Pieri type rules and GL(2|2) tensor products
aa r X i v : . [ m a t h . R T ] M a y PIERI TYPE RULES AND GL (2 | TENSOR PRODUCTS
THORSTEN HEIDERSDORF AND RAINER WEISSAUER
Abstract.
We derive a closed formula for the tensor product of a family ofmixed tensors using Deligne’s interpolating category
Rep ( GL ). We use thisformula to compute the tensor product of a family of irreducible GL ( n | n )-representations. This includes the tensor product of any two maximal atypicalirreducible representations of GL (2 | Introduction
For the classical group GL ( n ) the tensor product decomposition L ( λ ) ⊗ L ( µ ) = M ν c νλµ L ( ν )between two irreducible representations is given by the Littlewood-Richardson rulefor the Littlewood-Richardson coefficients c νλµ . Contrary to this case the analogousdecomposition between two irreducible representation of the General Linear Super-group GL ( m | n ) is poorly understood. A classical result from Berele and Regev[BR87] and Sergeev [Ser85] shows that the fusion rule between direct summandsof tensor powers V ⊗ r of the standard representation V ≃ k m | n is again given bythe Littlewood-Richardson rule. The first more general results were achieved in[Hei14] where we obtained a decomposition law for tensor products between anytwo mixed tensors, direct summands in a mixed tensor space V ⊗ r ⊗ ( V ∨ ) ⊗ s , r, s ∈ N .This result is based on the tensor product decomposition in Deligne’s interpolatingcategory Rep ( GL δ ) [Del07]. Due to the universal property of Deligne’s category,we have for δ = m − n a tensor functor F m | n : Rep ( GL m − n ) → Rep ( GL ( m | n )).Since the decomposition of the tensor product of two indecomposable elements isknown for Rep ( GL m − n ) by results of Comes and Wilson [CW11], we obtain ananalogous decomposition law once we describe the image F m | n ( X ) of an arbitraryindecomposable object X in Rep ( GL m − n ). This was achieved in [Hei14] based onresults by Brundan and Stroppel [BS12b] on the interplay between Khovanov al-gebras and Walled Brauer algebras. Since any Kostant module [BS10a] and anyprojective representation is a mixed tensor (up to some Berezin twist) [Hei14], theseresults give a decomposition law for their tensor products, covering in particularthe decomposition between any two irreducible GL ( m | The main results.
For m, n ≥ GL ( n | n ) is a mixed tensor: the superdimension of V vanishes in the GL ( n | n )-case, but the superdimension of any maximal atypical irreducible representation isnever zero [Ser10] [Wei10]. It is well-known that the weight of a maximal atypicalrepresentation is of the form λ = ( λ , . . . , λ n | − λ n , . . . , − λ ), and we denote Mathematics Subject Classification : 17B10, 17B20. the corresponding irreducible representation by [ λ , . . . , λ n ]. We also denote theirreducible representation [ i, , . . . ,
0] by S i for i ≥
0. In this paper we obtain analmost complete picture for the tensor product S i ⊗ S j for any i, j and any n . Thisis the only known case of a formula apart from the GL ( m | S i ⊗ S j onto the maximal atypical blockΓ decomposes into two indecomposable representations if i = j and into a singleindecomposable representation if i = jpr Γ ( S i ⊗ S j ) ∼ = δ ij Ber i − ⊕ M ij . These indecomposable representations have non vanishing superdimension, and soremarkably no maximal atypical summand has superdimension 0. We do not com-pute the composition factors of these indecomposable summands for n ≥
3, but weformulate a conjecture 8.1 for the socle of these summands. In the GL (2 | S i ⊗ S j which are notmaximal atypical. By corollary 7.5 these are all irreducible with degree of atypical-ity n − G = GL ( n ) × GL ( n ). Since tensoring with the Berezin super determinant Ber ≃ [1 , . . . ,
1] is a flat functor and
Ber i ⊗ Ber j ≃ Ber i + j these decompositionlaws extend to Berezin twists of the S i and of course to their duals. We stress theremarkable fact that we get a very restricted picture for summands of nonvanishingsuperdimension (one or two summands); and everything else is semisimple. We alsopoint out that while the article uses a fair amount of computation, its approachis rather conceptual and uses a lot of theory: Deligne’s interpolating categories Rep ( GL m − n ), the description of the functor F n : Rep ( GL ) → Rep ( GL (2 | R ( λ ) with k ( λ ) ≤ The GL (2 | -case. It is worth summarizing the situation in the n = 2-case.In the GL (2 | GL (2 | GL (2 |
2) is of the form [ a, b ] and any such representationis a Berezin twist of one of the S i = [ i,
0] for the Berezin determinant
Ber , ourresult covers the entire maximal atypical GL (2 | psl (2 | Summary of the proof.
The essential ingredients in our proof are resultson the Loewy structure and the tensor products of mixed tensors [Hei14] and theformalism of cohomological tensor functors developed in [HW14]. Every S i can IERI TYPE RULES AND GL (2 |
2) TENSOR PRODUCTS 3 be realised as the unique constituent of highest weight in a mixed tensor baptised A S i . The composition factors and the socle filtration of the A S i are known [Hei14].We split the computation of S i ⊗ S j into two parts. We first project onto themaximal atypical block Γ and then compute the remaining summands afterwards.We derive a closed formula for the projection of A S i ⊗ A S j on Γ in section 4.Now we specialise to the GL (2 | K . A S i ⊗ A S j splits into representations of the form A S k forsome k and mixed tensors R ( a, b ). The composition factors of the R ( a, b ) areknown in the GL (2 | A S i ⊗ A S j ] ∈ K the tensor product S i ⊗ S j occursexactly once, and all other tensor products are of the form Ber p ( S k ⊗ S l ) withboth k and l less or equal to i and j and some Berezin power p . This allows us tocompute the maximal atypical composition factors of S i ⊗ S j recursively in lemma5.2. In order to determine the decomposition into maximal atypical indecomposablerepresentations we use the theory of cohomological tensor functors [HW14]. Herewe consider the tensor functor DS : Rep ( GL (2 | → Rep ( GL (1 | DS ( L ) for any irreducible representationand we get DS ( S i ) = Ber i ⊕ Π − i Ber − where Π denotes the parity shift functor.This gives us strict estimates on the number of indecomposable summands andtheir superdimension which is enough to determine the indecomposable summandsin theorem 7.6. Once we have understood the GL (2 | DS to reduce the computation of the maximal atypical part of S i ⊗ S j tothis case as explained in section 6. In section 7 we compute the indecomposablesummands which are not maximal atypical. The remaining composition factors in A S i ⊗ A S j are all ( n − K -decomposition is already enough for the computation. These methodsallow in principle to compute the decomposition S i ⊗ S j for any n . However it isvery difficult to determine the composition factors of the maximal atypical mixedtensors R ( a, b ) for n ≥
3. We end the article with a conjecture concerning thedecomposition of S i ⊗ S j and its socle for arbitrary n .2. The superlinear groups
Let k be an algebraically closed field of characteristic zero. Let g = gl ( m | n ) = g ⊕ g be the general linear superalgebra and GL ( m | n ) the general linear super-group. By definition a finite dimensional super representation ρ of gl ( m | n ) definesa representation ρ of GL ( m | n ) if its restriction to g comes from an algebraic rep-resentation of G = GL ( m ) × GL ( n ), also denoted ρ . We denote the category offinite-dimensional representations with parity-preserving morphisms by T = T m | n .For M ∈ T we denote by M ∨ the ordinary dual and by M ∗ the twisted dual. Forsimple and for projective objects M of T we have M ∗ ∼ = M [HW14]. The category R n . Fix the morphism ε : Z / Z → G = GL ( n ) × GL ( n ) whichmaps − diag ( E n , − E n ) ∈ GL ( n ) × GL ( n ) denoted ǫ nn . We write ǫ n = ǫ nn . Note that Ad ( ǫ nn ) induces the parity morphism on the Lie superalgebra gl ( n | n ) of G . We define the abelian subcategory R n of T n as the full subcategory ofall objects ( V, ρ ) in T n with the property p V = ρ ( ǫ nn ); here ρ denotes the underlyinghomomorphism ρ : GL ( n ) × GL ( n ) → GL ( V ) of algebraic groups over k and p V the parity automorphism of V . The subcategory R n is stable under the dualities ∨ and ∗ . The irreducible representations are indexed by weights with respect to the THORSTEN HEIDERSDORF AND RAINER WEISSAUER standard Borel subalgebra of upper triangular matrices. We denote by L ( λ ) theirreducible representation with highest weight λ = ( λ , . . . , λ n | λ n +1 , . . . , λ n ). TheBerezin determinant of GL ( n | n ) defines a one dimensional representation B = Ber with weight (1 , . . . , | − , . . . , − M ∈ T n is called negligible, ifit is the direct sum of indecomposable objects M i in T n with superdimensions sdim ( M i ) = 0. The thick ideal of negligible objects is denotes N or N n . Atypicality. If L ( λ ) is projective, the weight λ is called typical. If not, λ iscalled atypical. The atypicality of a weight can be measured by a number between0 and n [Kac78]. If the atypicality is n , we say the weight is maximal atypical.An example is the Berezin determinant Ber of dimension 1. More generally anirreducible representation is maximal atypical if and only if λ is of the form λ =( λ , . . . , λ n | − λ n , . . . , − λ ). In this case we often write [ λ , . . . , λ n ] for L ( λ ). Thesuperdimension of an irreducible representation is non-zero if and only if L ( λ ) ismaximal atypical [Ser10] [Wei10].The abelian categories T n | n and R n decomposeinto blocks and the degree of atypicality is a block-invariant.3. Mixed tensors
Let MT denote the full subcategory of mixed tensors in R n whose objects aredirect sums of the indecomposable objects in R n that appear in a decomposition V ⊗ r ⊗ ( V ∨ ) ⊗ s for some natural numbers r, s ≥
0, where V ∈ R n denotes thestandard representation. By [BS12b] and [CW11] the indecomposable objects in MT are parametrized by ( n | n )-cross bipartitions (see below). Let R n ( λ ) (or R ( λ )if the dependency on n is clear) denote the indecomposable representation in R n corresponding to the bipartition λ = ( λ L , λ R ) under this parametrization. Wesometimes write R ( λ L ; λ R ) to avoid brackets. To any bipartition we attach a weightdiagram in the sense of [BS11], i.e. a labelling of the numberline Z according tothe following dictionary. Put I ∧ ( λ ) := { λ L , λ L − , λ L − , . . . } and I ∨ ( λ ) := { − λ R , − λ R , . . . } . Now label the integer vertices i on the numberline by the symbols ∧ , ∨ , ◦ , × accord-ing to the rule ◦ if i / ∈ I ∧ ∪ I ∨ , ∧ if i ∈ I ∧ , i / ∈ I ∨ , ∨ if i ∈ I ∨ , i / ∈ I ∧ , × if i ∈ I ∧ ∩ I ∨ . To any such data one attaches a cup-diagram as in [CW11] or [BS11] and we definethe following three invariants rk ( λ ) = number of crosses d ( λ ) = number of cups k ( λ ) = rk ( λ ) + d ( λ ) . A bipartition is ( n | n )-cross if and only if k ( λ ) ≤ n . By [BS12b] the modules R ( λ L , λ R ) have irreducible socle and cosocle equal to L ( λ † ) where the highest weight λ † can be obtained by a combinatorial algorithm from λ . Let θ : Λ → X + ( n ) denotethe resulting map λ λ † between the set of ( n | n )-cross bipartitions Λ and the set X + ( n ) of highest weights of R n . IERI TYPE RULES AND GL (2 |
2) TENSOR PRODUCTS 5
Theorem 3.1. [Hei14, Corollary 5.4, Theorem 5.12] R = R ( λ L , λ R ) is an inde-composable module of Loewy length d ( λ )+1 . It is projective if and only if k ( λ ) = n in which case we have R = P ( λ † ) .Deligne’s interpolating category . For every δ ∈ k we denote by Rep ( GL δ ) theinterpolating category defined in [Del07]. This is a k -linear pseudoabelian rigidtensor category. By construction it contains an object st of dimension δ , calledthe standard representation. For this category we have a tensor functor F n = F n | n : Rep ( GL ) → R n by mapping the standard representation of Rep ( GL ) tothe standard representation of GL ( n | n ) in R n . Every mixed tensor is in the imageof this tensor functor [CW11, 8.13]. The indecomposable objects in Rep ( GL δ ) areparametrized by bipartitions [CW11] and we denote by R ( λ ) the indecomposableelement associated to the bipartition λ . Then F n : R ( λ ) R n ( λ ) if k ( λ ) ≤ n and F n ( R ( λ )) = 0 if k ( λ ) > n [CW11][8.3]. The atypicality of R n ( λ ) is given by n − rk ( λ ) [Hei14]. Note that the superdimension of every nontrivial mixed tensorvanishes since sdim ( V ) = 0.4. The symmetric and alternating powers
We define as in [Hei14] the following indecomposable modules in R n A S i = R ( i ; 1 i ) and A Λ i = ( A S i ) ∨ = R (1 i ; i ) . We define S i = [ i, , . . . ,
0] for integers i ≥
1. We denote the trivial representation S by . Furthermore we denote the projective cover of [ λ ] by P [ λ ]. Lemma 4.1. [Hei14, Lemma 13.3]
The Loewy structure of the A S i is given by( n ≥ ) A S = ( , S , ) A S i = ( S i − , S i ⊕ S i − , S i − ) 1 < i = n A S n = ( S n − , S n ⊕ S n − ⊕ B − , S n − ) . We remark that mixed tensors are always rigid [Hei14, Corollary 5.4]. Theserepresentations are maximal atypical for any n . We now derive a closed formula forthe tensor products A S i ⊗ A S j . It turns out that the maximal atypical summandsare not irreducible whereas all other summands are irreducible. Therefore we splitthe computations in two parts: we first compute the projection to the maximalatypical block of A S i ⊗ A S j and deal with the remaining easier case later in section7. In the following formulas we often project to the maximal atypical block. Recallfrom [Hei14, Proposition 11.1] that a mixed tensor R ( λ L , λ R ) is maximal atypicalif and only if λ R = ( λ L ) ∗ where λ ∗ denotes the conjugate partition. In this case wesimply use the notation R ( λ L ), e.g. A S i = R ( i ) and A Λ i = R (1 i ). Lemma 4.2.
The atypical mixed tensors in R are the A S i and their duals A Λ j .They are the projective covers A S i = P [ i − and A Λ j = P [ − j + 1] .Proof. They are projective since k ( λ ) = 1. The statement about the top followsfrom an explicit computation of the map θ : Λ → X + [Hei14, 6.1]. (cid:3) THORSTEN HEIDERSDORF AND RAINER WEISSAUER
Corollary 4.3. In R A S i ⊗ A Λ j = A S |− i + j | +2 ⊕ A S |− i + j | +1 ⊕ A S |− i + j | A S i ⊗ A S j = A S i + j ⊕ · A S i + j − ⊕ A S i + j − Proof.
This is just rewriting the known formula ( a, b ∈ Z ) P [ a ] ⊗ P [ b ] = P [ a + b + 1] ⊕ P [ a + b ] ⊕ P [ a + b − (cid:3) Let us assume from now on that n ≥ Lemma 4.4.
After projection to the maximal atypical block ( n ≥ ) A S i ⊗ A Λ j = A S |− i + j | +2 ⊕ A S |− i + j | +1 ⊕ A S |− i + j | ⊕ R A S i ⊗ A S j = A S i + j ⊕ · A S i + j − ⊕ A S i + j − ⊕ R where R and R are direct sums of modules which do not contain any A S i or A Λ j .Proof. This follows from the GL (1 | GL (1 |
1) [GQS07] P [ a ] ⊗ P [ b ] = P [ a + b − ⊕ P [ a + b ] ⊕ P [ a + b + 1] . Hence this formula holds for the corresponding A S i respectively A Λ j . It then holdsin Rep ( GL ) up to summands in the kernel of the functor F : Rep ( GL ) → Rep ( GL (1 | R ( λ ) with k ( λ ) >
1. By[Hei14, Lemma 13.1] a maximal atypical mixed tensor satisfies d ( λ ) = 1 (and hence k ( λ ) = 1) if and only if and only if λ = ( i ; 1 i ) or λ = (1 i ; i ). Hence this formulaholds in any Rep ( GL ( n | n )) up to contributions which lie in the kernel of F n | n : Rep ( GL ) → Rep ( GL ( n | n )) and which are not (1 | (cid:3) Tensor products in Deligne’s category.
In order to compute A S i ⊗ A S j we compute R ( i ) ⊗ R ( j ) in Rep ( GL ). We then push the result to Rep ( GL ( n | n ))using F n . We recall the tensor product decomposition in Rep ( GL ). Caps.
We attach to the weight diagram of a bipartition a cap-diagram as in[BS11] [CW11]. We denote the degree P i λ i of a partition by | λ | . If | λ | = n wewrite λ ⊢ n . If λ = ( λ L , λ R ) is a bipartition we denote its degree ( | λ L | , | λ R | ) by | λ | and we write λ ⊢ ( r, s ) if | λ L | = r and | λ R | = s . Let us fix a bipartition λ andconsider the associated weight and cup diagram. For integers i < j we say that( i, j ) is a ∨∧ -pair if they are joined by a cap. For λ, µ ∈ Λ we say that µ is linkedto λ if there exists an integer k ≥ ν ( n ) for 0 ≤ n ≤ k such that ν (0) = λ, ν ( k ) = µ and the weight diagramm of ν ( n ) is obtained from the one of ν ( n − by swapping the labels of some pair ∨∧ -pair. Then we put D λ,µ = ( µ is linked to λ D λ,λ = 1 for all λ . Further D λ,µ = 0 unless µ = λ or | µ | =( | λ L | − i, | λ R | − i ) for some i >
0. Let t be an indeterminate and R δ respective R t the Grothendieck rings of Rep ( GL δ ) over k respective of Rep ( GL t )) over thefraction field k ( t ). We follow the notation of [CW11] and denote by ( λ ) or simply λ the element R ( λ ) in R t or R δ . Now define lift δ : R δ → R t as the Z -linear map IERI TYPE RULES AND GL (2 |
2) TENSOR PRODUCTS 7 defined by lift δ ( λ ) = P µ D λ,µ µ where the sum runs over all bipartitions µ . By[CW11, Theorem 6.2.3] lift δ is a ring isomorphism for every δ ∈ k . Tensor products.
By [CW11, Theorem 7.1.1] the following decomposition holdsfor arbitrary bipartitions in R t : λµ = X v ∈ P × P Γ νλµ ν with the numbersΓ νλµ = X α,β,η,θ ∈ P ( X κ ∈ P c λ L κα c µ R κβ ) ( X γ ∈ P c λ R γη c µ L γθ ) c ν L αθ c ν R βη , see [CW11, Theorem 5.1.2]. Here c νλµ denotes the Littlewood-Richardson coefficientand P the set of all partitions. In particular if λ ⊢ ( r, s ), µ ⊢ ( r ′ , s ′ ), then Γ νλµ = 0unless | ν | ≤ ( r + r ′ , s + s ′ ). So to decompose tensor products in Rep ( GL δ ) applythe following three steps: Determine the image of the lift lift δ ( λµ ) in R t , use theformula above and then take lift − δ .4.2. Computations in R t . We continue to use our notation for the maximal atyp-ical case and write ( i ) instead of ( i ; 1 i ). Clearly lift( i ) = ( i )+( i − i ) = (1 i )+(1 i − ). Hence in order to compute the tensor product R ( i ) ⊗ R ( j ) we have to com-pute the tensor product ( i ) ⊗ ( j ) ⊕ ( i ) ⊗ ( j − ⊕ ( i − ⊗ ( j ) ⊕ ( i − ⊗ ( j −
1) in R t . Wederive first a closed formula for ( i ) ⊗ ( j ) in R t , i.e. for (( i, , . . . ) , (1 i )) ⊗ ( j, , . . . ) , (1 j ). • We analyze the sum P γ ∈ P c λ R γ,θ c µ L γ,η . Here λ R = (1 i ) and µ L = ( j, , . . . ).We need to find all pairs of partitions ( a, b ) such that c µ L a,b is non-zero. Wedenote this by ( µ L ) − . Now the Pieri rule gives ( µ L ) − = (0 , j ) , (1 , j − , . . . , ( j − , , ( j,
0) and ( λ R ) − = (0 , i ) , (1 , i − ) , . . . , (1 i , c λ R α,θ c µ L β,η is zero unless ( γ, θ ) and ( γ, η ) are of the form (0 , i ) and (0 , j ) orare of the form (1 , i −
1) and (1 , j − ). • The contribution P κ ∈ P c λ L κ,α c µ R κ,β : Here µ R = (1 j ) , λ L = ( i ). Similarly tothe previous case this gives only the possibilities c i ,i c j , j and c i ,i − c j , j − .Hence the sum X α,β,η,θ ( X κ ∈ P c λ L κ,α c µ R κ,β )( X γ ∈ P c λ R γ,η c µ L γ,θ )collapses to ( c i ,i c j , j + c i ,i − c j , j − ) ( c i , i c j ,j + c i , i − c j ,j − ) . This corresponds to the choices • (A) α = i, β = 1 j • (B) α = i − , β = 1 j − • (C) η = 1 i , θ = j • (D) η = 1 i − , θ = j − AC, AD, BC, BD can there be a summand ( ν ) withnonvanishing Γ νλµ = c ν L α,θ c ν R β,η . From now on we only consider bipartitions ν with ν L = ( ν R ) ∗ and identify such a bipartition with the partition ν L . THORSTEN HEIDERSDORF AND RAINER WEISSAUER • The AC-case: c ν L i,j c ν R j , i ( ν L , ν R ). By the Pieri rule ν L can be any of ( i + j ) , ( i + j − , , ( i + j − , , . . . and ν R any of (1 i + j ) , (2 , i + j − , . . . , ( i, | i − j | ). Hence the following partitions ν (i.e. bipartitions of the form ( ν L ; ( ν L ) ∗ )appear with multiplicity 1:( i + j ) , ( i + j − , , . . . , (( max ( i, j ) , min ( i, j )) . • The AD-case: c ν L i,j − c ν R j , i − . Restricting to ν L = ( ν R ) ∗ we obtain ν ∈ { ( i + j − , ( i + j − , , . . . , (( max ( i, j ) , min ( i, j ) − } . • The BC-case: c ν L i − ,j c ν R j − , i . Here ν is any of ν ∈ { (( i + j − , ( i + j − , , . . . , (( max ( i, j ) , min ( i, j ) − } . • The BD-case: c ν L i − ,j − c ν R j − , i − . Here ν ∈ { (( i + j − , ( i + j − , , . . . , ( max ( i − , j − , min ( i − , j − . } Hence in the Grothendieck ring R t ( i ) ⊗ ( j ) =( i + j ) + ( i + j − ,
1) + . . . + (( max ( i, j ) , min ( i, j ))+( i + j −
1) + ( i + j − ,
1) + . . . + (( max ( i, j ) , min ( i, j ) − i + j −
1) + ( i + j − ,
1) + . . . + (( max ( i, j ) , min ( i, j ) − i + j −
2) + ( i + j − ,
1) + . . . + ( max ( i − , j − , min ( i − , j − . Going back to
Rep ( GL ) . We calculate now the inverse lift − to get thedecomposition in Rep ( GL ).In the special case j = 1 , i > j −
1) = 0 and hence lift(( i ) ⊗ (1)) =( i ) ⊗ (1) + ( i ) + ( i −
1) + ( i − ⊗ (1). In R t we have( i ) ⊗ (1) = ( i + 1) + ( i,
1) + 2( i ) + ( i − i ) ⊗ (1)) = ( i + 1) + ( i,
1) + 4( i ) + ( i − ,
1) + 4( i −
1) + ( i − . After removing the contributions which will lead to A S i +1 ⊕ A S i ⊕ A S i − we areleft with ( i,
1) + ( i ) + ( i − ,
1) + ( i − i,
1) and hence theindecomposable module R ( i,
1) appears as a direct summand.
Lemma 4.5. In Rep ( GL ) we have for i ≥ A S i ⊗ A S = A S i +1 ⊕ A S i ⊕ A S i − ⊕ R ( i, . In the general case we add up the contributions (( i ) + ( i − · (( j ) + ( j − i )( j ) + ( i )( j −
1) + ( i − j ) + ( i − j − a, a, b ), a > b > a, a ) , a >
0. We havelift( a, b ) =( a, b ) + ( a, b −
1) + ( a − , b ) + ( a − , b − , a > b > a, a ) =( a, a ) + ( a, a −
1) + ( a − , a −
2) + ( a − , a − . IERI TYPE RULES AND GL (2 |
2) TENSOR PRODUCTS 9
After removing the contributions in R t which will give the A S i + j ⊕ · A S i + j − ⊕ A S i + j − and applying successively the liftings from above we get the following de-compositions. For i > , j = 2 we get A S i ⊗ A S = A S i +2 ⊕ · A S i +1 ⊕ A S i ⊕ R ( i + 1 , ⊕ R ( i, ⊕ · R ( i, ⊕ R ( i − , i > , j ≥ i > j . Then A S i ⊗ A S j = A S i + j ⊕ · A S i + j − ⊕ A S i + j − ⊕ R ( i + j − , ⊕ R ( i + j − , ⊕ · R ( i + j − , ⊕ R ( i + j − , ⊕ · R ( i + j − , ⊕ R ( i + j − , ⊕ R ( i + j − , ⊕ · R ( i + j − , ⊕ R ( i + j − , ⊕ R ( i + j − , ⊕ . . . ⊕ R ( i, j ) ⊕ · R ( i, j − ⊕ R ( i, j − ⊕ R ( i − , j − . Now assume i = j . For i = j = 2 we get A S ⊗ A S = A S ⊕ · A S ⊕ A S ⊕ R (3 , ⊕ R (2 , ⊕ · R (2 , . For i = j > i = j while omitting the last factor ⊕ R ( i − , j − Example 4.6.
We obtain in
Rep ( GL ) A S ⊗ A S = A S ⊕ A S ⊕ A S ⊕ R (4 , ⊕ R (3 , ⊕ · R (3 , ⊕ R (2 , . If we apply F n for n ≥
2, we obtain the same decomposition in T since all sum-mands R ( λ ) in this decomposition satisfy k ( λ ) ≤
2. The highest weights appearingin the socle and head of these indecomposable modules are [3 , , . . . ,
0] for λ = (4 , , , , . . . ,
0] for λ = (3 , , , . . . ,
0] for λ = (3 ,
1) and [1 , , . . . ,
0] for λ = (2 , Remark 4.7.
In the same way one can compute a closed formula for the projectionon the maximal atypical block of the tensor product A S i ⊗ A Λ j . This is not neededfor the GL (2 |
2) calculations, hence we skip the calculations. The final result is thefollowing. In the statement we may assume that j > A S i ⊗ A Λ = A S i ⊗ A S .We may also assume that i ≥ j since ( A S i ⊗ A Λ j ) ∨ = A Λ i ⊗ A S j . For i = j = 2 weobtain A S ⊗ A Λ = A S + ⊕ A S ⊕ A Λ ⊕ R (3 , ⊕ R (2 , ) ⊕ R (2 , i > j = 2 we obtain A S i ⊗ A Λ = A S i ⊕ A S i − ⊕ A S i − ⊕ R ( i + 1 , ⊕ R ( i, ) ⊕ R ( i, ⊕ R ( i − , . The general formula is for i > j > A S i ⊗ A Λ j = A S |− i + j | +2 ⊕ A S |− i + j | +1 ⊕ A S |− i + j | ⊕ R ( i + j − ( j − , j − ) ⊕ R ( i + j − j, j ) ⊕ R ( i, j − ) ⊕ R ( i, j − ) . . . ⊕ R ( i + j − k, k ) ⊕ · R ( i + j − k, k − ) ⊕ R ( i + j − k, k − ) ⊕ . . . ⊕ R ( i − j + 2 , ) ⊕ · R ( i − j + 2 , ⊕ R ( i − j + 1 , . For i = j > R ( i − j + 1 , GL (2 | tensor products - the maximal atypical part We compute the decomposition of the tensor product of any two maximal atypi-cal irreducible modules in R . In this section we compute only the direct summandswhich are maximal atypical. The remaining summands are computed in section 7.The basic idea is to look at our formulas for A S i ⊗ A S j in the Grothendieck groupand use these to compute the composition factors of S i ⊗ S j recursively startingwith the obvious tensor product S i ⊗ S . We then determine the decomposition intoindecomposable summands using results on cohomological tensor functors [HW14]and case-by-case distinctions.5.1. The R -case: Setup. Every maximally atypical irreducible representation L ( λ ) = [ λ , λ ] (in the notation of section 2) is a Berezin twist of a representationof the form S i := [ i,
0] for i ∈ N . Since tensoring with Ber is a flat functor,it is therefore enough to decompose the tensor product S i ⊗ S j . The Ext-quiverof the maximal atypical block Γ of R can be easily determined from [BS10a].It has been worked out by [Dro09]. For all irreducible modules in Γ we have dimExt ( L ( λ ) , L ( µ )) = dimExt ( L ( µ ) , L ( λ )) = 0 or 1. The Ext-quiver can bepicturised as follows where a line segment between two irreducible modules denotesa non-trivial extension class between these two modules and where an irreduciblemodule [ x, y ] is represented as a point in Z . . . .B j +3 ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ B j +3 S . . .B j +2 ❣❣❣❣❣❣❣❣❣❣❣❣❣❣ B j +2 S B j +2 S . . .B j +1 ❣❣❣❣❣❣❣❣❣❣❣❣❣ B j +1 S B j +1 S B j +1 S . . .B j ✐✐✐✐✐✐✐✐✐✐✐ B j S B j S B j S B j S . . .. . . ❥❥❥❥❥❥❥❥❥ . . . . . . . . . . . . . . . . . . The Loewy structure of the projective covers of a maximally atypical irreduciblemodule can also be computed from [BS12a] or be taken from Drouot: For [ a, b ] , a = IERI TYPE RULES AND GL (2 |
2) TENSOR PRODUCTS 11 b + k, k ≥ P [ a, b ] = B a − k SkB a − k S k +1 B a − k S k − B a − k − S k +1 B a − k +1 S k − B a − k S k B a − k − S k +2 B a − k − S k B a − k +2 S k − B a − k S k +1 B a − k S k − B a − k − S k +1 B a − k +1 S k − B a − k S k . For [ a, b ] , a = b + 2 the Loewy structure is P [ a, b ] = B a − S B a − S B a − S B a − S B a − S B a − S B a − S B a − S B a − S B a − B a − B a − S B a − S B a − S B a − S B a − S . For [ a, b ] , a = b + 1 the Loewy structure is P [ a, b ] = B a − S B a − S B a − B a − S B a B a − B a − S B a − S B a − S B a S B a − S B a − B a − S B a B a − B a − S . For [ a, b ] , a = b the Loewy structure is P [ a, b ] = B a B a S B a − S B a +1 S B a B a − B a − B a − S B a S B a +1 B a +2 B a S B a − S B a +1 S B a . The R -case: Mixed tensors. We specialise our decomposition of A S i ⊗ A S j to the R -case. All formulas hold only after projection to Γ. It is easy to see thatthe R ( a, b ) satisfy k ( λ ) = 2 and hence are projective covers. The top and socle ofthese covers can be easily computed using the map θ : Λ → X + . For small j we get A S ⊗ A S = A S ⊕ · A S ⊕ A ∨ S A S i ⊗ A S = A S i +1 ⊕ · A S i ⊕ A S i − ⊕ P [ i − , . A S i ⊗ A S = A S i +2 ⊕ · A S i +1 ⊕ A S i ⊕ P ([ i, ⊕ P ([ i − , ⊕ · P ([ i − , ⊕ P ([ i − , i > i >
2. Assume now i > , j ≥ i > j . A S i ⊗ A S j = A S i + j ⊕ · A S i + j − ⊕ A S i + j − ⊕ P [ i + j − , ⊕ P [ i + j − , ⊕ · P [ i + j − , ⊕ P [ i + j − , ⊕ · P [ i + j − , ⊕ P [ i + j − , ⊕ P [ i + j − , ⊕ · P [ i + j − , ⊕ P [ i + j − , ⊕ P [ i + j − , ⊕ . . . ⊕ P [ i − , j − ⊕ · P [ i − , j − ⊕ P [ i − , j − ⊕ P [ i − , j − . For i = j = 2 A S ⊗ A S = A S ⊕ A S ⊕ A S ⊕ P [2 , ⊕ P [0 , ⊕ P [1 , . For i = j > P [ i − , j − The R -case: K -decomposition. The tensor product decomposition ofthe A S i ⊗ A S j along with the knowledge of the composition factors of the indecom-posable summands permits to give recursive formulas for the K -decomposition ofthe tensor products S i ⊗ S j in the Grothendieck ring K = K ( R n ). Due to theasymmetry of the formulas and the asymmetry of the K -decompositions for A S i and P [ a, b ] for small i and a − b we compute the tensor products for small i and j first. The K -decomposition S ⊗ S follows immediately from the A S ⊗ A S -decomposition and we get S ⊗ S = 2 + 2 S + B + B − + B − S + S . Similarly one computes S ⊗ S = 2 S + S + B − S + S + BS S ⊗ S = S + B − S + 2 S + S + BS + 2 BS + + 2 B + B . Lemma 5.1.
We have P [ i,
0] = 2 A S i +1 + B − A S i +2 + B A S i for i ≥ in K ( R ) .Proof. This is just a direct inspection of the Loewy structures above. (cid:3)
Lemma 5.2.
For all i > j we have in the Grothendieck group K ( R ) S i ⊗ S j =2( S i + j − + BerS i + j − + · · · + Ber j − S i − j +1 )+ S i + j (1 + Ber − ) + · · · + Ber b S i − j (1 + Ber − ) . For i = j we get S i ⊗ S i =2( S i − + BerS i − + · · · + Ber i − S )+ S i (1 + Ber − ) + · · · + Ber i (1 + Ber − ) + B i − + B i − . Proof.
We first consider the cases S i ⊗ S and S i ⊗ S for i > i > S i ⊗ S , i >
1: For the induction start i = 2 see above. Put C i = S i ⊗ S in K ( R n ). For i ≥ S i ⊗ S +2 S i − ⊗ S + S i − ⊗ S =( S i +1 + 2 S i + S i − ) + ( S i − + 2 S i − + S i − ) + (2 C i − + Ber − S i +1 + Ber − S i − + BerS i − + BerS i − ). Hence using the induction assumption S i − ⊗ S = 2 S i − + S i − + Ber − S i − + S i − + BerS i − we get S i ⊗ S = 2 S i + S i +1 + S i − + Ber − S i +1 + BerS i − , and this proves the induction step. Likewise for S i ⊗ S . Now assume i > j >
2. Then for A S i ⊗ A S j we get the regular formula in K ( R ) A S i ⊗ A S j = S i ⊗ S j + 4( S i − ⊗ S j − ) + 2( S i − ⊗ S j )+2( S i − ⊗ S j − ) + 2( S i ⊗ S j − ) + S i ⊗ S j − +2( S i − ⊗ S j − ) + S i − ⊗ S j + S i − ⊗ S j − . All tensor products except S i ⊗ S j are known by induction. On the other hand thissum equals A S i + j + 2 A S i + j + A S i + j − + P [ i + j − ,
0] + 2 P [ i + j − ,
0] + P [ i + j − , B ) + 2 BP [ i + j − ,
0] + BP [ i + j − , B ) + . . . + 2 B j − P [ i − j +1 ,
0] + B j − P [ i − j, B ). Plugging in P [ a,
0] = 2 A S a +1 + B − A S a +2 + A S a forall a ≥ B -power on both sides finishes theproof. The case i = j works exactly the same way. (cid:3) IERI TYPE RULES AND GL (2 |
2) TENSOR PRODUCTS 13
The R -case: Socle Estimates. We say w ( M ) = k for a module M , if M ∨ ∼ = Ber − k M . Examples: w ( S i ) = i − w ( Ber ) = 2, and therefore w ( S i ⊗ S j ) = i + j − . On the other hand for ∗ -selfdual modules M we have soc ( M ) ∼ = cosoc ( M ) , since ∗ -duality is trivial on semisimple modules. On the other hand w ( M ) = k implies soc ( M ) ∨ ∼ = Ber − k cosoc ( M ), so that both conditions together imply w ( soc ( M )) = k . Hence being semi-simple, it is a direct sum of modules soc ( M ) ∼ = soc ′ ( M ) ⊕ M ν ∈ Z m ( ν ) · Ber ν S k +1 − ν with S i = 0 for i < m ( ν ), plus a sum soc ′ ( M ) of modulesof type (cid:0) Ber ν ⊕ Ber k − ν − j +1 (cid:1) S j for certain ν ∈ Z and certain natural numbers j with k − ν − j + 1 = ν . Proposition 5.3.
For n ≥ and for i > j ≥ we have soc ′ ( M ) = 0 for M = S i − ⊗ S j − and in K ( R n ) soc ( S i − ⊗ S j − ) ֒ → · S i + j − + 2 · BerS i + j − + · · · + 2 · Ber j − S i − j +1 . For i = j ≥ we have in K ( R n ) soc ( S i − ⊗ S i − ) ֒ → · S i − + 2 · BerS i − + · · · + 2 · Ber i − S + B i − . Proof.
Assume i > j . Note that soc ( M ) ֒ → soc ( A S i ⊗ A S j ) and by the aboveformulas the latter is S i + j − +3 S i + j − + 3 S i + j − + ( Ber + ) S i + j − + 2 BerS i + j − +( Ber + ) BerS i + j − + 2 Ber S i + j − + · · · +( Ber + ) Ber j − S i − j + 2 Ber j − S i − j +1 . Since k = w ( M ) = ( i − − j − − i + j −
4, this implies the as-sertion soc ′ ( M ) = 0. Indeed the terms S i + j − + 3 S i + j − and also N = ( Ber + ) Ber ν S i + j − − ν cannot contribute to soc ′ ( M ), since N ∨ =( Ber − + ) Ber − ν Ber − i − j +3+2 ν S i + j − − ν =( Ber − + ) Ber − i − j +3+ ν S i + j − − ν and Ber − k N = Ber − k ( Ber + ) Ber ν S i + j − − ν =( Ber + Ber ) Ber − i − j +3+ ν S i + j − − ν have no common irreducible summand. Hence soc ( M ) is contained in 3 · S i + j − +2 · BerS i + j − + · · · + 2 · Ber j − S i − j +1 . The proof is analogous for i = j . (cid:3) The Duflo-Serganova functor DS . We recall some constructions from thearticle [HW14].
An embedding . We view G n − = GL ( n − | n −
1) as an ‘outer block matrix’in G n = GL ( n | n ) and G as the ‘inner block matrix’ at the matrix positions n ≤ i, j ≤ n + 1. Fix the following element x ∈ g , x = (cid:18) y (cid:19) for y = . . .
00 0 . . . . . . . . . . We furthermore fix the embedding ϕ n, : G n − × G ֒ → G n defined by (cid:18) A BC D (cid:19) × (cid:18) a bc d (cid:19) A B a b c d C D . We use this embedding to identify elements in G n − and G with elements in G n .In this sense ǫ n = ǫ n − ǫ holds in G n , for the corresponding elements ǫ n − and ǫ in G n − resp. G , defined in section 2. Two functors . One has a functor (
V, ρ ) V + = { v ∈ V | ρ ( ǫ )( v ) = v } + : R n → R n − where V + is considered as a G n − -module using ρ ( ǫ ) ρ ( g ) = ρ ( g ) ρ ( ǫ ) Similarlydefine V − = { v ∈ V | ρ ( ǫ )( v ) = − v } . With the grading induced from V = V ⊕ V this defines a representation V − of G n − in Π R n − . Obviously( V, ρ ) | G n − = V + ⊕ V − . Cohomological tensor functors . Since x is an odd element with [ x, x ] = 0, we get2 · ρ ( x ) = [ ρ ( x ) , ρ ( x )] = ρ ([ x, x ]) = 0for any representation ( V, ρ ) of G n in R n . Notice d = ρ ( x ) supercommutes with ρ ( G n − ). Furthermore ρ ( x ) : V ± → V ∓ holds as a k -linear map, an immediateconsequence of dρ ( ε ) = − ρ ( ε ) d , i.e. of Ad ( ε )( x ) = − x . Since ρ ( x ) is an odd morphism, ρ ( x ) induces the following even morphisms (morphisms in R n − ) ρ ( x ) : V + → Π( V − ) and ρ ( x ) : Π( V − ) → V + . The k -linear map ∂ = ρ ( x ) : V → V is a differential and commutes with the actionof G n − on ( V, ρ ). Therefore ∂ defines a complex in R n − ∂ / / V + ∂ / / Π( V − ) ∂ / / V + ∂ / / · · · Since this complex is periodic, it has essentially only two cohomology groups de-noted H + ( V, ρ ) and H − ( V, ρ ) in the following. This defines two functors (
V, ρ ) D ± n,n − ( V, ρ ) = H ± ( V, ρ ) D ± n,n − : R n → R n − . IERI TYPE RULES AND GL (2 |
2) TENSOR PRODUCTS 15
For the categories T = T n resp. T n − (for the groups G n resp. G n − ) considerthe tensor functor of Duflo and Serganova in [DS05] DS n,n − : T n → T n − defined by DS n,n − ( V, ρ ) = V x := Kern ( ρ ( x )) /Im ( ρ ( x )). Then for ( V, ρ ) ∈ R n H + ( V, ρ ) ⊕ Π( H − ( V, ρ )) = DS n,n − ( V ) . Indeed, the left side is DS n,n − ( V ) = V x for the k -linear map ∂ = ρ ( x ) on V = V + ⊕ V − . Hence H + is the functor obtained by composing the tensor functor DS n,n − : R n → T n − with the functor T n − → R n − that projects the abelian category T n − onto R n − using T n = R n ⊕ Π R n . The ring homomorphism d . As an element of the Grothendieck group K ( R n − )we define for a module M ∈ R n d ( M ) = H + ( M ) − H − ( M ) . The map d is additive by [HW14]. Notice K ( T n ) = K ( R n ) ⊕ K ( R n [1]) = K ( R n ) ⊗ ( Z ⊕ Z · Π) . We have a commutative diagram K ( T n ) DS (cid:15) (cid:15) / / K ( R n ) d (cid:15) (cid:15) K ( T n − ) / / K ( R n − )where the horizontal maps are surjective ring homomorphisms defined by Π
7→ − . Since DS induces a ring homomorphism, d defines a ring homomorphism.5.6. The R -case: Indecomposability. If we display the maximal atypical com-position factors [ x, y ] of S i ⊗ S j in the Z -lattice, we get the following picture. Here (cid:3) denotes composition factors occuring with multiplicity 2 and the ◦ appear withmultiplicity 1. The socle is contained in the subset of composition factors denotedby (cid:3) . ◦◦ (cid:3) ◦◦ (cid:3) ◦ . . . . . . . . . ◦ (cid:3) ◦◦ with the two ◦ to the upper left at position B j S i − j and B j − S i − j and the onesto the lower right at position B − S i + j and S i + j . The picture in the i = j -case issimilar ◦⊙ (cid:3) ◦◦ ♦♦♦♦♦♦♦♦ ◦ (cid:3) ◦ . . . . . . with the composition factor ⊙ at position B i − appearing with multiplicity 2 andthe additional ◦ at position B i − .We now make use of the cohomological tensor functors DS . In the GL (1 | S i ≃ B i and hence S i ⊗ S j = S i + j . We know from [HW14] that DS ( S i ) = S i +Π − i B − and DS ( B ) = Π − B . Hence DS ( S i ⊗ S j ) splits into four indecomposablesummands each of superdimension 1 or each of superdimension -1: DS ( S i ⊗ S j ) = ( S i ⊕ B − ) ⊗ ( S j ⊕ Π − j B − )= B i + j ⊕ Π − j B i − ⊕ Π − i B j − ⊕ Π − i − j B − . Hence M = S i ⊗ S j splits into at most four indecomposable summands of sdim = 0. Lemma 5.4.
Every atypical direct summand is ∗ -invariant.Proof. If I is a direct summand which is not ∗ -invariant, M contains I ∗ as a directsummand and [ I ] = [ I ∗ ] in K ( R n ). However any summand of length > ◦ which occur in M only with multiplicity 1, a contradiction. (cid:3) Corollary 5.5.
The superdimension of any maximally atypical summand is = 0 .Proof. M does not contain any projectice cover (look at composition factors). If sdim ( I ) = 0, DS ( I ) = 0. However ker ( DS ) = AKac [HW14] (the modules with afiltration by AntiKac-modules) which are not *-invariant, unless they are projective. (cid:3)
Assume i > j . By ∗ -invariance the Loewy length of a direct summand is either1 or 3. If I is irreducible, then necessarily I = (cid:3) for a composition factor ofthe socle. By socle considerations both (cid:3) will split as direct summands. Theremaining module would have superdimension zero, hence the Loewy length of adirect summand is 3. Fix a composition factor of type (cid:3) . The multiplicity of (cid:3) inthe socle cannot be 2. If the multiplicity of (cid:3) in the socle is zero, then (cid:3) has to bein the middle Loewy layer. But this would force composition factors of type ◦ tobe in the socle. Contradiction. Hence Corollary 5.6.
For n ≥ and i > jsoc ( S i ⊗ S j ) = S i + j − ⊕ BerS i + j − ⊕ · · · ⊕ Ber j − S i − j +1 . We conclude that the superdimension of a direct summand is either 2 or 4.Hence M is either indecomposable or splits into two summands M = I L I ofsuperdimension 2. If M would split, it would split in the following way: IERI TYPE RULES AND GL (2 |
2) TENSOR PRODUCTS 17 ◦◦ (cid:3) ◦ ⊕ ◦ (cid:3) ◦◦ (cid:3) ◦ ◦ (cid:3) ◦◦ (cid:3) ◦ ◦ Now we use the ring homomorphism d : K ( R n ) → K ( R n − ) defined by d ( M ) = H + ( M ) − H − ( M ) as above. We know d ( S i ⊗ S j ) = B i + j + ( − − j B i − + ( − − i B j − + ( − − i − j B − . Since DS maps Anti-Kac modules to zero, d applied to any square with edges B k S i , B k +1 S i − , B k +1 S i , B k S i +1 is zero. Hence d ( I ) is given by applying d to the hookin the lower right d ( S i + j + S i + j − + B − S i + j ) and to ( B v S i + j +1 − v + B v S i + j − v )from the upper left of I . We get d ( I ) = B i + j + ( − i − j B − + ( − v B i + j +1 − v +( − v B i + j − v with the two additional summands ( − v B i + j +1 − v + ( − v B i + j − v .Contradiction, hence M is indecomposable.Now assume i = j . By the socle estimates for S i ⊗ S i and ∗ -duality either B i − splits as a direct summand or both B i − lie in the middle Loewy layer. Notethat Hom ( B i − , S i ⊗ S i ) = Hom ( B i − ⊗ ( S i ) ∨ , S i ) = End ( S i ) = k , hence thelast case cannot happen. Hence B i − splits as a direct summand. We show thatthe remaining module M ′ in S i ⊗ S i = B i − ⊕ M ′ is indecomposable. As in the i > j -case the Loewy length of any direct summand of M ′ must be 3. As before weobtain for i = jsoc ( S i ⊗ S i ) = S i − ⊕ BerS i − ⊕ · · · ⊕ Ber i − S ⊕ B i − . The remaining part M ′ can either split into three indecomposable modules of su-perdimension one each, in a direct sum of two modules of superdimension onerespectively two or is indecomposable. One cannot split the upper left ˜ I ◦◦ (cid:3) ◦ ♦♦♦♦♦♦♦ as a direct summand since its superdimension is −
1. Similarly one cannot split ◦◦ (cid:3) ◦◦ ♦♦♦♦♦♦♦ ◦ as a direct summand since the remaining module would have superdimension zero.Since all composition factors except the B ’s have superdimension ± M ′ could split only into M ′ = I ⊕ I with sdim ( I ) = 1 and sdim ( I ) = 2 with I as above.We argue now as in the i > j -case. In the Grothendieck ring K ( R n ) d ( M ) = B i + ( − − i B i − + ( − − i B i − , but d ( I ) has four summands as in the i > j -case. Contradiction, hence M isindecomposable. Corollary 5.7.
Up to summands which are not in the maximal atypical blockwe obtain S i ⊗ S j ≃ M ( i > j ) where M is indecomposable with Loewy structure S i + j − BerS i + j − · · · Ber j − S i − j +1 S i + j (1 + Ber − ) · · · Ber j S i − j (1 + Ber − ) S i + j − BerS i + j − · · · Ber j − S i − j +1 and S i ⊗ S i = B i − ⊕ M where M is indecomposable with Loewy structure S i − BerS i − · · · Ber i − S S i (1 + Ber − ) · · · Ber i S (1 + Ber − ) B i − S i − + BerS i − · · · Ber i − S . We remark that the summand
Ber i − in (Π i S i ) ⊗ belongs to Λ (Π i S i ) andthe summand M to Sym (Π i S i ), see also [HW15]. Note also that Λ (Π( V )) = Sym ( V ) for V ∈ R n .6. Reduction to the GL (2 | -case We do not calculate the maximal atypical composition factors of S i ⊗ S j for n ≥
3. Nonetheless we can determine the number of indecomposable summandsand their superdimension. We assume n ≥ i ≥ j . Lemma 6.1.
The Loewy length of a direct summand in S i ⊗ S j or S i ⊗ ( S j ) ∨ is ≤ .Proof. Since S i is in the socle and top of A S i +1 we have a surjection A S i +1 ⊗ A S j +1 → S i ⊗ S j . By the explicit formulas for A S i +1 ⊗ A S j +1 , the maximal Loewy lengthof a summand in A S i +1 ⊗ A S j +1 is ≤
5. For that recall that the Loewy length ofa mixed tensor R ( λ ) equals 2 d ( λ ) + 1, and it is easy to check that ( a, b ) satisfies d ( a, b ) = 2. Hence the quotient S i ⊗ S j has Loewy length at most 5. The case S i ⊗ ( S j ) ∨ is proved in the same manner. (cid:3) Since the Loewy length of a maximal atypical projective cover in R n is 2 n + 1we get Corollary 6.2.
For all n no maximal atypical projective cover appears in thedecompositions S i ⊗ S j resp. S i ⊗ ( S j ) ∨ .Proof. For n = 2 we saw this by brute force computations. For n ≥ n + 1 > (cid:3) We say a direct sum is clean if none of the summands is negligible. We say anegligible module in R n is potentially projective of degree r if DS n − r ( N ) ∈ T r isprojective and DS i ( N ) is not for i ≤ n − r . Lemma 6.3.
Every maximal atypical negligible summand in a tensor product L ( λ ) ⊗ L ( µ ) is potentially projective of degree at least 3. IERI TYPE RULES AND GL (2 |
2) TENSOR PRODUCTS 19
We proved in [HW15] that the kernel of DS equals P roj if we restrict DS tothe full subcategory T ± n of indecomposable modules occuring as direct summandsin an iterated tensor product of irreducible modules. Proof.
The decomposition of S i ⊗ S j in R is clean. Further DS sends negligiblemodules in T ± n to negligible modules in T ± n − [HW15] and the kernel of DS on T ± n consists of the projective elements. Since DS n − ( L ( λ ) ⊗ L ( µ )) ∈ T splits into adirect sum of irreducible representations of the form B a S b for some a, b ∈ Z by our GL (2 | DS n − ( N ) = 0. (cid:3) Lemma 6.4.
For all n the projection of S i ⊗ S j or S i ⊗ ( S j ) ∨ on the maximalatypical block is clean.Proof. We know that this is true for n = 2. If N is a maximal atypical summand in S i ⊗ S j , we apply DS several times until N becomes projective. Since DS ( S i ) = S i for i < n − DS ( S i ) = S i ⊕ Π n − − i Ber − for i ≥ n −
1, the tensor product DS ◦ . . . ◦ DS ( S i ⊗ S j ) splits into a tensor product of S i ’s and Berezin powers.The projective summand coming from N gives now a contradiction to 6.2. In the S i ⊗ ( S j ) ∨ -case we can argue in the same way using DS (( S j ) ∨ ) = DS ( S j ) ∨ . (cid:3) Nonvanishing superdimension.
In this part we refer extensively to resultsfrom [HW15]. We conclude from the previous paragraph that all maximal atyp-ical summands in S i ⊗ S j have non-vanishing superdimension. Hence the directsummands can be seen in the quotient R n / N by the modules of superdimension 0.According to [Hei15] [HW15] the tensor subcategory generated by the image of anirreducible element L in this quotient is of the form Rep ( H L , ǫ ) for some algebraicsupergroup H L and some twist ǫ as in section 2. We apply this to the representa-tions S i . By abuse of notation we denote the image of S i in the quotient still by S i . We show in [HW15] that the connected derived group ( H S i ) der of S i alwayssatisfies ( H S i ) der ≃ SL ( i + 1) for i ≤ n − H S i ) der ≃ SL ( n ) for i ≥ n − . Furthermore the restriction of S i (seen as a representation of H S i ) to ( H S i ) der remains irreducible. By superdimension reasons this restriction corresponds to thestandard representation of SL ( i + 1) or SL ( n ) respectively. The derived group ofthe group corresponding to the tensor category < S i , S j >, j = i generated by S i and S j in the quotient is the direct product of the derived groups corresponding to S i and S j . If S i ⊗ S j would decompose as M ⊕ M (up to superdimension 0) , therestriction to the derived group H der ≃ ( H S i ) der × ( H S j ) der would give a decomposition Res ( M ) ⊕ Res ( M ) of the tensor product Res ( S i ) ⊗ Res ( S j ). But this tensor product is the external tensor product of the standardrepresentation of the first factor with the standard representation of the secondfactor. Hence S i ⊗ S j is indecomposable for i = j up to superdimension 0. If i = j then the tensor product S i ⊗ S i behaves up to summands of superdimension 0 likethe tensor product of the standard representation of SL ( n ) with itself. Since thistensor product splits into the two irreducible representations of weight (2 , , . . . , and (1 , , , . . . , S i ⊗ S i has two indecomposable summands of non-vanishingsuperdimension. Corollary 6.5.
The tensor product S i ⊗ S j in R n has a single indecompos-able maximal atypical summand for i = j and decomposes in two indecomposablesummands for i = j . Remark 6.6.
In other words, once the know the GL (2 | S i -case since this can be treated in an adhoc way[HW15, Section 9].7. The lower atypical summands in R n We compute the remaining direct summands of the tensor product S i ⊗ S j in R n for n ≥
2. These direct summands are all irreducible which will follow fromthe fact that all summands of atypicality < n in an A S i ⊗ A S j tensor product areirreducible and have vanishing Ext with each other. Lemma 7.1. A S i ⊗ A S j is a direct sum of maximally atypical summands andirreducible representations of atypicality n − and likewise for A Λ i ⊗ A Λ j .Proof. In the decomposition of lift(( i ; 1 i ) ⊗ ( j ; 1 j )) in R t , the bipartitions which willnot contribute to the maximal atypical block are of the form( ( i + j − k, k ); (2 r , i + j − r ) )for some k, r ≥ k = r . We have I ∧ = { i + j − k, k − , − , − , − , . . . } I ∨ = {− , , , . . . , r − , r, r + 1 , . . . , i + j − r − , i + j − r + 1 , . . . } Since k = r , neither one of the two conditions i + j − k = i + j − r , k − r − ∨∧ -pair, hence the corresponding modules are irreducible. (cid:3) Lemma 7.2.
The composition factors of S i ⊗ S j in R n which are not maximallyatypical are given by the set R (( i + j − k, k ); (2 r , i + j − r )) , k, r = 0 , , . . . , min ( i, j ) , k = r. All these modules have atypicality n − and are irreducible.Proof. This is again a recursive determination from the A S i ⊗ A S j tensor products.As before the S i ⊗ S and S i ⊗ S -cases for i ≥ i ≥ S i ⊗ S j , i, j ≥ A S i ⊗ A S j =( S i + 2 S i − + S i − ) ⊗ ( S j + 2 S j − + S j − )= S i ⊗ S j + lower termswhere the lower terms are known by induction. In the A S i ⊗ A S j tensor productthe R ( , )’s from above cannot occur (for degree reasons) in any tensor product A S p ⊗ A S q for p ≤ i, q ≤ j where either p < i or q < j . Hence they cannot occur inany tensor product decomposition of any S p ⊗ S q for p, q as above, hence they haveto occur in the S i ⊗ S j -decomposition. The number of these modules is ( min ( i, j ) − min ( i, j ). Substracting the inductively known numbers of not maximally atypical IERI TYPE RULES AND GL (2 |
2) TENSOR PRODUCTS 21 contributions in S p ⊗ S q in the A S i ⊗ A S j -tensor product from the number of allsuch contributions in A S i ⊗ A S j we get min ( i, j ) − min ( i, j ) remaining modules.Hence there are no other summands in S i ⊗ S j . (cid:3) Lemma 7.3.
The irreducible representation R (( i + j − k, k ); (2 r , i + j − r )) isisomorphic to L ( i + j − k, k, , . . . , | , . . . , , − r, − i − j + r ) . Proof. . Let m denote the maximal coordinate of a cross or circle in the weightdiagram of the bipartition. To obtain the weight diagram of the highest weight wehave to switch all labels to the right of this coordinate as well as the first M − n + 2labels to its left which are not labelled × or ◦ by the explicit description of θ in[Hei14, 6.1]. Since we have four symbols × and ◦ this amounts to switching all thelabels at positions ≥ − < M (all of them ∨ ’s) and the n − ∧ ’s at positions − , . . . , − n + 1 to ∨ ’s. The crosses are at the positions i + j − k, k − i + j − r, r −
1. The result follows. (cid:3)
Lemma 7.4.
The lower atypical direct summands of S i ⊗ S j in R n are givenby the set R (( i + j − k, k ); (2 r , i + j − r )) , k, r = 0 , , . . . , min ( i, j ) , k = r. Proof.
For any irreducible mixed tensors R ( λ ) , R ( µ ) we have Ext ( R ( λ ) , R ( µ )) = 0since every block contains a unique irreducible mixed tensor by [Hei14]. (cid:3) For a maximally atypical weight ( λ , . . . , λ n | − λ n , . . . , − λ ) denote by L ( λ , . . . , λ n ) ⊠ L ( − λ n , . . . , − λ )the underlying irreducible GL ( n ) × GL ( n )-module. Denote by π the followingadditive map from irreducible GL ( n ) × GL ( n ) modules to irreducible GL ( n | n )-modules: π (( L ( λ , . . . , λ n ) ⊠ L ( µ , . . . , µ n ))= ( L ( λ , . . . , λ n | µ , . . . , µ n ) ∈ Γ L ( λ , . . . , λ n | µ , . . . , µ n ) else. Corollary 7.5.
The not maximally atypical contributions to S i ⊗ S j are givenby π ( ( L ( i, , . . . , ⊠ L (0 , . . . , , − i ) ) ⊗ ( L ( j, , . . . , ⊠ L (0 , . . . , , − j ) ) . Corollary 7.6.
For n = 2 the tensor product S i ⊗ S j ( i > j ) decomposes as S i ⊗ S j ≃ S i + j − BerS i + j − · · · Ber j − S i − j +1 S i + j (1 + Ber − ) · · · Ber j S i − j (1 + Ber − ) S i + j − BerS i + j − · · · Ber j − S i − j +1 ⊕ π ( ( L ( i, , . . . , ⊠ L (0 , . . . , , − i ) ) ⊗ ( L ( j, , . . . , ⊠ L (0 , . . . , , − j ) ) . The tensor product S i ⊗ S i decomposes as S i ⊗ S i ≃ B i − ⊕ S i − BerS i − · · · Ber i − S S i (1 + Ber − ) · · · Ber i S (1 + Ber − ) B i − S i − + BerS i − · · · Ber i − S ⊕ π ( ( L ( i, , . . . , ⊠ L (0 , . . . , , − i ) ) ⊗ ( L ( i, , . . . , ⊠ L (0 , . . . , , − i ) ) . The GL (3 | -case and a conjecture The method applied to compute the S i ⊗ S j tensor products in the GL (2 | n . Note that the results on the A S i ⊗ A S j tensor products are valid for any n . Furthermore we determined the part of S i ⊗ S j which is not maximal atypical for any n ≥
2, hence we restrict here to the maximalatypical part. The obstacle to use the method of the R -case effectively is thatthe composition factors of the modules R ( a, b ) appearing in the A S i ⊗ A S j -case aredifficult to compute. Decomposing a few R ( a, b ) for small a and b in the n = 3-case and then computing the composition factors of the S i ⊗ S j tensor productsrecursively, we arrive at the following tensor products (Λ = ( S ) ∨ ). Here wealways project to the maximal atypical block. S ⊗ S ≃ ⊕ S S + ( S ) ∨ S S ⊗ S ≃ S S [2 , , S B − S S ⊗ S ≃ S S [3 , , S S S ⊗ S ≃ [1 , , ⊕ S [2 , , S [3 , ,
0] [2 , ,
0] ( S ) ∨ [2 , − , −
1] [0 , − , − S S [2 , , S ⊗ S ≃ S [3 , , S [4 , ,
0] [3 , , S [2 , ,
0] [3 , − , − S [3 , , S ⊗ S ≃ [2 , , ⊕ S [4 , ,
0] [3 , , S [5 , ,
0] [4 , , B − S [3 , , S [3 , ,
0] [1 , ,
0] [2 , , S [4 , ,
0] [3 , , Conjecture 8.1.
For n ≥ , S i ⊗ S j = M is indecomposable if i = j (corollary6.5). S i ⊗ S i splits as [ i − , i − , . . . , i − , ⊕ M. The socle of M is for i ≥ jsoc ( M ) = [ i + j − , , . . . , ⊕ [ i + j − , , , . . . , ⊕ . . . ⊕ [ i, j − , . . . , and M has Loewy length 3. Note that since A S i → A S j ։ S i ⊗ S j and the maximal Loewy length of a directsummand R ( a, b ) in A S i → A S j is 5, the Loewy length of M is at most 5. IERI TYPE RULES AND GL (2 |
2) TENSOR PRODUCTS 23
References [BR87] A. Berele and A. Regev. Hook Young diagrams with applications to combinatorics and torepresentations of Lie superalgebras.
Adv. Math. , 64:118–175, 1987.[BKN09a] Boe, B. D. and Kujawa, J. R. and Nakano, D. K.,
Complexity and module varieties forclassical Lie superalgebras , Int. Math. Res. Not., 2011, (2009)[BKN09b] Boe, B. D. and Kujawa, J. R. and Nakano, D. K.,
Cohomology and support varietiesfor Lie superalgebras II , Proc. Lond. Math. Soc. (3), 98, 1, (2009)[BKN10] Boe, B. D. and Kujawa, J. R. and Nakano, D. K.,
Cohomology and support varietiesfor Lie superalgebras , Trans. Am. Math. Soc., 362, 12, (2010)[Bru03] Brundan, J.,
Kazhdan-Lusztig polynomials and character formulae for the Lie superalge-bra gl ( m | n ), J. Am. Math. Soc., 16, 1, (2003)[BS10a] Brundan, J. and Stroppel, C., Highest weight categories arising from Khovanov’s diagramalgebra. II: Koszulity , Transform. Groups, 15, 1, (2010)[BS11] Brundan, J. and Stroppel, C.,
Highest weight categories arising from Khovanov’s diagramalgebra. I: cellularity , Mosc. Math. J., 11, 4, (2011)[BS12a] Brundan, J. and Stroppel, C.,
Highest weight categories arising from Khovanov’s diagramalgebra. IV: the general linear supergroup , J. Eur. Math. Soc. (JEMS), 14, 2, (2012)[BS12b] Brundan, J. and Stroppel, C.,
Gradings on walled Brauer algebras and Khovanov’s arcalgebra , Adv. Math., 231, 2, (2012)[CW11] Comes, J. and Wilson, B.,
Deligne’s category
Rep ( GL δ ) and representations of generallinear supergroups , Represent. Theory, 16, (2012)[Del07] Deligne, P., La cat´egorie des repr´esentations du groupe sym´etrique S t , lorsque t n’est pasun entier naturel , Mehta, V. B. (ed.), Algebraic groups and homogeneous spaces. Proceedingsof the international colloquium, Mumbai, India, January 6–14, 2004. 209-273, (2007)[Dro09] Drouot, Fr., Quelques proprietes des representations de la super-algebre der Lie gl ( m, n ),PhD thesis, (2009)[DS05] Duflo M. and Serganova V., On associated variety for Lie superalgebras ,arXiv:math/0507198v1, (2005)[Ger98] Germoni, J.,
Indecomposable representations of special linear Lie superalgebras , J. Alge-bra , 209, 2, (1998)[GKPM11] Geer, N. and Kujawa, J. and Patureau-Mirand, B.,
Generalized trace and modifieddimension functions on ribbon categories , Sel. Math., New Ser., 17, 2, (2011)[GQS07] Goetz, G. and Quella, Th. and Schomerus, V.,
Representation theory of sl (2 | Tensor products of psl (2 , representa-tions . ArXiv e-prints: hep-th/0506072, (2005)[Hei14] Heidersdorf, Th., Mixed tensors of the General Linear Supergroup , J.Algebra, 491, (2017)[HW14] Heidersdorf, Th. and Weissauer, R.,
Cohomological tensor functors on representations ofthe General Linear Supergroup , ArXiv e-prints: 1406.0321, to appear in: Mem. Am. Math.Soc. (2014)[HW15] Heidersdorf, Thorsten and Weissauer, Rainer,
On classical tensor categories attached tothe irreducible representations of the General Linear Supergroup GL ( n | n ), preprint, (2015)[Hei15] Heidersdorf, T., On supergroups and their semisimplified representation categories , ArXive-prints, 1512.03420, (2015)[Kac78] Kac, V.,
Representations of classical Lie superalgebras , Differ. geom. Meth. math. Phys.II, Proc., Bonn 1977, Lect. Notes Math. 676, 597-626 (1978)[Kuj11] Kujawa, J.,
The generalized Kac-Wakimoto conjecture and support varieties for the Liesuperalgebra osp ( m | n ), Recent developments in Lie algebras, groups and representation the-ory. 2009-2011 Southeastern Lie theory workshop series, (2012)[Sch79] Scheunert, M., The theory of Lie superalgebras. An introduction , 1979, Lecture Notes inMathematics. 716. Berlin-Heidelberg-New York: Springer-Verlag. X, 271 p. (1979)[Ser06] Serganova, V.
Blocks in the category of finite-dimensional representations of gl ( m, n ),2006[GS10] Gruson, C. and Serganova, V., Cohomology of generalized supergrassmannians and char-acter formulae for basic classical Lie superalgebras.
Proc. Lond. Math. Soc. (3) 101 (2010)[Ser06] Serganova, V.,
Blocks in the category of finite-dimensional representations of gl ( m | n ),preprint, (2006) [Ser10] Serganova, V., On the superdimension of an irreducible representation of a basic classicalLie super algebra , Supersymmetry in mathematics and physics. UCLA Los Angeles, USA2010. Papers based on the presentations at the workshop, Februar 2010., Berlin: Springer(2011)[Ser85] A.N. Sergeev. The tensor algebra of the identity representation as a module over the Liesuperalgebras G l ( n, m ) and Q(n). Math. USSR, Sb. , 51:419–427, 1985.[Wei10] Weissauer, R.,
Monoidal model structures, categorial quotients and representations ofsuper commutative Hopf algebras II: The case Gl ( m, n ), arXiv e-prints, (2010) T.H.: Department of Mathematics, The Ohio State UniversityT. H.: Max-Planck Institut f¨ur Mathematik, Bonn
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