˜ f e − b ˜ f e − b +1 · · · ˜ f a − ˜ f a − · ˜ f a − · ˜ f a ˜ f a +1 . . . ˜ f e − · ˜ f a − ˜ f a − · · · ˜ f µ if a + b > e .More explicitly on the bipartitions appearing in Theorem 5.4, F a,b : (( ke + je ) , ∅ ) (cid:55)→ (( ke + je + a, b ) , ( a, b )) and F a,b : (( ke + je − re, re − e +1) , ( e − (cid:55)−→ (cid:40) (( ke + je − re + a, re − e + 1 + a ) , ( e − b = 0,(( ke + je − re + a, re − e + 1 + a, b − ) , ( e, b )) if b > −F a,b as above but with all residue subscripts replaced by their negatives. Example. Let e = 4. Then Theorem 5.2 tells us that S ((4) , (4)) ∼ = D ((4 , , (3)) (cid:104) (cid:105) ⊕ D ((8) , ∅ ) (cid:104) (cid:105) . We getfrom ((4) , (4)) to ((6 , , (6 , F , = ˜ f ˜ f ˜ f , depicted as follows. ˜ f (cid:55)−→ ˜ f (cid:55)−→ ˜ f (cid:55)−→ Theorem 5.4 tells us that S ((6 , , (6 , ∼ = D ((6 , , (4 , (cid:104) (cid:105) ⊕ D ((10 , , (2 , (cid:104) (cid:105) . Now we focus on the firstsummand, letting µ = ((4 , , (3)). It follows from above that the first summand of S ((6 , , (6 , isobtained by applying F , to µ , depicted as follows. ˜ f (cid:55)−→ ˜ f (cid:55)−→ ˜ f (cid:55)−→ ecomposable Specht modules indexed by bihooks II F , = ˜ f ˜ f ˜ f ˜ f to obtain ((6 , ) , (6 , )), depicted as follows. ˜ f (cid:55)−→ ˜ f (cid:55)−→ ˜ f (cid:55)−→ ˜ f (cid:55)−→ Now Theorem 5.4 tells us that S ((6 , ) , (6 , )) ∼ = D ((6 , , ) , (4 , )) (cid:104) (cid:105) ⊕ D ((10 , ) , (2 , )) (cid:104) (cid:105) , so the first sum-mand of S ((6 , ) , (6 , )) is obtained by applying F , to µ , depicted as follows. ˜ f (cid:55)−→ ˜ f (cid:55)−→ ˜ f (cid:55)−→ ˜ f (cid:55)−→ We next introduce and extend some notation from [Sut20, Definition 3.1] in order to work withbihooks that are the transpose of those handled in Theorem 5.4. Definition 5.6. For x ∈ N , we define the following weakly decreasing sequence of e − x . { x } := (cid:22) x + e − e − (cid:23) , (cid:22) x + e − e − (cid:23) , . . . , (cid:22) xe − (cid:23) If λ = ( λ , λ , . . . , λ r ) is a partition, then we define { λ } := ( { λ } , { λ } , . . . , { λ r } ), and analogously if λ = ( λ (1) , λ (2) ) ∈ P n , then { λ } := ( { λ (1) } , { λ (2) } ). Example. Let e = 3, λ = ((5 e ) , ∅ ) = ((15) , ∅ ) and µ = ((3 e, e + 1) , ( e − 1) = ((9 , , (2)). Then { λ } = ((8 , , ∅ ) and { µ } = ((5 , , ) , (1 )). Corollary 5.7. Let λ = (( b + 1 , je + a − ) , ( b + 1 , ke + a − )) , for some k (cid:62) j (cid:62) , with < a (cid:54) e and (cid:54) b < e with a + b (cid:54) = e , or for a = b = 0 and set n := | λ | . Suppose that p does not divide any of theintegers k + j, k + j − , . . . , k − j + 2 , or else that p = 2 = j and k ≡ . Then, S λ ∼ = j (cid:77) r =1 D −F a,b { (( n − re, ( r − e +1) , ( e − } (cid:104) k + j (cid:105) ⊕ D −F a,b { (( n ) , ∅ ) } (cid:104) k + j (cid:105) . Proof. We assume that a = b = 0, and observe that by [KMR12, Theorems 7.25 and 8.5], we have((S (( ke ) , ( je )) ) sgn ) (cid:126) (cid:104) k +2 j (cid:105) ∼ = S ((1 je ) , (1 ke )) (cid:126) (cid:104) k +2 j (cid:105) ∼ = S ((1 je ) , (1 ke )) , and so S ((1 je ) , (1 ke )) ∼ = (cid:77) ν D m e,κ ( ν ) (cid:104) k + j (cid:105) , where the sum is over all ν appearing in Theorem 5.2.Thus we must compute the images of the bipartitions ν under the Mullineux map. First, it issimple to check that m e,κ ((( n ) , ∅ )) = ( ˜ f ˜ f . . . ˜ f − ˜ f ) k + j ∅ = { (( n ) , ∅ ) } . Robert Muth, Liron Speyer & Louise Sutton Likewise, it’s not so difficult to see that m e,κ (((( k + j − r ) e, ( r − e + 1) , ( e − f ˜ f . . . ˜ f − ˜ f ) r ( ˜ f ˜ f . . . ˜ f − ˜ f ) k + j − r ∅ . Note that ˜ f . . . ˜ f − ˜ f ( ˜ f ˜ f . . . ˜ f − ˜ f ) k + j − r ∅ = { ((( k + j − r + 1) e − , (1 e − )) } and that applying ( ˜ f ˜ f . . . ˜ f − ˜ f ) r − ˜ f to this adds cogood nodes alternately between one of thefirst e − e − −F a,b (((1 je ) , (1 ke ))) = λ , and that −F a,b is well-behaved on all bipartitions weare applying it to in the statement of the corollary; by this, we mean that at each step we add themaximal number of conormal nodes, so that the corresponding sequence of i -induction functors sendsa simple module D ν to a simple module D −F a,b ( ν ) . The result follows. Remark. In the corollary above, and its proof, we essentially applied the twist by sign and then applied −F a,b . We could have done these the other way round, taking our bipartitions in Theorem 5.2 andapplying F a,b first, followed by the sign-twist. j Here, we further examine some of the non-semisimple cases, once again starting with the Specht mod-ules S (( ke ) , ( je )) . Once the structures of these modules are determined, we can argue as in Theorem 5.4and Corollary 5.7 to determine the structure of the other decomposable Specht modules that are ob-tained by applying divided power functors, and then a sign-twist. We already completely determinedthe structure of S (( ke ) , ( je )) when j = 1 in Proposition 5.1, and for j > (( ke ) , ( je )) is not semisimple. Of course, we have more bihooks corresponding to decomposableSpecht modules for this ‘ j = 1 situation’. We can apply the argument from Theorem 5.4 to the j = 1case in Proposition 5.1 to yield the following. Corollary 5.8. Suppose k (cid:62) , and let λ = (( ke + a, b ) , ( e + a, b )) , for some k (cid:62) , with < a (cid:54) e and (cid:54) b < e with a + b (cid:54) = e , or for a = b = 0 . Then if p (cid:45) k + 1 , S λ is semisimple, and moreover S λ ∼ = D (( ke + e + a ) , ( a )) (cid:104) (cid:105) ⊕ D (( ke + a,a +1) , ( e − (cid:104) (cid:105) if b = 0 , D (( ke + e + a, b ) , ( a, b )) (cid:104) (cid:105) ⊕ D(( ke + a,a +1 , b − ) , ( e, b ) ) (cid:104) (cid:105) if b (cid:62) . If p | k + 1 , then S λ is non-semisimple, and moreover S λ ∼ = D (( ke + e + a ) , ( a )) (cid:104) (cid:105) | D (( ke + a,a +1) , ( e − (cid:104) (cid:105) | D (( ke + e + a ) , ( a )) (cid:104) (cid:105) if b = 0 , D (( ke + e + a, b ) , ( a, b )) (cid:104) (cid:105) | D(( ke + a,a +1 , b − ) , ( e, b ) ) (cid:104) (cid:105) | D (( ke + e + a, b ) , ( a, b )) (cid:104) (cid:105) if b (cid:62) . Proof. If p (cid:45) k + 1, this is just a special case of Theorem 5.4. If p | k + 1, then we apply divided powerfunctors as in the proof of Theorem 5.4, noting that they are exact functors that send simple modulesto simple modules.Thus in the non-semisimple cases we may apply the corresponding (composition of) functors to themodule S (( ke ) , ( je )) ∼ = D (( n ) , ∅ ) (cid:104) (cid:105) | D (( ke, , ( e − (cid:104) (cid:105) | D (( n ) , ∅ ) (cid:104) (cid:105) to obtain the result, using the followingsimple observation.Note that if we have a module M ∼ = A | B | A , for simple modules A and B , and an exact functor F such that F ( A ) and F ( B ) are simple modules, then we may apply F to the short exact sequences0 → A → M → B | A → → A | B → M → A → → F ( A ) → F ( M ) → F ( B | A ) → → F ( A | B ) → F ( M ) → F ( A ) → . ecomposable Specht modules indexed by bihooks II F ( A | B ) ∼ = F ( A ) |F ( B ) and F ( B | A ) ∼ = F ( B ) |F ( A ). It follows that F ( M )must have submodules isomorphic to F ( A ) and F ( A ) |F ( B ) and quotients isomorphic to F ( A ) and F ( B ) |F ( A ), and thus that F ( M ) ∼ = F ( A ) |F ( B ) |F ( A ). The result now follows by applying this(with F being the appropriate composition of divided power functors, as in the proof of Theorem 5.4)to S (( ke ) , ( je )) ∼ = D (( n ) , ∅ ) (cid:104) (cid:105) | D (( ke, , ( e − (cid:104) (cid:105) | D (( n ) , ∅ ) (cid:104) (cid:105) , noting that F (D (( n ) , ∅ ) (cid:104) (cid:105) ) ∼ = (cid:40) D (( ke + e + a ) , ( a )) (cid:104) (cid:105) if b = 0 , D (( ke + e + a, b ) , ( a, b )) (cid:104) (cid:105) if b (cid:62) F (D (( ke, , ( e − (cid:104) (cid:105) ) ∼ = (cid:40) D (( ke + a,a +1) , ( e − (cid:104) (cid:105) if b = 0 , D(( ke + a,a +1 , b − ) , ( e, b ) ) (cid:104) (cid:105) if b (cid:62) . The result now follows.As in Corollary 5.7 and the remark thereafter, we may twist our Specht modules by sign to obtainthe result for the conjugate bipartitions. Corollary 5.9. Suppose k (cid:62) , and let λ = (( b + 1 , e + a − ) , ( b + 1 , ke + a − )) , for some k (cid:62) , with < a (cid:54) e and (cid:54) b < e with a + b (cid:54) = e , or for a = b = 0 . Then S λ is semisimple if and only if p (cid:45) k + 1 , and the structure of S λ is given by applying the Mullineux map m e,κ to all of the bipartitionsappearing in Corollary 5.7.Remark. When j = 1, observe that the set of all decomposable Specht modules indexed by bihookscoincides with those that are semisimple.The next natural case to handle is j = 2, since this will leave us with a uniform condition for thenon-semisimplicity in the remaining cases. The following lemma will enable us to apply results forthe Schur algebra to our Specht modules when they are not semisimple. Lemma 5.10. Over an arbitrary field F , we have L ( µ ) ∼ = M (D T ( µ ) ) for all two-column partitions µ ∈ P n .Proof. Consider the simple R nδ, C -module D T ( µ ) , C . Following [BK09b, § Z -form M ⊂ D λ, C such that M ⊗ Z C = D T ( µ ) , C , and define the R nδ, F -module J T ( µ ) = M ⊗ Z F . By [BK09b, Theorem5.17], we have [J T ( µ ) : D ν, F ] = (cid:40) ν = T ( µ );0 if ν (cid:82) T ( µ ) . (5.11)As D T ( µ ) , C ∼ = (cid:99) M (L ( µ ) C ), it follows from [KM17a, Lemma 7.1] that all simple factors of J T ( µ ) are ofthe form (cid:99) M (L( γ ) F ), where [J T ( µ ) : (cid:99) M (L ( γ ) F )] = (cid:40) γ = µ ;0 if γ (cid:82) µ. (5.12)Now, going by induction on dominance order on two-column partitions, we show that (cid:99) M (L ( µ ) F ) ∼ =D T ( µ ) , F . Make the induction assumption on µ . By (5.12), the simple factors of J T ( µ ) are (cid:99) M (L ( µ ) F )and modules of the form (cid:99) M (L ( γ ) F ) ∼ = D T ( γ ) , F for γ (cid:67) µ . But we have D T ( µ ) , F (cid:54)∼ = D T ( γ ) , F when γ (cid:67) µ , so it follows from (5.11) that (cid:99) M (L ( µ ) F ) ∼ = D T ( µ ) , F , as required. Remark. If we combine Theorem 2.13, Corollary 3.11 and Lemma 5.10 and the fact that ∆(1 k ) ⊗ ∆(1 j )has a filtration by the Weyl modules ∆(1 k + j ) , ∆(2 , k + j − ) , . . . , ∆(2 j , k − j ), we now have a formulafor readily computing all composition factors of S (( ke ) , ( je )) , and their multiplicities. We remark thatby Lemma 3.10 and Theorem 3.5, all simple factors of S (( ke ) , ( je )) are relatively unshifted with respectto each other; i.e., if one simple is shifted by j , all shifts must be by j .6 Robert Muth, Liron Speyer & Louise Sutton Proposition 5.13. Suppose k (cid:62) , and let n = ke + 2 e . Then(i) if p (cid:54) = 2 and p does not divide any of the integers k + 2 , k + 1 , k , of if p = 2 and k ≡ ,then S (( ke ) , (2 e )) ∼ = D (( ke,e +1) , ( e − (cid:104) (cid:105) ⊕ D (( ke + e, , ( e − (cid:104) (cid:105) ⊕ D (( n ) , ∅ ) (cid:104) (cid:105) ; (ii) if (cid:54) = p | k + 2 , then S (( ke ) , (2 e )) ∼ = D (( ke,e +1) , ( e − (cid:104) (cid:105) ⊕ (cid:0) D (( n ) , ∅ ) (cid:104) (cid:105) | D (( ke + e, , ( e − (cid:104) (cid:105) | D (( n ) , ∅ ) (cid:104) (cid:105) (cid:1) ; (iii) if (cid:54) = p | k + 1 , or if p = 2 and k ≡ then S (( ke ) , (2 e )) ∼ = D (( ke + e, , ( e − (cid:104) (cid:105) ⊕ (cid:0) D (( n ) , ∅ ) (cid:104) (cid:105) | D (( ke,e +1) , ( e − (cid:104) (cid:105) | D (( n ) , ∅ ) (cid:104) (cid:105) (cid:1) ; (iv) if (cid:54) = p | k , then S (( ke ) , (2 e )) ∼ = D (( n ) , ∅ ) (cid:104) (cid:105) ⊕ (cid:0) D (( ke + e, , ( e − (cid:104) (cid:105) | D (( ke,e +1) , ( e − (cid:104) (cid:105) | D (( ke + e, , ( e − (cid:104) (cid:105) (cid:1) ; (v) if p = 2 and k ≡ , then S (( ke ) , (2 e )) ∼ = D (( n ) , ∅ ) (cid:104) (cid:105)⊕ (cid:0) D (( ke + e, , ( e − (cid:104) (cid:105) | D (( n ) , ∅ ) (cid:104) (cid:105) | D (( ke,e +1) , ( e − (cid:104) (cid:105) | D (( n ) , ∅ ) (cid:104) (cid:105) | D (( ke + e, , ( e − (cid:104) (cid:105) (cid:1) ; (vi) if p = 2 and k ≡ , then S (( ke ) , (2 e )) is indecomposable, and S (( ke ) , (2 e )) (cid:104)− (cid:105) ∼ = D (( n ) , ∅ ) D (( ke + e, , ( e − D (( ke,e +1) , ( e − D (( ke + e, , ( e − D (( n ) , ∅ ) Proof. Part (i) is completely handled in Theorem 5.2. For part (ii), we apply Theorem 2.13 to M (S (( ke ) , (2 e )) ) ∼ = ∆(1 k ) ⊗ ∆(1 ), which has a filtration by the Weyl modules ∆(1 k +2 ) = L (cid:0) k +2 (cid:1) ,∆(2 , k ), and ∆(2 , k − ). We may check that ∆(2 , k ) ∼ = L (cid:0) k +2 (cid:1) | L (cid:0) , k (cid:1) and ∆(2 , k − ) =L (cid:0) , k − (cid:1) . Given that S (( ke ) , (2 e )) is decomposable and that each summand is self-dual, it followsthat ∆(1 k ) ⊗ ∆(1 ) ∼ = L (cid:16) , k − (cid:17) ⊕ (cid:16) L (cid:16) k +2 (cid:17) | L (cid:16) , k (cid:17) | L (cid:16) k +2 (cid:17)(cid:17) . Since M (D (( n ) , ∅ ) ) = L (cid:0) k +2 (cid:1) , M (D (( ke + e, , ( e − ) = L (cid:0) , k (cid:1) , and M (D (( ke,e +1) , ( e − ) = L (cid:0) , k − (cid:1) ,by Lemma 5.10, the ungraded result follows.To see the grading shifts, we note that S (( ke + e ) , ( e )) ∼ = D (( n ) , ∅ ) (cid:104) (cid:105) | D (( ke + e, , ( e − (cid:104) (cid:105) | D (( n ) , ∅ ) (cid:104) (cid:105) by Theorem 5.2. It is easy to check that dim(im α k +1 , ) > 1, and thus that im α k +1 , is injective.Propositions 4.5 and 4.7 thus give us the required shifts for all simple factors.Parts (iii) and (iv) are almost identical, and the details of the ungraded result are left to thereader. For the grading, we now have that S (( ke + e ) , ( e )) ∼ = D (( n ) , ∅ ) (cid:104) (cid:105) ⊕ D (( ke + e, , ( e − (cid:104) (cid:105) in both cases,so that Proposition 4.5 provides us the necessary grading shift for the non-trivial simple submodule(which is a summand in the case of part (iii), but not for part (iv)). It is difficult to directlycheck that the homomorphism α k, is injective in this case, so a priori, it may kill the trivial factor.Applying Proposition 4.7 gives the grading shift for D (( ke,e +1) , ( e − . In part (iii), this also gives usthe grading shift on the simple submodule, and the grading shift for the remaining simple factor (theone-dimensional simple head of the non-simple summand) follows by Proposition 3.13, as in the proofof Proposition 5.1. In part (iv), Proposition 4.7 gives the grading shift for D (( ke,e +1) , ( e − and thenon-trivial simple module below it, from which we can once again deduce the grading shift on thesimple head of the non-simple summand. In order to obtain the grading shift on the trivial summand ecomposable Specht modules indexed by bihooks II α k, ◦ α k +1 , : S (( n ) , ∅ ) → S (( ke ) , (2 e )) is a non-zero degree 2 homomorphism –this can be seen by noting that the leading term is ( k +2)( k +1) z (( ke ) , (2 e )) , and that this cannot appearwhen reducing any of the products of Ψ terms that arise when composing the two homomorphisms.For part (v), one can check using Theorem 3.12 that the Schur functor maps ∆(1 k ) ⊗ ∆(1 ) toM ( k, p = 2 and k ≡ , k ) has a copy of L (cid:0) k +2 (cid:1) as a submodule, and ∆(2 , k − ) hasboth L (cid:0) , k (cid:1) and L (cid:0) k +2 (cid:1) appearing as composition factors. There is a unique way to combine thesemodules to make one with two summands, each of which is self-dual. Namely,∆(1 k ) ⊗ ∆(1 ) ∼ = L (cid:16) k +2 (cid:17) ⊕ (cid:16) L (cid:16) , k (cid:17) | L (cid:16) k +2 (cid:17) | L (cid:16) , k − (cid:17) | L (cid:16) k +2 (cid:17) | L (cid:16) , k (cid:17)(cid:17) . Matching labels as in Theorem 5.2 and applying Lemma 5.10 yields the ungraded version of the statedresult. For the grading shifts, Propositions 4.5 and 4.7 give us the required information as in part (ii).In part (vi), one can check using Theorem 3.12 that M ( k, 2) is indecomposable when p = 2 and k ≡ , k ) has a copy of L (cid:0) k +2 (cid:1) as asubmodule, and ∆(2 , k − ) has a copy of L (cid:0) , k (cid:1) as a submodule. With the fact that M ( k, 2) has afiltration by the Weyl modules ∆(1 k +2 ), ∆(2 , k ), and ∆(2 , k − ), it follows that there is once againa unique way to construct this indecomposable self-dual module with these factors, namely∆(1 k ) ⊗ ∆(1 ) ∼ = L (cid:0) k +2 (cid:1) L (cid:0) , k (cid:1) L (cid:0) , k − (cid:1) L (cid:0) , k (cid:1) L (cid:0) k +2 (cid:1) The ungraded result follows as before. The gradings follow from Propositions 4.5 and 4.7 as before. Remark. For the Specht modules in parts (v) and (vi) of the above proposition, we did not determinein [SS20] whether or not they were decomposable, so that even this coarser information is new.Applying the map F a,b to the labels of the above decomposable Specht modules, the observationin the proof of Corollary 5.8 immediately yields the following. Corollary 5.14. Suppose k (cid:62) , and let n = ke + 2 e . For λ = (( ke + a, b ) , (2 e + a, b )) , for some k (cid:62) , with < a (cid:54) e and (cid:54) b < e with a + b (cid:54) = e , or for a = b = 0 , the structure of S λ isobtained from Proposition 5.13 by replacing each simple module D µ (cid:104) (cid:105) therein with D F a,b ( µ ) (cid:104) (cid:105) , andthe structure of S λ (cid:48) is obtained from Proposition 5.13 by replacing each simple module D µ (cid:104) (cid:105) thereinwith D −F a,b ( { µ } ) (cid:104) (cid:105) = D m e,κ ◦F a,b ( µ ) (cid:104) (cid:105) . With the Specht modules for j = 1 and j = 2 completely understood, we will shift away fromexamining cases of small j . First, we will build on parts (v) and (vi) of Proposition 5.13. Obvservethat the condition in the next result is remarkably similar to that of Murphy [Mur80, Theorem 4.5],which determines the decomposability of level 1 Specht modules indexed by hooks. Proposition 5.15. Let k (cid:62) j (cid:62) , p = 2 , and let l be such that l − (cid:54) j < l . Then S (( ke ) , ( je )) isdecomposable if and only if k (cid:54)≡ j (mod 2 l ) .Proof. Using the Morita equivalence M as before, it suffices to show that M ( k, j ) ∼ = Y ( k, j ) if andonly if k ≡ j (mod 2 l ). In other words, we will show that if 0 (cid:54) m < j , some Y ( k + j − m, m ) is asummand of M ( k, j ) if and only if k (cid:54)≡ j (mod 2 l ). We use Theorem 3.12 at length, and we proceedto determine if j − m (cid:54) k + j − m .Set m = j − a for a ∈ { , , . . . , l − } . In order for Y ( k + j − m, m ) to not be a summand ofM ( k, j ), we must have k − j ≡ , , . . . , a − a +1 ). In order for this to hold simultaneouslyfor all a ∈ { , , . . . , l − } , we must have k − j ≡ l ). We thus have shown that if k (cid:54)≡ j (mod 2 l ), then S (( ke ) , ( je )) is decomposable.8 Robert Muth, Liron Speyer & Louise Sutton It remains to show that when k ≡ j (mod 2 l ), no Y ( k + j − m, m ) appears as a summand ofM ( k, j ), for m not of the form considered above. If k ≡ j (mod 2 l ), then k + j − m ≡ j − m )(mod 2 l ). It follows immediately that j − m (cid:54) (cid:54) k + j − m and the result follows.By applying i -induction functors, we deduce the following corollary. Corollary 5.16. Let p = 2 , λ = (( ke + a, b ) , ( je + a, b )) or (( b + 1 , je + a − ) , ( b + 1 , ke + a − )) , forsome j, k > , < a (cid:54) e and (cid:54) b < e with a + b (cid:54) = e , or for a = b = 0 , and let l be such that l − (cid:54) j < l . Then S λ is decomposable if and only if k (cid:54)≡ j (mod 2 l ) .Remark. The above corollary fills a gap in [SS20, Theorem 4.1], where we were unable to determinethe decomposability of S λ when p = 2 and j + k is even. This also strengthens [SS20, Conjecture 4.2]:if e (cid:54) = 2, we conjecture that [SS20, Theorems 3.8 and 4.1] and Corollary 5.16 provide a complete listof decomposable Specht modules indexed by bihooks. Next, we examine cases that are ‘close to being semisimple’, in the sense that every direct summandcontains few simple factors. We saw in Theorem 5.2 that S (( ke ) , ( je )) is semisimple when p does notdivide any of the integers k + j, k + j − , . . . , k − j + 2. The next case we look at is when p dividesexactly one of these integers. Proposition 5.17. Let k (cid:62) j > , m ∈ { , . . . , j } , λ = (2 m , n − m ) where n = k + j , and supposethat p divides exactly one of the integers n, n − , . . . , n − j +2 , so that n ∈ { ap, ap +1 , . . . , ap +2 j − } for some a ∈ N . We write n = ap + i for i ∈ { , , . . . , j − } .(i) If ( m, j ) (cid:54) = ( p, p ) , then ∆( λ ) ∼ = (cid:40) L ( λ ) if m ∈ { , , . . . , (cid:98) i +12 (cid:99)} ∪ { i + 2 , i + 3 , . . . , j } ;L (cid:0) i − m +1 , n − i +2 m − (cid:1) | L ( λ ) if m ∈ {(cid:98) i +12 (cid:99) + 1 , . . . , i + 1 } . (ii) If i = j − and ( m, j ) = ( p, p ) , then ∆( λ ) ∼ = (cid:40) L ( λ ) if k + 1 ≡ p (mod p );L (1 n ) | L ( λ ) if k + 1 ≡ p ) . Proof. Let µ = (2 l , n − l ) for 0 (cid:54) l (cid:54) m and compute [∆( λ ) : L ( µ )]. Since [∆( λ ) : L ( λ )] = 1, wesuppose that l < m and apply Theorem 2.13 to determine [∆( λ ) : L ( µ )], which is either 0 or 1.Since p divides exactly one of n, n − , . . . , n − j + 2, it follows that p (cid:62) j (cid:62) m > l (cid:62) 0. Thus,with the exception of the special case where p = j = m , l = 0, we have that (cid:98) m − lp (cid:99) = 0 (cid:52) p (cid:98) n − l +1 p (cid:99) and p (cid:45) m − l . Hence [∆( λ ) : L ( µ )] = 1 if and only if p | n − m − l + 1. The exceptional case only occurswhen p | k + 1 (so that i = j − n ) occurs as a compositionfactor of ∆(2 j , n − j ). It is easy to see that in this case it is a composition factor precisely when p | ( k + 1). This situation occurs in case (ii) of the proposition statement, which completes the proofof that case. We now treat the remaining case.First suppose that i ∈ { , , . . . , j − } . Then i − m − l + 1 (cid:54) i − m + 1 (cid:54) j − − m < j − m < j (cid:54) p and i − m − l + 1 (cid:62) i − m + 2 (cid:62) − m + 2 (cid:62) − j + 2 > − p. It follows that − p < i − m − l + 1 < p , and thus p | n − m − l + 1 if and only if l = i − m + 1 or l = i − m + 1 + p . Since i ∈ { , , . . . , j − } , the condition that p divides k + j − i and no other integersin [ k − j + 2 , k + j ] implies that p (cid:62) j − i − 1, and hence that i − m + 1 + p (cid:62) m . Since 0 (cid:54) l < m ,it follows that p | n − m − l + 1 can only happen when l = i − m + 1 and m ∈ {(cid:98) i +12 (cid:99) + 1 , . . . , i + 1 } . ecomposable Specht modules indexed by bihooks II i ∈ { j − , j, . . . , j − } . Then i − m − l + 1 < i + 1 (cid:54) j − < p and i − m − l + 1 (cid:62) j − m − l (cid:62) − l > − p. So − p < i − m − l + 1 < p , and thus p | n − m − l + 1 if and only if l = i − m + 1 or l = i − m + 1 − p .Since i ∈ { j − , j, . . . , j − } , the condition that p divides k + j − i and no other integers in[ k − j + 2 , k + j ] implies that p (cid:62) i + 1, and hence that i − m + 1 − p (cid:54) − m (cid:54) − 1. Once again,we use that 0 (cid:54) l < m to see that p | n − m − l + 1 can only happen when l = i − m + 1 and m ∈ {(cid:98) i +12 (cid:99) + 1 , . . . , i + 1 } ∩ { , , . . . , j } = {(cid:98) i +12 (cid:99) + 1 , . . . , j } .Hence for any i ∈ { , , . . . , j − } , we have [∆( λ ) : L ( µ )] = 1 if and only if l = i − m + 1 and m ∈ {(cid:98) i +12 (cid:99) + 1 , . . . , i + 1 } . Remark. Note that the case ( m, j ) = ( p, p ) only occurs when i = j − 1, which follows from the lowerbounds on p given in the above proof.Next, we decompose M ( n − j, j ) into its indecomposable summands. Proposition 5.18. Let k (cid:62) j > , k + j = n = ap + i for i ∈ { , . . . , j − } , and assume that p divides exactly one of n, n − , . . . , n − j + 2 . Then M ( n − j, j ) ∼ = i − j (cid:77) m =0 Y ( n − m, m ) ⊕ j (cid:77) m = (cid:100) i +12 (cid:101) Y ( n − m, m ) , unless j = p , i = j − (so that p | k + 1 ), and k + 1 ≡ p (mod p ) , in which case M ( n − j, j ) ∼ = Y ( n ) ⊕ j (cid:77) m = (cid:100) i +12 (cid:101) Y ( n − m, m ) . Proof. We apply Theorem 3.12 to determine if Y ( n − m, m ) is a direct summand of M ( n − j, j ) forall m ∈ { , . . . , j } .It is clear that Y ( n − j, j ) is a summand of M ( n − j, j ), so we suppose that m < j . Now,1 (cid:54) j − m (cid:54) j (cid:54) p . Then the p -adic expansion of j − m is [ j − m, , , . . . ] except for when j = p (andthus p | k + 1 and i = j − 1) and m = 0, in which case j − m has p-adic expansion [0 , , , , . . . ].If i ∈ { , , . . . , j − } , then the condition that p divides k + j − i and no other integers in[ k − j + 2 , k + j ] implies that p (cid:62) j − i − 1, and therefore that p + i − m (cid:62) j − − m (cid:62) . Similarly, if i ∈ { j − , j, . . . , j − } , we have that p (cid:62) i + 1, and therefore p + i − m (cid:62) i + 1 − m (cid:62) i − j ) + 3 . Thus if i ∈ { , , . . . , j − } and m ∈ { i + 1 , i + 2 , . . . , j − } , we have 0 < p + i − m (cid:54) p − 2. So the p -adic expansion of p + i − m is [ p + i − m, , , . . . ], and j − m (cid:54) p p + i − m (cid:54) p n − m . It followsthat Y ( n − m, m ) is always a summand of M ( n − j, j ) when m ∈ { i + 1 , i + 2 , . . . , j − } .Next, suppose that i ∈ { , , . . . , j − } and m ∈ { , , . . . , i } . Then 0 (cid:54) i − m (cid:54) i < j , so − m (cid:54) i − m < j − m (cid:54) p .If m (cid:54) (cid:98) i/ (cid:99) , then 0 (cid:54) i − m so that the p -adic expansion of i − m = n − ap − m is [ i − m, , , . . . ],and it follows that j − m (cid:54) (cid:54) p n − m , except in the case that j = p and m = 0. In this exceptional case,we must look at the second entry in the p -adic expansion of n . In particular, if k + 1 ≡ p ),then n ≡ p − p ), so that the first two entries of the p -adic expansion of n are p − j − m (cid:54) (cid:54) p n − m again. But if k + 1 ≡ p (mod p ), then n ≡ p − p ), so that thefirst two entries of the p -adic expansion of n are p − j − m (cid:54) p n − m in this case.This is the exceptional case in the statement of the proposition.0 Robert Muth, Liron Speyer & Louise Sutton If m > i/ 2, then 1 (cid:54) p + i − m < p , so that the p -adic expansion of p + i − m = n − ( a − p − m is [ p + i − m, , , . . . ]. Since j (cid:54) p (cid:54) p + i − m , it follows that j − m (cid:54) p + i − m , so that j − m (cid:54) p p + i − m (cid:54) p n − m .Finally, suppose that i ∈ { j, j + 1 , . . . , j − } and m ∈ { , , . . . , i } . Then, as above, p (cid:62) i + 1, sothat we have − p < − i (cid:54) i − m (cid:54) i < p or, equivalently, 0 < p + i − m < p. If m (cid:54) i/ 2, then the above becomes 0 (cid:54) i − m < p , so that the p -adic expansion of i − m is[ i − m, , , . . . ], so that j − m (cid:54) p n − m if and only j − m (cid:54) p i − m , if and only if m (cid:54) i − j . Ifinstead, m > i/ 2, then the above becomes 0 < p + i − m < p , and the result follows as before. Thiscompletes the proof.We now combine the above results to obtain the decomposition of the tensor product of Weylmodules. In the following theorem, we abuse notation and write L ( x ) to mean L (cid:0) x , n − x (cid:1) . Theorem 5.19. Let k (cid:62) j > and suppose that p divides exactly one of the integers k + j, k + j − , . . . , k − j + 2 , so that n = k + j ∈ { ap, ap + 1 , . . . , ap + 2 j − } for some a ∈ N . We write n = ap + i for i ∈ { , , . . . , j − } . For r ∈ { , , . . . , (cid:98) i/ (cid:99)} , define the uniserial module M r = L ( (cid:98) i/ (cid:99) − r ) | L ( (cid:100) i/ (cid:101) + 1 + r ) | L ( (cid:98) i/ (cid:99) − r ) . First we assume that we are not in the case that j = p and k + 1 ≡ p (mod p ) .(i) Suppose that i ∈ { , , . . . , j − } .(a) If i is odd, then ∆( k ) ⊗ ∆( j ) ∼ = L (cid:18) i + 12 (cid:19) ⊕ (cid:98) i (cid:99) (cid:77) r =0 M r ⊕ j (cid:77) m = i +2 L ( m ) (b) If i is even, then ∆( k ) ⊗ ∆( j ) ∼ = (cid:98) i (cid:99) (cid:77) r =0 M r ⊕ j (cid:77) m = i +2 L ( m ) (ii) Suppose that i ∈ { j, j + 1 , . . . , j − } .(a) If i is odd, then ∆( k ) ⊗ ∆( j ) ∼ = i − j (cid:77) m =0 L ( m ) ⊕ L (cid:18) i + 12 (cid:19) ⊕ j −(cid:100) i (cid:101)− (cid:77) r =0 M r (b) If i is even, then ∆( k ) ⊗ ∆( j ) ∼ = i − j (cid:77) m =0 L ( m ) ⊕ j −(cid:100) i (cid:101)− (cid:77) r =0 M r Finally, in the exceptional case that j = p , i = j − , k + 1 ≡ p (mod p ) . Then ∆( k ) ⊗ ∆( j ) ∼ = L (0) ⊕ L (1) ⊕ L (2) if j = p = 2 , L (0) ⊕ (cid:98) i (cid:99)− (cid:77) r =0 M r ⊕ L ( j ) if j = p > . ecomposable Specht modules indexed by bihooks II Proof. We know from Proposition 5.17 that the Weyl module ∆( λ ) is simple if m ∈ { , , . . . , (cid:98) i +12 (cid:99)} ∪{ i + 2 , i + 3 , . . . , j } , whereas if m ∈ {(cid:98) i +12 (cid:99) + 1 , . . . , i + 1 } , then ∆( λ ) ∼ = L ( µ ) | L ( λ ), where µ =(2 i − m +1 , n − i +2 m − ). Moreover, we obtain a self-dual module by stacking the Weyl modules ∆( λ )and ∆( µ ) = L ( µ ) to give L ( µ ) | L ( λ ) | L ( µ ). Examining the five cases in turn, along with knowingthe number of summands we must obtain – by Proposition 5.18 – the result follows.Applying the map T from Definition 5.3 and the observation in the proof of Corollary 5.8 imme-diately yields the following. Theorem 5.20. Let k (cid:62) j > and suppose that p divides exactly one of the integers k + j, k + j − , . . . , k − j + 2 , so that n = k + j ∈ { ap, ap + 1 , . . . , ap + 2 j − } for some a ∈ N . We write n = ap + i for i ∈ { , , . . . , j − } .Then the structure of S (( ke ) , ( je )) is obtained from Theorem 5.19 by replacing each simple module L ( µ ) therein with D T ( µ ) (cid:104) j (cid:105) . Moreover, for λ = (( ke + a, b ) , ( je + a, b )) , for some k (cid:62) j (cid:62) , with < a (cid:54) e and (cid:54) b < e with a + b (cid:54) = e , or for a = b = 0 , the structure of S λ is obtained fromTheorem 5.19 by replacing each simple module L ( µ ) therein with D F a,b ◦ T ( µ ) (cid:104) j (cid:105) , and the structure of S λ (cid:48) is obtained from Theorem 5.19 by replacing each simple module L ( µ ) therein with D −F a,b ( { T ( µ ) } ) (cid:104) j (cid:105) =D m e,κ ◦F a,b ◦ T ( µ ) (cid:104) j (cid:105) .(i) Then the structure of S (( ke ) , ( je )) is obtained from Theorem 5.19 by replacing each simple module L ( µ ) therein with D T ( µ ) (cid:104) j (cid:105) .(ii) Let λ = (( ke + a, b ) , ( je + a, b )) , for some k (cid:62) j (cid:62) , with < a (cid:54) e and (cid:54) b < e with a + b (cid:54) = e , or for a = b = 0 . Then the structure of S λ is obtained from Theorem 5.19 by replacingeach simple module L ( µ ) therein with D F a,b ◦ T ( µ ) (cid:104) j (cid:105) (iii) Let λ = (( b + 1 , je + a − ) , ( b + 1 , ke + a − )) , for some k (cid:62) j (cid:62) , with < a (cid:54) e and (cid:54) b < e with a + b (cid:54) = e , or for a = b = 0 . Then the structure of S λ is obtained from Theorem 5.19 byreplacing each simple module L ( µ ) therein with D −F a,b ( { T ( µ ) } ) (cid:104) j (cid:105) = D m e,κ ◦F a,b ◦ T ( µ ) (cid:104) j (cid:105) . We finish with an example to illustrate that even when all of the relevant Weyl modules have atmost two composition factors, we do not have to go so far to see some difficult module structuresappear. With this example in mind, it is unclear how much further one can hope to push the resultsof Theorems 5.19 and 5.20. Example 5.21. Let p = 3, k = 7, and j = 3. Using Theorems 2.13 and 3.12, we deduce that∆(1 ) ⊗ ∆(1 ) (and therefore S (( ke ) , ( je )) ) has two summands, composed from the simple Weyl modules∆(1 ) = L (cid:0) (cid:1) and ∆(2 , ) = L (cid:0) , (cid:1) , as well as the Weyl modules∆(2 , ) ∼ = L (cid:0) (cid:1) | L (cid:0) , (cid:1) and ∆(2 , ) ∼ = L (cid:0) , (cid:1) | L (cid:0) , (cid:1) . By checking residues, one can see that ∆(2 , ) lies in one block, while the other three Weyl moduleslie in another one. Thus one summand of ∆(1 ) ⊗ ∆(1 ) is simple, and isomorphic to L (cid:0) , (cid:1) , whilethe other one comprises of the remaining three Weyl modules, or five simple modules, and must beself-dual. It follows that this summand has the following structure.L (2)L (0) L (3) L (0)L (2)2 Robert Muth, Liron Speyer & Louise Sutton References [Alp80] J. L. Alperin, Diagrams for modules , J. Pure Appl. Algebra (1980), no. 2, 111–119.[Page 19.][BBS19] C. Bessenrodt, C. Bowman, and L. 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