Delignes category Rep(GL_delta) and representations of general linear supergroups

DELIGNE’S CA TEGOR Y Rep( GL δ ) AND REPRESENT A TIONS OF GENERAL LINEAR SUPER GR OUPS JONA THAN COMES AND BENJAMIN WILSON Abstract. W e classify indecomposable summands of mixed tensor powers of the natural representation for the general linear supergroup up to isomorphism. W e also giv e a formula for the c haracters of these summands in terms of com- posite sup ersymmetric Sch ur p olynomials, and give a method for decomp osing their tensor products. Along the wa y , we describ e indecomposable ob jects in Rep( GL δ ) and explain how to decomp ose their tensor products. Contents 1. In tro duction 1 2. Category-theoretic preliminaries 4 3. The category Rep( GL δ ) 12 4. Indecomp osable ob jects in Rep( GL δ ) 16 5. Connection to representations of the general linear group 24 6. The lifting map 27 7. Decomp osing tensor products in Rep( GL δ ) 33 8. Represen tations of the general linear sup ergroup 35 References 45 1. Introduction 1.1. Classical Sch ur-W eyl duality concerns the comm uting actions of the sym- metric group Σ r and the general linear group GL d on the tensor p o wer V ⊗ r of the natural represen tation V of GL d . It enables the lab elling of the isomorphism classes of indecomp osable summands of the tensor p o w er b y partitions λ ` r with heigh t l ( λ ) ≤ d (this is Weyl’s Strip The or em ), and the description of the c harac- ters of these summands in terms of Sc hur p olynomials. Sch ur-W eyl duality for the general linear sup ergroup GL ( m | n ), established by Sergeev [Ser2] and b y Berele and Regev [BR], pro vides similar insights into structure of the tensor pow er V ⊗ r of the natural represen tation V of GL ( m | n ). In this case, the isomorphism classes of the indecomposable summands of V ⊗ r are parametrized by those partitions λ ` r that are ( m | n ) -ho ok , that is, partitions whose Y oung diagram can b e cov ered by an m -wide, n -high ho ok, and the characters of the summands are describ ed by the so-called sup ersymmetric Sc h ur p olynomials. An analogue of Sc h ur-W eyl duality for the mixe d tensor p owers T ( r , s ) = T V ,V ∗ ( r , s ) = V ⊗ r ⊗ ( V ∗ ) ⊗ s Date : Octob er 28, 2018. 1 2 JONA THAN COMES AND BENJAMIN WILSON of GL d w as dev elop ed by [BCH + , Ste, Koi, T ur]. Here, the walled Brauer algebras B r,s ( δ ), with δ = d , replace the group algebra of the symmetric group as the generic cen tralizer of the GL d -action. It is kno wn, in particular, that the indecomposable summands are lab elled up to isomorphism by certain bipartitions (i.e. pairs of partitions). Sch ur-W eyl duality for mixed tensor p o w ers also holds for the general linear sup ergroup GL ( m | n ), where δ = m − n is the sup er dimension of the natural represen tation V . Ho wev er, in this case, man y fundamental questions remain unre- solv ed. In this pap er, w e classify for the ﬁrst time the indecomp osable summands of the mixed tensor p o w ers for the general linear sup ergroups up to isomorphism, and derive a c haracter formula for the indecomp osable summands in terms of com- p osite sup ersymmetric Sch ur p olynomials. In addition, we describ e a metho d for the decomp osition of tensor pro ducts of these indecomp osable summands. 1.2. W e work o v er a ﬁeld K of characteristic zero throughout, identify ﬁnite di- mensional represen tations of GL ( m | n ) with integral representations of the Lie su- p eralgebra gl ( m | n ), and write Rep( GL d ) and Rep( gl ( m | n )) for the categories of ﬁnite-dimensional representations of GL d and gl ( m | n ), resp ectiv ely . F undamental to our approac h is the tensor category Rep( GL δ ), deﬁned by Deligne [Del1-2], that p ermits the sim ultaneous study of the mixed tensor p o w ers for the general linear groups and the general linear sup ergroups. This category , which we refer to as Deligne’s c ate gory , is constructed as the additive and Karoubi en v elope of a “sk ele- ton” tensor category Rep 0 ( GL δ ) (cf. § 3.5). Up to isomorphism, the ob jects w r,s of Rep 0 ( GL δ ) are parametrized by pairs ( r, s ) of non-negativ e in tegers, represent- ing the p otencies of a mixed tensor p ow er, and the morphism spaces are spanned b y w alled Brauer diagrams of the appropriate sizes. The structure of Deligne’s category dep ends upon the parameter δ ∈ K so that, in particular, the endomor- phism algebras are the w alled Brauer algebras B r,s ( δ ). The univ ersal prop ert y of Deligne’s category guaran tees that for any rigid δ -dimensional ob ject V in a tensor category T satisfying h ypotheses familiar from classical Sc h ur-W eyl duality (cf. § 4.7), there exists a full tensor functor F : Rep( GL δ ) → T suc h that for any r , s ≥ 0, F ( w r,s ) = T ( r , s ) is the corresp onding mixed tensor p o w er. In particular, for any d > 0 and for any m, n ≥ 0 there exist full tensor functors F d : Rep( GL d ) → Rep( GL d ) , F m | n : Rep( GL m − n ) → Rep( gl ( m | n )) deﬁned b y the natural represen tations of GL d and of GL ( m | n ), respectively . Recen t results in the represen tation theory of the walled Brauer algebra also pla y crucial roles. F or any bipartition λ , let λ • , λ ◦ b e the partitions deﬁned b y λ = ( λ • , λ ◦ ), let l ( λ ) = l ( λ • ) + l ( λ ◦ ), and write λ ` ( | λ • | , | λ ◦ | ). It was shown in [CDDM] that the w alled Brauer algebras B r,s ( δ ) are cellular, with standard mo dules parametrized b y the set of bipartitions Λ r,s = { λ | λ ` ( r − i, s − i ) , 0 ≤ i ≤ min( r , s ) } . A recursive formula for the decomp osition n um b ers for the w alled Brauer algebras w as describ ed in [CD], using cap diagrams introduced by Brundan and Stropp el [BS2-5]. The classiﬁcation of the simple mo dules for the walled Brauer algebras up to isomorphism obtained in [CDDM] enables the parametrization of the indecom- p osable ob jects of Deligne’s category b y bipartitions (cf. Theorem 4.6.2). W riting L ( λ ) for the indecomp osable ob ject corresp onding to the bipartition λ , w e ha v e the DELIGNE’S Rep( GL δ ) AND REPRESENT A TIONS OF gl ( m | n ) 3 follo wing description for the decomp osition of tensor pro ducts of indecomp osable ob jects of Rep( GL δ ) for generic δ (see Theorem 7.1.1 for a precise statemen t). Theorem 1.2.1. F or generic values of δ ∈ K , L ( λ ) ⊗ L ( µ ) ∼ = M ν L ( ν ) ⊕ Γ ν λ,µ wher e the sum is over bip artitions ν and the c o eﬃcients Γ ν λ,µ ar e given in terms of Littlewo o d-R ichar dson c o eﬃcients (se e (23) ). This theorem is derived using Koik e’s Theorem [Koi], whic h gives the decom- p osition of a tensor pro duct of irreducible rational representations for the general linear group. In the case of general linear sup ergroups, it is compatible with the result of Sergeev [Ser2]. The calculation of the decomp osition num b ers for the w alled Brauer algebras [CD] and the cap diagrams of Brundan and Stropp el permit the deﬁnition of a “lifting isomorphism” in the spirit of Comes and Ostrik [CO]. The lifting isomorphism relates the additiv e Grothendiec k rings of Rep( GL δ ) in the singular and generic cases, thus enabling the decomp osition of any tensor pro duct of indecomp osable ob jects of Rep( GL δ ) for any v alue of δ . The known decomposition of the mixed tensor pow ers for GL d [BCH + , Ste, Koi, T ur] ﬁnds the following expression in terms of the functor F d : Theorem 1.2.2. (se e The or em 5.2.2) F or any d > 0 and any bip artition λ , F d ( L ( λ )) is an inde c omp osable obje ct of Rep( GL d ) and is non-zer o if and only if l ( λ ) ≤ d . Mor e over, any non-zer o inde c omp osable summand of a mixe d tensor p ower T ( r , s ) in Rep( GL d ) is isomorphic to F d ( L ( λ )) for pr e cisely one bip artition λ ∈ Λ r,s with l ( λ ) ≤ d . The Y oung diagr am of a bip artition λ is obtained by superimp osing the Y oung diagrams for the partitions λ • and λ ◦ so that their top and left edges coincide, and then rotating the Y oung diagram for λ ◦ 180-degrees ab out its upper-left corner (cf. § 4.1). A bipartition is ( m | n ) -cr oss 1 if its Y oung diagram can be cov ered with an m -high, n -wide cross (cf. § 8.7). The decomposition of the mixed tensor pow ers for gl ( m | n ) can b e describ ed in terms of the functor F m | n as follows. Theorem 1.2.3. F or any m, n ≥ 0 and any bip artition λ , F m | n ( L ( λ )) is an inde- c omp osable obje ct of Rep( gl ( m | n )) and is non-zer o if and only if λ is ( m | n ) -cr oss. Mor e over, any non-zer o inde c omp osable summand of a mixe d tensor p ower T ( r , s ) in Rep( gl ( m | n )) is isomorphic to F m | n ( L ( λ )) for pr e cisely one ( m | n ) -cr oss bip arti- tion λ ∈ Λ r,s . The same result was recently obtained via a diﬀeren t approach b y Brundan and Stropp el [BS1]. Their approac h yields additional information ab out the mo dules F m | n ( L ( λ )); for instance, the irreducible so cles and heads are computed explicitly . In the case when s = 0, a bipartition λ ∈ Λ r, 0 is ( m | n )-cross if and only if λ • is ( m | n )-ho ok. Th us, in this case, Theorem 1.2.3 gives the decomp osition of the co v ariant tensor p o w er V ⊗ r familiar from the work of Sergeev [Ser2] and of Berele and Regev [BR] (similarly , in the contra v ariant case, i.e. when r = 0). On the other hand, if n = 0 then a bipartition λ is ( m | n )-cross if and only if l ( λ ) ≤ m . Th us Theorem 1.2.2 can b e seen as the sp ecial case of Theorem 1.2.3 where n = 0. 1 In [MV2], the term gl ( m | n ) -standar d is used. 4 JONA THAN COMES AND BENJAMIN WILSON F or any bipartition µ , let s µ denote the corresp onding composite sup ersymmet- ric Sch ur p olynomial (see e.g. [MV2]). Then w e hav e the following formula for the characters of the indecomp osable summands of the mixed tensor p o w ers for GL ( m | n ). Theorem 1.2.4. (se e The or em 8.5.2) L et m, n ≥ 0 and δ = m − n . Then for any bip artition λ , c h F m | n ( L ( λ )) = X µ D λ,µ ( δ ) s µ , wher e the D λ,µ ar e the de c omp osition numb ers for the wal le d Br auer algebr as. When | λ ◦ | = 0, the decomp osition num b er D λ,µ is 1 if λ = µ and 0 otherwise, and s λ is the (non-comp osite) sup ersymmetric Sch ur p olynomial asso ciated to λ • . Th us, if λ = ( λ • , ∅ ) and λ • is ( m | n )-ho ok, then the character form ula of Theorem 1.2.4 reduces to that of Sergeev [Ser2] and Berele and Regev [BR]. 1.3. The paper is organized as follo ws. W e b egin in § 2 with a review of the category-theoretic notions necessary for the deﬁnition of Deligne’s category and deriv ation of its univ ersal prop ert y in § 3. The indecomposable ob jects of Deligne’s category are then classiﬁed in § 4 using the cellular structure of the walled Brauer algebras as describ ed in [CDDM]. In § 5, the representation theory of the general linear group is brieﬂy recalled, and Koik e’s Theorem on the decomposition of tensor pro ducts of indecomp osable represen tations is review ed. The lifting isomorphism is deﬁned in § 6, relating the additive Grothendieck rings of Deligne’s category in the singular and generic cases, and it is shown that the deﬁning co eﬃcien ts are the decomp osition num b ers of the w alled Brauer algebras. The analogue of Koike’s Theorem in Deligne’s category for generic v alues of δ is presen ted in § 7, and it is sho wn that the lifting isomorphism enables the decomp osition of tensor products of indecomp osables in all cases. In § 8, comp osite supersymmetric Sch ur p olynomials are in tro duced and the results of § 6 and § 7 are emplo yed to derive the character form ula for the indecomposable summands of the mixed tensor p o wers of the general linear sup ergroup. Next, the c haracter formula is used to prov e the classiﬁcation of these indecomposable summands in terms of ( m | n )-cross bipartitions, as describ ed b y Theorem 1.2.3. Finally , we illustrate how to decompose tensor pro ducts of these indecomp osable summands with an explicit example. 1.4. Ac kno wledgemen ts. The ma jorit y of the work for this paper w as done while the authors shared an oﬃce at the T ec hnisc he Universit¨ at M ¨ unc hen. W e would lik e to thank the universit y for providing us with the time and freedom to explore this pro ject. W e are also grateful to Maud De Vissc her for explaining the results of [CD] in the case δ = 0. The second author would like to thank Victor Ostrik and Jon Brundan for man y v aluable con v ersations concerning this pap er. 2. Ca tegor y-theoretic preliminaries Let K denote a ﬁeld of c haracteristic zero. A category is said to be K -line ar if the Hom sets are equipp ed with the structure of v ector spaces ov er the ﬁeld K in suc h a wa y that comp osition of morphisms is bilinear. DELIGNE’S Rep( GL δ ) AND REPRESENT A TIONS OF gl ( m | n ) 5 2.1. Monoidal categories. F or any category C , let σ C : C × C → C × C denote the functor ( X , Y ) 7→ ( Y , X ). A monoidal c ate gory is a tuple ( C , ⊗ , 1 , c) where C is a category , T := − ⊗ − : C × C → C is a bifunctor and 1 is a distinguished ob ject (the unit for T ), satisfying T ◦ ( T × id C ) = T ◦ (id C × T ) , 1 ⊗ − = id C = − ⊗ 1 , and c : T ⇒ T ◦ σ C is a natural isomorphism (the symmetric br aiding ) satisfying (id Y ⊗ c X,Z ) ◦ (c X,Y ⊗ id Z ) = c X,Y ⊗ Z , c Y ,X ◦ c X,Y = id X ⊗ Y , (c X,Z ⊗ id Y ) ◦ (id X ⊗ c Y ,Z ) = c X ⊗ Y ,Z , for all ob jects X, Y , Z of C . When no confusion arises, we may write C for b oth the underlying category and monoidal category ( C , ⊗ C , 1 C , c C ) and omit the subscript C when it is implicit. A monoidal category , in our sense, is elsewhere called a strict, symmetric monoidal category . A monoidal functor is a tuple ( F , η , α ) where F : C → D is a functor, C , D are monoidal categories, η : F ◦ T C ⇒ T D ◦ ( F × F ) is a natural transformation and α : F ( 1 C ) → 1 D is an isomorphism suc h that η Y ,X ◦ F (c X,Y ) = c F X,F Y ◦ η X,Y , id F X ⊗ ( α ◦ η X, 1 ) = id F X , ( η X,Y ⊗ id F Z ) ◦ η X ⊗ Y ,Z = (id F X ⊗ η Y ,Z ) ◦ η X,Y ⊗ Z . The monoidal functor is strict if F ( X ⊗ Y ) = F X ⊗ F Y and η X,Y = id F X ⊗ F Y for all ob jects X , Y of C , F 1 = 1 and α = id 1 . A monoidal functor, in our sense, is elsewhere called a non-strict symmetric monoidal functor. Let C , D b e monoidal categories and let F = ( F , η , α ) , F 0 = ( F 0 , η 0 , α 0 ) : C → D b e monoidal functors. A monoidal natur al tr ansformation  : F ⇒ F 0 is a natural transformation of the underlying functors F ⇒ F 0 suc h that  1 = ( α 0 ) − 1 ◦ α and η 0 X,Y ◦  X ⊗ Y =  X ⊗  Y ◦ η X,Y , for all ob jects X, Y . As in the following examples and throughout, we use monoidal categories in place of their familiar, non-strict, counterparts when con venien t. This is without loss of generality , by Maclane’s Coherence Theorem (see e.g. [Mac2]). Example 2.1.1. F or ﬁnite-dimensional v ector spaces U , V ov er K , write U ⊗ V for the usual tensor pro duct. Deﬁne c U,V : U ⊗ V → V ⊗ U , b y u ⊗ v 7→ v ⊗ u for all u ∈ U and v ∈ V , and let 1 denote a one-dimensional vector space. Let V ect K denote the category of ﬁnite-dimensional vector spaces and linear maps ov er K , mo dulo the identiﬁcation (1) U ⊗ ( V ⊗ W ) = ( U ⊗ V ) ⊗ W for all ob jects U , V and W . Then (V ect K , ⊗ , 1 , c) is a monoidal category . Example 2.1.2. Recall that a sup erspace ov er K is a Z / 2 Z -graded v ector space U = U ¯ 0 ⊕ U ¯ 1 . Elements of U ¯ 0 and U ¯ 1 are said to be even and o dd , resp ectiv ely . An elemen t of U is said to b e pur e if it is either ev en or odd. F or u ∈ U ¯ i , write ¯ u = i for the p arity of u . A morphism of sup erspaces is simply a morphism of vector spaces. 6 JONA THAN COMES AND BENJAMIN WILSON If ϕ : U → V is a sup erspace morphism, then declare ϕ to b e pure and of parit y ¯ ϕ if ϕ : U ¯ i → V ¯ i + ¯ ϕ , i = 0 , 1 . Th us the vector space of all sup erspace morphisms U → V b ecomes itself super- space. Given sup erspaces U, V , let U ⊗ V denote the their tensor product as vector spaces, considered as a sup erspace with the grading ( U ⊗ V ) ¯ i = M ¯ j + ¯ k = ¯ i U ¯ j ⊗ V ¯ k , and deﬁne c U,V : U ⊗ V → V ⊗ U via c U,V : u ⊗ v 7→ ( − 1) ¯ u ¯ v v ⊗ u (this is the so-called rule of signs ). W rite 1 for a one-dimensional purely ev en sup erspace. Finally , write SV ect K for the category of all ﬁnite-dimensional super- spaces and their morphisms, mo dulo the identiﬁcation (1) for all sup erspaces U , V and W . Then (SV ect K , ⊗ , 1 , c) is a monoidal category . 2.2. T ensor categories. Let C b e a monoidal category and X an ob ject of C . A dual of X is a tuple ( X ∗ , ev X , coev X ) where X ∗ is an ob ject of C and ev X , co ev X are morphisms ev X : X ∗ ⊗ X → 1 , co ev X : 1 → X ⊗ X ∗ , of C such that (2) (id X ⊗ ev X ) ◦ (co ev X ⊗ id X ) = id X , (ev X ⊗ id X ∗ ) ◦ (id X ∗ ⊗ co ev X ) = id X ∗ . The category C is rigid if every ob ject has a dual. A tensor c ate gory is a rigid K -linear monoidal category such that End 1 = K and − ⊗ − is a bilinear bifunctor. A (strict) tensor functor is a (strict) monoidal functor that is preadditiv e. Let ϕ : X → Y be a C -morphism. Duals ( X ∗ , ev X , coev X ) and ( Y ∗ , ev Y , coev Y ) for X and Y deﬁne a dual morphism ϕ ∗ : Y ∗ → X ∗ b y ϕ ∗ = ev Y ⊗ id X ∗ ◦ id Y ∗ ⊗ ϕ ⊗ id X ∗ ◦ id Y ∗ ⊗ co ev X , and one has that (3) ev X ◦ ( ϕ ∗ ⊗ id X ) = ev Y ◦ (id Y ∗ ⊗ ϕ ) , (id X ⊗ ϕ ∗ ) ◦ co ev X = ( ϕ ⊗ id Y ∗ ) ◦ co ev Y . No w let D b e another monoidal category , and F = ( F , η , α ) : C → D a monoidal functor. Then the functor F and the dual for X in C deﬁne a dual (4) ( F ( X ∗ ) , ev F X , coev F X ) for F X in D , where ev F X = α ◦ F (ev X ) ◦ η − 1 X ∗ ,X co ev F X = η X,X ∗ ◦ F (co ev X ) ◦ α − 1 . Example 2.2.1. F or any v ector space U in V ect K , let U ∗ denote the usual linear dual, and deﬁne ev U : U ∗ ⊗ U → 1 , ev U : λ ⊗ u 7→ λ ( u ) , for all λ ∈ U ∗ , u ∈ U . Cho ose any basis { u i } of U , let { λ i } denote the basis of U ∗ orthonormal to it, and deﬁne co ev U : 1 → U ⊗ U ∗ , 1 7→ X i u i ⊗ λ i , DELIGNE’S Rep( GL δ ) AND REPRESENT A TIONS OF gl ( m | n ) 7 (this map is indep enden t of the choice of basis). Then ( U ∗ , ev U , coev U ) is a dual for U , and th us V ect K is a tensor category . Example 2.2.2. F or an y sup erspace V in SV ect K , write U ∗ for the superspace deﬁned by ( U ∗ ) ¯ i = ( U ¯ i ) ∗ , i = 0 , 1 and deﬁne ev U , co ev U exactly as in example (1) ab o v e, only c ho osing the basis { u i } to consist of pure elements. Then ( U ∗ , ev U , coev U ) is a dual for U , and so SV ect K is a tensor category . The following proposition will b e useful later. Prop osition 2.2.3. L et C b e a rigid monoidal c ate gory, D a monoidal c ate gory and  : ( F, η , α ) ⇒ ( F 0 , η 0 , α 0 ) a monoidal natur al tr ansformation of monoidal functors C → D . Then  is a natur al isomorphism, and for al l obje cts X in C ,  X ∗ = ((  X ) ∗ ) − 1 = (  − 1 X ) ∗ with r esp e ct to the duals for F X and F 0 X deﬁne d by (4) . Pr o of. Let X b e an ob ject of C and ( X ∗ , ev X , coev X ) a dual for X . With resp ect to the duals for F X and F 0 X deﬁned b y (4), one has (  X ) ∗ = (ev F 0 X ⊗ id F X ∗ ) ◦ (id F 0 X ∗ ⊗  X ⊗ id F X ∗ ) ◦ (id F 0 X ∗ ⊗ co ev F X ) . The second equalit y of the claim follo ws from the ﬁrst b y functoriality . W e demon- strate the ﬁrst. Using that  is b oth monoidal and natural, one sees that (5) ev F 0 X ◦ (  X ∗ ⊗  X ) = ev F X , (  X ⊗  X ∗ ) ◦ co ev F X = co ev F 0 X . Th us (  X ) ∗ ◦  X ∗ = (  X ) ∗ ◦ (  X ∗ ⊗ id 1 ) = (ev F 0 X ⊗ id F X ∗ ) ◦ (  X ∗ ⊗  X ⊗ id F X ∗ ) ◦ (id F X ∗ ⊗ co ev F X ) = id F X . One sho ws that  X ∗ ◦ (  X ) ∗ = (id 1 ⊗  X ∗ ) ◦ (  X ) ∗ = id F 0 X in a similar fashion.  2.3. Categorical dimension. Suppose that C is a rigid monoidal category . F or an y ob ject X of C and ϕ ∈ End C X , deﬁne the c ate goric al tr ac e tr ϕ b y tr ϕ = ev X ◦ c X,X ∗ ◦ ( ϕ ⊗ id X ∗ ) ◦ co ev X ∈ End C 1 , and deﬁne the c ate goric al dimension of X b y dim X = tr(id X ). The categorical trace and dimension do not depend upon the choice of dual for X , and are preserv ed b y any monoidal functor. One has, moreo ver, that dim( X ⊗ Y ) = dim X · dim Y , for all ob jects X, Y . If C is a tensor category , then one has additionally that tr X : End X → End 1 = K is a homomorphism of ab elian groups for any ob ject X , and dim( X ⊕ Y ) = dim X + dim Y , whenev er the bipro duct of ob jects X and Y exists. 8 JONA THAN COMES AND BENJAMIN WILSON Example 2.3.1. The categorical trace and dimension in V ect K coincide with their elemen tary counterparts. Example 2.3.2. The categorical trace and dimension in SV ect K coincide with sup ertrace and sup erdimension, respectively . That is, if ϕ ∈ End U is an endomor- phism in SV ect K , then tr ϕ = tr ϕ ¯ 0 − tr ϕ ¯ 1 , where the summands are traces of vector space endomorphisms, and so dim U = dim K U ¯ 0 − dim K U ¯ 1 , where the summands are vector space dimensions. When C = V ect K is the category of ﬁnite-dimensional v ector spaces, categorical trace and dimension coincide with their elementary counterparts. As describ ed in § 8.1, when C = SV ect K is the category of sup er vector spaces, the categorical dimension of an ob ject coincides with the sup erdimension. 2.4. F unctor categories. F or categories C , D , deﬁne the following categories whose ob jects are functors C → D of the sp eciﬁed type and whose morphisms are natural transformations of the speciﬁed t yp e: category C , D functors nat. trans. H om ( C , D ) an y an y an y H om ⊗ ( C , D ) monoidal monoidal monoidal H om + ( C , D ) preadditiv e preadditiv e an y H om ⊗ + ( C , D ) tensor tensor monoidal H om ⊗ -str + ( C , D ) tensor strict tensor monoidal 2.5. The additiv e en v elope. Let C b e a preadditive category . An additive en- velop e of C is a pair ( C add , ι ) where C add is an additive category and ι : C → C add is a fully-faithful preadditiv e functor such that for an y additive category D , the “restriction functor” (6) H om + ( C add , D ) ∼ − → H om + ( C , D ) , F 7→ F ◦ ι, ( η : F ⇒ F 0 ) 7→ η ι , is an equiv alence of categories 2 . Thus an additiv e env elop e is unique up to equiv a- lence of categories, when it exists, and the category C may b e iden tiﬁed with a full sub category of C add via the functor ι . The additive env elope may be constructed as follo ws. Let C add denote the cat- egory with ob jects X = ( X j ) given by ﬁnite-length tuples of ob jects from C and morphisms ϕ : ( X j ) → ( Y i ) , ϕ = ( ϕ i,j ) , ϕ i,j ∈ Hom C ( X j , Y i ) , giv en b y “matrices” of morphisms from C , comp osed via matrix m ultiplication, i.e. ( ϕ ◦ ψ ) i,j = X k ϕ i,k ◦ ψ k,j . Addition of morphisms in C add is deﬁned component-wise by addition in C . Con- catenation of tuples deﬁnes a bipro duct on C add where the injection and pro jection maps are matrices built from identit y and zero morphisms of C in a straightforw ard manner. The empty tuple is a zero ob ject for this bipro duct, and so C add is an additiv e category . The functor ι : C → C add deﬁned by sending any C -ob ject X to the length-1 tuple ( X ) and any C -morphism to the 1 × 1-matrix ( ϕ ) is fully-faithful, and is such that the univ ersal property (6) holds. It is straightforw ard to sho w that 2 preadditive functors necessarily preserve biproducts, when they exist. DELIGNE’S Rep( GL δ ) AND REPRESENT A TIONS OF gl ( m | n ) 9 if C is a tensor category , then C add can also b e endow ed with the structure of a tensor category such that ι is a tensor functor and for an y additive tensor category D , (7) H om ⊗ + ( C add , D ) ∼ − → H om ⊗ + ( C , D ) , F 7→ F ◦ ι, ( η : F ⇒ F 0 ) 7→ η ι , is an equiv alence of categories. 2.6. Splittings of Idemp oten ts & the Karoubi en v elop e. Let C b e a category , X an ob ject of C and e 2 = e ∈ End C ( X ); one sa ys that e is an idemp otent of C . A splitting of e is a tuple (im e, ι e , π e ) where im e is an ob ject of C and ι e : im e → X , π e : X → im e, are morphisms of C suc h that (1) e = ι e ◦ π e , (2) id im e = π e ◦ ι e . One says that the idemp otent e splits , and calls im e the image of e . Given e 2 = e and an y tuple satisfying part (1), part (2) is equiv alent to ι e and π e b eing mono- and epi-morphisms, resp ectiv ely . A category is said to b e Kar oubi if every idemp oten t of the category splits. A Kar oubi envelop e of a category C is a tuple ( C k ar , ι ) where C k ar is Karoubi and ι : C → C k ar is a fully-faithful functor such that for any Karoubi category D , the “restriction functor” (8) H om ( C k ar , D ) ∼ − → H om ( C , D ) , F 7→ F ◦ ι, ( η : F ⇒ F 0 ) 7→ η ι , is an equiv alence of categories. Thus a Karoubi en v elop e is unique upto equiv alence of categories, when it exists, and the category C may b e iden tiﬁed with a full sub category of C k ar via the functor ι . The Karoubi en v elope of any category C can be constructed as follo ws. Let C k ar denote the category whose ob jects are tuples ( X , e ) where X is an ob ject of C and e ∈ End C X is an idemp oten t, and morphisms Hom C k ar (( X, e ) , ( Y , f )) = { ϕ ∈ Hom C ( X, Y ) | f ◦ ϕ = ϕ = ϕ ◦ e } . W rite ϕ 0 for ϕ : ( X, e ) → ( Y , f ) considered as a morphism of C . Then C k ar - morphisms ϕ, ψ are equal if and only if they ha ve the same source and target in C k ar and ϕ 0 = ψ 0 . The comp osition of morphisms in C k ar is inherited from C , that is, ϕ ◦ ψ is deﬁned b y the source of ψ , the target of ϕ , and ( ϕ ◦ ψ ) 0 = ϕ 0 ◦ ψ 0 ; one has that (id ( X,e ) ) 0 = e . Any idempotent ϕ ∈ End C k ar ( X, e ) has a splitting (im ϕ, ι ϕ , π ϕ ) deﬁned by im ϕ = ( X , ϕ ) and ( ι ϕ ) 0 = ( π ϕ ) 0 = ϕ 0 . Thus C k ar is a Karoubi category . The functor ι : C → C k ar deﬁned by ι ( X ) = ( X, id X ) and ( ι ( ϕ )) 0 = ϕ is fully-faithful fully-faithful, and is such that the univ ersal prop ert y (8) holds. It can b e shown that if C is a tensor category , then C k ar can also be endo w ed with the structure of a tensor category suc h that ι is a tensor functor and for any Karoubi tensor category D (9) H om ⊗ + ( C k ar , D ) ∼ − → H om ⊗ + ( C , D ) , F 7→ F ◦ ι, ( η : F ⇒ F 0 ) 7→ η ι , is an equiv alence of categories. Moreov er, if C is an additive tensor category , then so is C k ar . Idemp oten ts e, e 0 of a ring are said to b e ortho gonal if ee 0 = e 0 e = 0. An idemp oten t is said to b e primitive if it is non-zero and can not b e written as the 10 JONA THAN COMES AND BENJAMIN WILSON sum of t w o non-zero orthogonal idemp oten ts. The pro of of the following lemma is elemen tary . Lemma 2.6.1. L et C b e a K -line ar Kar oubi c ate gory and let X an obje ct of C . Then X is inde c omp osable if and only if id X is a primitive idemp otent. Prop osition 2.6.2. L et C denote a c ate gory c onsider e d as a ful l sub c ate gory of its Kar oubi envelop e C k ar , and cho ose splittings for the idemp otents of C . Then any obje ct of C k ar is isomorphic to the image of an idemp otent of C . Pr o of. The claim is true of the Karoubi env elope constructed explicitly ab o ve, with its constructed splittings. Moreov er, this Karoubi env elope is equiv alen t to any other Karoubi env elope of C via a functor compatible with the identiﬁcations of C as a full subcategory and the choices of splittings.  2.7. Krull-Sc hmidt categories. A K -line ar Krul l-Schmidt category is a cate- gory that is K -linear, additive and Karoubi, with ﬁnite-dimensional Hom-spaces. Th us, if C is any K -linear category with ﬁnite-dimensional Hom-spaces, then the Karoubi env elop e of the additiv e en v elope ( C add ) k ar is an example of a K -linear Krull-Sc hmidt category . Recall that an ob ject X of a preadditive category C is inde c omp osable if for any bipro duct decomposition X = X 1 ⊕ X 2 with asso ciated maps ι i : X i → X, π i : X → X i , there exists i ∈ { 1 , 2 } with ι i ◦ π i = 0 ∈ End X . As indicated by the following prop osition, ob jects in Krull-Schmidt categories possess essen tially unique biproduct decomp ositions into indecomposable summands, as in the familiar case of ﬁnitely-generated mo dules o v er a ﬁnite-dimensional algebra. Prop osition 2.7.1. L et C b e a K -line ar c ate gory c onsider e d as a ful l sub c ate gory of its Kar oubi envelop e C k ar , let A b e an obje ct of C and e, e 0 , e 00 ∈ End C A b e idemp otents with splittings chosen in C k ar . Then: (1) im e is inde c omp osable if and only if e is a primitive idemp otent, and up to isomorphism, al l inde c omp osables ar e so obtaine d. (2) if e = e 0 + e 00 and e 0 , e 00 ar e ortho gonal, then im e ∼ = im e 0 ⊕ im e 00 . (3) Supp ose further that End C A is ﬁnite dimensional. Then im e ∼ = im e 0 if and only if e and e 0 ar e c onjugate in End C A . Pr o of. T o prov e part (1), suppose that e = e 0 + e 00 is a sum of orthogonal idemp o- ten ts in C . Then, since C k ar is K -linear, (10) π e ◦ e ◦ ι e = π e ◦ e 0 ◦ ι e + π e ◦ e 00 ◦ ι e . If, instead, id im e = f 0 + f 00 ∈ End im e is a sum of orthogonal idempotents, then similarly (11) ι e ◦ id im e ◦ π e = ι e ◦ f 0 ◦ π e + ι e ◦ f 00 ◦ π e , since C k ar is K -linear. By the splittings relations 2.6, equations (10) and (11) giv e decomp ositions of id im e and e , resp ectiv ely , as sums of orthogonal idemp oten ts. Moreo ver, the substitution of the summands of (10) for f 0 , f 00 in (11) yields the original decomp osition e = e 0 + e 00 , and inv ersely . Th us there is a bijectiv e corre- sp ondence b et w een orthogonal idemp oten t decomp ositions of e in C and id im e in C k ar that is linear in b oth summands. Thus the claim follo ws from lemma 2.6.1 and prop osition 2.6.2. P art (2) follows immediately from C k ar b eing both K -linear and Karoubi and the deﬁnition of a biproduct. DELIGNE’S Rep( GL δ ) AND REPRESENT A TIONS OF gl ( m | n ) 11 T o prov e part (3), supp ose that ϕ ∈ End C ( A ) and e 0 = ϕeϕ − 1 . Then π e 0 ◦ ϕ ◦ ι e : im e → im e 0 is an isomorphism with inv erse π e ◦ ϕ − 1 ◦ ι e 0 . Con v ersely , supp ose that Φ : im e → im e 0 is an isomorphism. Then the map (End A ) e 0 → (End A ) e : α 7→ α ◦ ι e 0 ◦ Φ ◦ π e ◦ e is an isomorphism of left End A -mo dules with inv erse β 7→ β ◦ ι e ◦ Φ − 1 ◦ π e 0 ◦ e 0 b y the splitting relations 2.6. Since End A is a ﬁnite-dimensional algebra, it follows that e and e 0 are conjugate 3 .  Recall that a ring R is semip erfe ct if R/J is semisimple and idemp oten ts of R /J lift to R , where J = J ( R ) denotes the radical. In particular, any ﬁnite-dimensional algebra is semip erfect (see e.g. [AF]). Prop osition 2.7.2. L et C b e a pr e additive c ate gory and X an obje ct of C . Then (1) If End X is lo c al, then X is inde c omp osable. (2) Supp ose further that C is Kar oubi and End X is semip erfe ct. Then if X is inde c omp osable, then End X is lo c al. Pr o of. Supp ose that X = X 1 ⊕ X 2 and write ι i : X i → X , π i : X → X i for the morphisms deﬁning the bipro duct decomp osition. Then e i = π i ◦ ι i is an idempotent in End X for i = 1 , 2. As End X is lo cal, it has no non-trivial idemp oten ts. Hence e 1 = 0 or e 2 = 0. Supp ose that C is Karoubi, that X is indecomposable and that R = End X is semip erfect. Then the ring R/J is semisimple, that is, is a semisimple mo dule ov er itself. Since idemp oten ts in C split, R = End X has no non-trivial idemp oten ts, and since R is semip erfect, the same is true of R /J . Hence R /J is a simple mo dule o ver itself, as any non-trivial summand deﬁnes a non-trivial idemp oten t of R/J . Th us J is a maximal left ideal of R . But J is the in tersection of all maximal left ideals of R , so J is the unique maximal left ideal. Therefore R is lo cal.  Corollary 2.7.3. L et C b e a K -line ar Krul l-Schmidt c ate gory and let X b e an obje ct of C . Then X is inde c omp osable if and only if End X is lo c al. Prop osition 2.7.4. L et C b e a K -line ar Krul l-Schmidt c ate gory, D a pr e additive c ate gory and F : C → D a ful l pr e additive functor. Then F X is inde c omp osable obje ct of D if X is an inde c omp osable obje ct of C . Mor e over, if X, Y ar e inde c om- p osable obje cts of C such that F X , F Y ar e non-zer o isomorphic obje cts of D , then X ∼ = Y . Pr o of. The ﬁrst part follows from corollary 2.7.3 since homomorphic images of lo cal rings are lo cal. T o prov e the second part, supp ose that X , Y are indecomp osable ob jects of C and that F X ∼ = F Y are non-zero in D . As F is full, there exist morphisms ϕ : X → Y , ψ : Y → X such that F ( ψ ◦ ϕ ) = id F X . Th us α = ψ ◦ ϕ is not nilp oten t. Hence α 6∈ J , where J is the radical of the ﬁnite-dimensional algebra End X . As End X is lo cal, J is the unique maximal left ideal, so it follows that (End X ) α = End X . Let β ∈ End X b e such that β ◦ α = β ◦ ψ ◦ ϕ = id X . Then 3 Recall that if Λ is a ﬁnite-dimensional algebra, then idemp oten ts e, e 0 ∈ Λ are conjugate if and only if Λ e ∼ = Λ e 0 as left Λ-modules. 12 JONA THAN COMES AND BENJAMIN WILSON ϕ ◦ β ◦ ψ is a non-zero idemp oten t in the lo cal algebra End Y , hence is equal to id Y . Th us X ∼ = Y .  2.8. The additive Grothendiec k ring R C . Let C denote a K -linear Krull-Schmidt category , let Z [ C ] denote the free Z -module generated b y the isomorphism classes of the ob jects of C , and let ( · , · ) C denote the bilinear form on Z [ C ] deﬁned by bilinear extension of the rule ([ U ] , [ V ]) C = dim K Hom C ( U, V ) . W rite R C for the quotient of Z [ C ] by the relations [ A ⊕ B ] − [ A ] − [ B ] for all ob jects A , B in C . Thus R C is the free Z -mo dule generated b y the iso classes of the indecomp osable ob jects of C . Since Hom( A ⊕ A 0 , B ) ∼ = Hom( A, B ) ⊕ Hom( A 0 , B ) , and similarly in the second argumen t, the deﬁning relations of R C are con tained in the left and right radicals of the bilinear form. By abuse of notation, we use the same notation for the bilinear form induced in this wa y on R C . W e call R C the additive Gr othendie ck gr oup of C . Note that in the case where C is semisimple, R C is the ordinary Grothendieck group and the bilinear form is non-degenerate with an orthonormal basis given by the isomorphism classes of the simple ob jects. No w suppose that C is a K -linear Krull Schmidt tensor category . F or an y ob ject A of C , the functors − ⊗ A and A ⊗ − are preadditiv e, and so setting [ A ][ B ] = [ A ⊗ B ] for all ob jects A, B of C deﬁnes a bilinear multiplication on R C . This multiplication is comm utativ e, since C carries a symmetric braiding, and has unit [ 1 ]. Th us R C is a commutativ e ring, called the additive Gr othendie ck ring of C . The duality on C deﬁnes an inv olutiv e ring automorphism ∗ of R C via [ A ] ∗ = [ A ∗ ] for all ob jects A of C . As duality deﬁnes a contra v arian t endofunctor, one has ([ A ] , [ B ]) C = ([ B ] ∗ , [ A ] ∗ ) C for all ob jects A, B . Moreo v er, the Hom-set adjunction Hom( A ⊗ B , C ) ∼ = Hom( A, C ⊗ B ∗ ) (an immediate consequence of equations (2)) gives the inv ariance relation ([ A ][ B ] , [ C ]) = ([ A ] , [ C ][ B ] ∗ ) for all ob jects A, B , C . 3. The ca tegor y Re p ( GL δ ) In this section we deﬁne Deligne’s tensor category Rep( GL δ ) and prov e that it satisﬁes a certain universal prop ert y (see Proposition 3.5.1). T o deﬁne Rep( GL δ ) w e ﬁrst diagrammatically deﬁne a smaller “skeleton category .” W e then tak e the additiv e en velope ( § 2.5) follow ed by the Karoubi env elope ( § 2.6) of this skeleton category to get Rep( GL δ ). T o start, we in troduce the diagrams we will use to deﬁne the skeleton category . DELIGNE’S Rep( GL δ ) AND REPRESENT A TIONS OF gl ( m | n ) 13 3.1. W ords and diagrams. Suppose w and w 0 are ﬁnite (p ossibly empty) w ords in tw o letters denoted • (blac k letter) and ◦ (white letter). A ( w, w 0 ) -diagr am is a graph which satisﬁes the following conditions: (i) The vertices are p ositioned in t w o (p ossibly empt y) horizontal ro ws. (ii) Eac h v ertex is dra wn as either • or ◦ so that the b ottom (resp. top) row of v ertices is the w ord w (resp. w 0 ). (iii) Eac h vertex is adjacent to exactly one edge. (iv) An edge is adjacen t to both a blac k and a white v ertex if and only if the v ertices adjacen t to that edge are either both in the top row or both in the b ottom ro w. An edge in a ( w , w 0 )-diagram is called a pr op agating e dge if it is adjacen t to a v ertex in the top row and a vertex in the b ottom ro w. Example 3.1.1. (1) Let 1 denote the empty w ord. The empt y graph is the unique ( 1 , 1 )-diagram. On the other hand, there are tw o ( • • ◦◦ , 1 )-diagrams: (2) There are six ( • ◦ ◦• , •◦ )-diagrams: Eac h of the top four ( • ◦ ◦• , •◦ )-diagrams hav e t w o propagating edges, whereas the b ottom t w o hav e no propagating edges. Remark 3.1.2. Suppose w (resp. w 0 ) is a word with r (resp. r 0 ) blac k letters and s (resp. s 0 ) whites letters. It is easy to show that a ( w , w 0 )-diagram exists if and only if r + s 0 = r 0 + s , in which case the num b er of ( w , w 0 )-diagrams is ( r + s 0 )!. Supp ose w , w 0 , and w 00 are ﬁnite words in the letters • and ◦ . Giv en a ( w , w 0 )- diagram X and a ( w 0 , w 00 )-diagram Y , w e let Y  X denote the graph obtained b y stacking Y atop X so that the top ro w of v ertices of X are identiﬁed with the b ottom ro w of v ertices of Y . Next, we let Y · X denote the ( w, w 00 )-diagram obtained b y restricting Y  X to its top and b ottom rows of vertices. Finally , let  ( X , Y ) denote the num b er of cycles in Y  X (i.e. the num b er of connected components of Y  X min us the num ber of connected comp onen ts of Y · X ). F or example, if . 14 JONA THAN COMES AND BENJAMIN WILSON 3.2. The skeleton category. Fix δ ∈ K . Using the setup from 3.1 w e can now deﬁne the skeleton category Rep 0 ( GL δ ). Deﬁnition 3.2.1. The category Rep 0 ( GL δ ) has Ob jects: ﬁnite w ords in the letters • and ◦ . Morphisms: Hom( w, w 0 ) is the K -v ector space on basis { ( w , w 0 )-diagrams } . Comp osition: Hom( w 0 , w 00 ) × Hom( w , w 0 ) → Hom( w, w 00 ) sending ( f , g ) 7→ f g is the K -bilinear map satisfying Y X = δ ` ( X,Y ) Y · X whenev er X is a ( w, w 0 )-diagram and Y is a ( w 0 , w 00 )-diagram. T o sho w Rep 0 ( GL δ ) is indeed a category , it is easy to c hec k that comp osition in Rep 0 ( GL δ ) is asso ciativ e. Also, if w is a ﬁnite word in • and ◦ , then the ( w , w )- diagram with eac h vertex in the b ottom row adjacen t to the vertex directly ab o v e it is the iden tit y morphism in End( w ). F or example, . 3.3. T ensor category structure of Rep 0 ( GL δ ) . W e will no w equip Rep 0 ( GL δ ) with the structure of a tensor category in the sense of § 2.2. Deﬁnition 3.3.1. The bifunctor − ⊗ − : Rep 0 ( GL δ ) × Rep 0 ( GL δ ) → Rep 0 ( GL δ ) is deﬁned as follo ws: On ob jects: Set w 1 ⊗ w 2 = w 1 w 2 (concatenation of w ords) for an y ob jects w 1 and w 2 in Rep 0 ( GL δ ). On morphisms: Assume w i and w 0 i are ﬁnite w ords in • and ◦ , and X i is a ( w i , w 0 i )-diagram for i = 1 , 2. Let X 1 ⊗ X 2 denote the ( w 1 w 2 , w 0 1 w 0 2 )-diagram ob- tained by placing X 1 directly to the left of X 2 . Now extend K -linearly in both argumen ts to deﬁne tensor pro ducts of arbitrary morphisms in Rep 0 ( GL δ ). Let c w 1 ,w 2 : w 1 ⊗ w 2 → w 2 ⊗ w 1 b e the ( w 1 w 2 , w 2 w 1 )-diagram such that the v ertex in the bottom row corresp onding to the i th letter in w 1 (resp. w 2 ) is adjacen t to the vertex in the top row corresp onding to the i th letter in w 1 (resp. w 2 ). F or example, . It is easy to see that Deﬁnition 3.3.1 gives Rep 0 ( GL δ ) the structure of a monoidal category with unit ob ject 1 (the empt y word) and symmetric braiding c . Next, we will show that Rep 0 ( GL δ ) is rigid. T o do so, given a ﬁnite word w in • and ◦ let w ∗ denote the w ord obtained from w by replacing all black letters with white letters and vice versa. No w deﬁne the morphism ev w : w ∗ ⊗ w → 1 (resp. co ev w : 1 → w ⊗ w ∗ ) to b e the ( w ∗ w , 1 )-diagram (resp. ( 1 , w w ∗ )-diagram) such that the i th letter in w ∗ is adjacent to the i th letter in w for all i . F or example, . It is easy to c hec k that ev w and coev w mak e w ∗ a dual to w . Since End( 1 ) = K and − ⊗ − is bilinear, it follows that Rep 0 ( GL δ ) is a tensor category . DELIGNE’S Rep( GL δ ) AND REPRESENT A TIONS OF gl ( m | n ) 15 3.4. Univ ersal prop ert y of Rep 0 ( GL δ ) . F ollowing § 2.3 w e can compute the cat- egorical dimension of an y ob ject in Rep 0 ( GL δ ). In particular, dim( • ) = ev • c • , ◦ co ev • = δ where the last equalit y follows from the fact that . Rep 0 ( GL δ ) is characterized as the universal tensor category generated by an ob- ject of dimension δ and its dual [Del1]. More precisely , it p ossesses the following univ ersal prop ert y: Prop osition 3.4.1. Given a tensor c ate gory T , let T δ denote the c ate gory of δ - dimensional obje cts in T and their isomorphisms. Then the fol lowing functor in- duc es an e quivalenc e of c ate gories: Θ : H om ⊗ -str + (Rep 0 ( GL δ ) , T ) → T δ F 7→ F ( • ) ( η : F ⇒ F 0 ) 7→ η • Pr o of. Since tensor functors preserve categorical dimension, Θ( F ) = F ( • ) is an ob ject of the category T δ , and by Proposition 2.2.3, Θ( η ) = η • is an isomorphism of F ( • ), hence a morphism of T δ . Let X be an ob ject of T δ and let ( X ∗ , ev X , coev X ) b e a dual for X in T . Let w, w 0 b e ﬁnite words in • and ◦ . By the coherence theorem for tensor categories (see for instance [Sel] and references therein), the image of an y ( w , w 0 )-diagram under a strict tensor functor is completely determined by the image of (12) id • , id ◦ , ev • , coev • . Since ev X ◦ c X,X ∗ ◦ co ev X = dim X = δ, there exists a unique, well-deﬁned, strict tensor functor F X : Rep 0 ( GL δ ) → T with F X : • 7→ X , ◦ 7→ X ∗ ev • 7→ ev X , co ev • 7→ coev X . Th us Θ is essen tially surjective. W e no w sho w that Θ is full. Let X , Y b e ob jects of T δ and write F X , F Y for the functors deﬁned X, Y as ab ov e. Suppose no w that ϕ : X → Y is a morphism in T δ ; so ϕ is in v ertible. Deﬁne a family of isomorphisms  = (  w : F X ( w ) ⇒ F Y ( w )) w indexed by ﬁnite w ords w , b y  • = ϕ,  ◦ = ( ϕ − 1 ) ∗ and  w ⊗ w 0 =  w ⊗  w 0 for all ﬁnite words w , w 0 . It remains to show that  is a natural transformation, that is, for all morphisms σ : w → w 0 in Rep 0 ( GL δ ), that F Y ( σ ) ◦  w =  w 0 ◦ F X ( σ ) . F or ﬁnite w ords w , w 0 , we ha v e F Y (c w,w 0 ) ◦  ww 0 = c F Y w,F Y w 0 ◦  ww 0 =  w 0 w ◦ c F X w,F X w 0 =  w 0 w ◦ F X (c w,w 0 ) . 16 JONA THAN COMES AND BENJAMIN WILSON Th us, in verifying naturality , the words may b e reordered. As  is monoidal b y construction, if suﬃces to chec k naturalit y in the cases where σ is one of the four morphisms (12). The cases of the identit y morphisms are trivial. In the cases of the latter tw o morphisms, the naturality relations are precisely ev X = ev Y ◦ ( ϕ − 1 ) ∗ ⊗ ϕ, ϕ ⊗ ( ϕ − 1 ) ∗ ◦ co ev X = co ev Y , whic h follow immediately from (3). Th us Θ is full. Finally , suppose that  is a morphism of H om ⊗ -str + (Rep 0 ( GL δ ) , T ). Since  is monoidal, it is determined by  • and  ◦ . By Prop osition 2.2.3,  ◦ = (  − 1 • ) ∗ , and so  is determined b y  • alone. Th us Θ is faithful.  3.5. Deﬁnition of Rep( GL δ ) . Let Rep( GL δ ) = (Rep 0 ( GL δ ) add ) k ar denote the Karoubi en velope of the additive env elope of Rep 0 ( GL δ ), as p er § 2.5 and § 2.6. Thus Rep 0 ( GL δ ) may be identiﬁed with a full sub category of Rep( GL δ ), and the tensor category structure of Rep( GL δ ) extends that of Rep 0 ( GL δ ). F or every idemp oten t e of Rep 0 ( GL δ ), ﬁx a splitting (im e, ι e , π e ) of e in Rep( GL d ). As a notational con venience, whenev er e ∈ End X and f ∈ End Y are idemp oten ts of ob jects X , Y of Rep 0 ( GL d ), identify Hom Rep( GL d ) (im e, im f ) = f Hom Rep 0 ( GL d ) ( X, Y ) e ⊂ Hom Rep 0 ( GL d ) ( X, Y ) via the morphisms ι e , ι f , π e , π f . F or an y tensor category T , let H om 0 (Rep( GL δ ) , T ) denote the full sub category of H om ⊗ + (Rep( GL δ ) , T ) whose ob jects are those func- tors whose restriction Rep 0 ( GL δ ) → T yields a strict tensor functor. The category Rep( GL δ ) has the follo wing universal property (see [Del1, Prop osition 10.3]). Prop osition 3.5.1. Supp ose that T is a tensor c ate gory and let T δ b e as in Pr op o- sition 3.4.1. Then the fol lowing functor induc es an e quivalenc e of c ate gories: H om 0 (Rep( GL δ ) , T ) → T δ F 7→ F ( • ) ( η : F ⇒ F 0 ) 7→ η • Pr o of. The universal prop erties of the additive en v elop e (7) and Karoubi env elop e (9), yield that H om ⊗ + (Rep( GL δ ) , T ) ∼ = H om ⊗ + (Rep 0 ( GL δ ) , T ) via the restriction functors there describ ed. As H om ⊗ -str + (Rep 0 ( GL δ ) , T ) is a full sub category of the latter, the result follo ws from Prop osition 3.4.1.  4. Indecomposable objects in Rep ( GL δ ) The main goal of this section is to classify isomorphism classes of indecomp os- able ob jects in Rep( GL δ ). By Proposition 2.7.1(1), these indecomposable ob jects corresp ond to primitive idempotents in endomorphism algebras of Rep 0 ( GL δ ). In ligh t of this, we ﬁrst describe the classiﬁcation of conjugacy classes of suc h primitive idemp oten ts. T o start, let us ﬁx some notation. F or nonnegative integers r and s , let w r,s denote the word with r black letters follo wed b y s white letters: w r,s = • · · · • | {z } r ◦ · · · ◦ | {z } s . DELIGNE’S Rep( GL δ ) AND REPRESENT A TIONS OF gl ( m | n ) 17 Using the symmetric braiding it is easy to see that every ob ject in Rep 0 ( GL δ ) is naturally isomorphic to w r,s for some r, s ≥ 0. Hence, w e will only consider endomorphisms of the w r,s ’s. W e will write K B r,s ( δ ) (or just B r,s ) for the endo- morphism algebra End( w r,s ). The algebras B r,s are the so-called wal le d Br auer algebr as (compare with [BCH + ], [Koi], [T ur]). It is well known that conjugacy classes of primitiv e idemp oten ts in an algebra A are in bijective corresp ondence with isomorphism classes of simple A -mo dules 4 , which in turn are in bijective cor- resp ondence with isomorphisms classes of pro jective indecomp osable A -mo dules, hereafter referred to as PIMs (see for example [Ben]). In this correspondence a primitiv e idempotent e ∈ A corresp onds to the PIM eA . The isomorphism classes of simples in the walled Brauer algebras are classiﬁed in [CDDM]. T o explain their classiﬁcation, we ﬁrst need to recall some prop erties of (bi)partitions and their relation to symmetric groups. 4.1. (Bi)partitions. A p artition is a tuple of nonnegativ e in tegers α = ( α 1 , α 2 , . . . ) whose p arts (i.e. α i ’s) are such that α i ≥ α i +1 for all i > 0, and α i = 0 for all but ﬁnitely many i . W e write | α | = P i> 0 α i for the size of α and w e write α ` | α | . W e deﬁne the length of α , written l ( α ), to be the smallest positive integer with α l ( α )+1 = 0. W e will sometimes write ( · · · 2 a 2 1 a 1 ) for the partition with a i parts equal to i . It will be conv enien t for us to identify a partition α with its Y oung diagr am which consists of l ( α ) left-aligned ro ws of b o xes, with α i b o xes in the i th ro w (reading from top to b ottom). F or example, . Next, w e let α t denote the tr ansp ose of α , i.e. α t i is the num b er of b o xes in the the i th column of α . F or example, (5 , 2 3 , 1 2 ) t = (6 , 4 , 1 3 ). Finally , we let P denote the set of all partitions. Elemen ts of P × P are called bip artitions . Giv en a bipartition λ , we let λ • and λ ◦ denote the partitions such that λ = ( λ • , λ ◦ ). W e write | λ | = ( | λ • | , | λ ◦ | ) for the size of λ and we write λ ` | λ | . Moreo v er, we write l ( λ ) := l ( λ • ) + l ( λ ◦ ) for the length of λ . W e deﬁne a partial order on sizes of bipartitions by declaring that ( a, b ) ≤ ( c, d ) whenev er a ≤ c and b ≤ d . In particular, we write | µ | < | λ | to mean | µ | ≤ | λ | and µ 6 = λ . Additionally , we set λ ∗ = ( λ • , λ ◦ ) ∗ = ( λ ◦ , λ • ). W e also ha v e a bipartition- v ersion of a Y oung diagram whic h w e get as follo ws: ﬁrst place the Y oung diagram of λ ◦ atop the Y oung diagram of λ • so that the upper left corners are o v erlapping, then rotate the Y oung diagram of λ ◦ 180 degrees ab out its upp er left corner. F or 4 It will b e convenien t for us to work with right mo dules. How ev er, the categories of right and left B r,s -modules are equiv alent via the anti-automorphism on B r,s given b y reading diagrams up rather than down the page. 18 JONA THAN COMES AND BENJAMIN WILSON example, the diagram associated to the bipartition ((4 , 3 , 1) , (2 2 , 1 2 )) is . 4.2. Symmetric groups. F or a nonnegative integer r , let Σ r denote the symmetric group on r -elements, and let K Σ r denote the corresponding group algebra 5 . It is well known that the primitive idemp oten ts in K Σ r (up to conjugation) are in bijectiv e corresp ondence with partitions of size r (see for example [FH]). Given α ` r , let z α ∈ K Σ r denote the corresp onding primitiv e idemp oten t. F or example, z ( r ) = 1 n ! P σ ∈ Σ r σ so that the partition ( r ) corresp onds to the trivial K Σ r -mo dule. W e now connect the theory of symmetric groups with that of the w alled Brauer algebras. Regardless of δ , the w alled Brauer algebras B r, 0 and B 0 ,r are isomorphic to the group algebra K Σ r . These isomorphisms are given b y B r, 0 ∼ ← − K Σ r ∼ − → B 0 ,r σ • ← [ σ 7→ σ ◦ where, giv en σ ∈ Σ r , σ • (resp. σ ◦ ) is the ( w r, 0 , w r, 0 )-diagram (resp. ( w 0 ,r , w 0 ,r )- diagram) whose i th b ottom v ertex is adjacent to its σ ( i )th top v ertex (reading left to right) for 1 ≤ i ≤ r . F or example, if σ ∈ Σ 5 is the 3-cycle 2 7→ 3 7→ 5 7→ 2, then . More generally , given nonnegativ e integers r and s , w e ha v e an inclusion of algebras K [Σ r × Σ s ]  → B r,s giv en b y ( σ, τ ) 7→ σ • ⊗ τ ◦ for all σ ∈ Σ r , τ ∈ Σ s (here K [Σ r × Σ s ] denotes the group algebra of the direct pro duct Σ r × Σ s ). Using this embedding w e can consider K [Σ r × Σ s ] as a subalgebra of B r,s , and we will do so for the rest of the pap er. No w, let J ⊂ B r,s denote the K -span of all ( w r,s , w r,s )-diagrams with less than r + s propagating edges. One can sho w that J is a tw o-sided ideal in B r,s with B r,s /J ∼ = K [Σ r × Σ s ] (see [CDDM, Proposition 2.3 and (2)]). Hence, we hav e a surjection of algebras π : B r,s  K [Σ r × Σ s ]. It is straigh tforw ard to show (13) π ( a ) = a for all a ∈ K [Σ r × Σ s ] ⊂ B r,s . 4.3. Deﬁnition of the idemp oten t e λ . Giv en a bipartition λ ` ( r , s ), w e set z λ := z • λ • ⊗ z ◦ λ ◦ ∈ K [Σ r × Σ s ] ⊂ B r,s . Note that z λ is an idemp otent deﬁned up to conjugation. The assignmen t λ 7→ z λ induces a bijection betw een bipartitions of size ( r , s ) and primitiv e idemp otents in K [Σ r × Σ s ] (up to conjugation). It is important to notice that while z λ is a primitive idemp oten t in K [Σ r × Σ s ], it will generally not b e primitive in the (usually) larger algebra B r,s . Let z λ = e 1 + · · · + e k b e a decomp osition of z λ in to m utually orthogonal primitive idemp oten ts in B r,s . Then π ( e 1 ) , . . . , π ( e k ) are mutually orthogonal idemp oten ts in K [Σ r × Σ s ] whose sum, b y (13), is z λ . As z λ is primitiv e in K [Σ r × Σ s ], there is a unique i ∈ { 1 , . . . , k } such 5 By con ven tion, Σ 0 denotes the trivial group of size 0! = 1, hence we iden tify K Σ 0 = B 0 , 0 = K . DELIGNE’S Rep( GL δ ) AND REPRESENT A TIONS OF gl ( m | n ) 19 that π ( e i ) 6 = 0. Set e λ = e i . Again, note that e λ ∈ B r,s is a primitiv e idemp oten t deﬁned up to conjugation. Example 4.3.1. (1) Let ∅ denote the empty partition (0 , 0 , . . . ). Then z ∅ is the iden tity element of K Σ 0 = B 0 , 0 . Hence, if λ ` ( r , 0), then z λ = z • λ • ⊗ id 1 = z • λ • . Moreo ver, since B r, 0 = K Σ r , we also ha v e e λ = z • λ • . Similarly , if λ ` (0 , s ) then e λ = z ◦ λ ◦ . As a sp ecial case, e ( ∅ , ∅ ) = id 1 (the empty graph). (2) Consider the bipartition ( 2 , 2 ). z • 2 = id • and z ◦ 2 = id ◦ whic h implies z ( 2 , 2 ) = id •◦ . If δ 6 = 0, then id •◦ decomp oses as e 1 + e 2 in B 1 , 1 where e 1 and e 2 are the following orthogonal primitiv e idemp oten ts: . In this case π ( e 1 ) = id •◦ and π ( e 2 ) = 0, hence e ( 2 , 2 ) = e 1 . If δ = 0, then id •◦ is primitiv e in B 1 , 1 , and hence is equal to e ( 2 , 2 ) . W e close this subsection with the following useful prop osition. F or a proof, we refer the reader to the proof of a completely analogous statement for Rep( S t ) found [CO, Prop osition 3.8]. Prop osition 4.3.2. The idemp otents e λ ar e absolutely primitive. In other wor ds, if K ⊂ K 0 is a ﬁeld extension then e λ ∈ K B r,s ( δ ) is primitive when viewe d as an idemp otent in K 0 B r,s ( δ ) . 4.4. Deﬁnition of the idempotent e ( i ) λ . Next, w e explain ho w to construct new idemp oten ts from the the e λ ’s. Consider the follo wing morphisms: ψ r,s = (id • ) ⊗ r ⊗ co ev • ⊗ (id ◦ ) ⊗ s , ˆ ψ r,s = (id • ) ⊗ r ⊗ ev ◦ ⊗ (id ◦ ) ⊗ s , φ r,s = (id • ) ⊗ r ⊗ ((ev ◦ ⊗ id ◦ )(id • ⊗ c ◦ , ◦ )) ⊗ (id ◦ ) ⊗ s − 1 ( s > 0) , ˆ φ r,s = (id • ) ⊗ r − 1 ⊗ ((id • ⊗ ev ◦ )( c • , • ⊗ id ◦ )) ⊗ (id ◦ ) ⊗ s ( r > 0) . F or example, The following iden tities easily follow from the deﬁnitions ab o v e: (14) φ r,s ψ r,s = id w r,s , ˆ φ r,s ψ r,s = id w r,s , ˆ ψ r,s ψ r,s = δ id w r,s . No w, giv en a bipartition λ ` ( r, s ) we set e (0) λ = e λ and deﬁne e ( i ) λ ∈ B r + i,s + i for i > 0 recursiv ely by e ( i ) λ =        ψ r + i − 1 ,s + i − 1 e ( i − 1) λ φ r + i − 1 ,s + i − 1 if s > 0 , ψ r + i − 1 ,s + i − 1 e ( i − 1) λ ˆ φ r + i − 1 ,s + i − 1 if s = 0 and r > 0 , 1 δ ψ i − 1 ,i − 1 e ( i − 1) λ ˆ ψ i − 1 ,i − 1 if λ = ( ∅ , ∅ ) and δ 6 = 0 . 20 JONA THAN COMES AND BENJAMIN WILSON Notice that e ( i ) λ is undeﬁned when i > 0, λ = ( ∅ , ∅ ) and δ = 0. How ev er, e ( i ) λ is deﬁned (up to conjugation) in all other cases. Example 4.4.1. (1) In Example 4.3.1(1) we found that e ( ∅ , ∅ ) is the empty graph. Hence for δ 6 = 0 we ha v e (2) By Example 4.3.1(1), for any δ ∈ K w e hav e . (3) If δ = 0, then it follo ws from Example 4.3.1(2) that On the other hand, if δ 6 = 0 then by Example 4.3.1(2) It follows from (14) that e ( i ) λ is an idempotent whenever it is deﬁned. Hence, the image of e ( i ) λ is an ob ject in Rep( GL δ ). Prop osition 4.4.2. Given a bip artition λ , the obje cts im e λ and im e ( i ) λ ar e iso- morphic in Rep( GL δ ) whenever e ( i ) λ is deﬁne d. Pr o of. It suﬃces to sho w im e ( i − 1) λ ∼ = im e ( i ) λ whenev er i > 0. Using (14) it is easy to c heck that the follo wing table lists m utually inv erse morphisms b et ween im e ( i − 1) λ and im e ( i ) λ in all desired cases: im e ( i − 1) λ → im e ( i ) λ im e ( i ) λ → im e ( i − 1) λ case: ψ r + i − 1 ,s + i − 1 e ( i − 1) λ e ( i − 1) λ φ r + i − 1 ,s + i − 1 s > 0 ψ r + i − 1 ,s + i − 1 e ( i − 1) λ e ( i − 1) λ ˆ φ r + i − 1 ,s + i − 1 s = 0 , r > 0 ψ i − 1 ,i − 1 e ( i − 1) λ 1 δ e ( i − 1) λ ˆ ψ i − 1 ,i − 1 λ = ( ∅ , ∅ ) , δ 6 = 0  DELIGNE’S Rep( GL δ ) AND REPRESENT A TIONS OF gl ( m | n ) 21 Corollary 4.4.3. e ( i ) λ is absolutely primitive whenever it is deﬁne d. Pr o of. By Proposition 4.3.2, e λ is absolutely primitive. Thus, b y Proposition 2.7.1(1), im e λ ∼ = im e ( i ) λ is absolutely indecomp osable, which implies e ( i ) λ is ab- solutely primitive.  4.5. Classiﬁcation of primitiv e idemp oten ts in w alled Brauer algebras. W e are no w in position to state the classiﬁcation of conjugacy classes of primitive idemp oten ts in walled Brauer algebras. The next theorem is merely a translation of the classiﬁcation of simple modules for walled Brauer algebras [CDDM, Theorem 2.7] to the language of primitive idempotents (see for instance [Ben]). Theorem 4.5.1. (1) If r 6 = s or δ 6 = 0 then { e ( i ) λ | λ ` ( r − i, s − i ) , 0 ≤ i ≤ min( r , s ) } is a c omplete set of p airwise non-c onjugate primitive idemp otents in B r,s . (2) If δ = 0 and r > 0 , then { e ( i ) λ | λ ` ( r − i, r − i ) , 0 ≤ i < r } is a c omplete set of p airwise non-c onjugate primitive idemp otents in B r,r . Corollary 4.5.2. Primitive idemp otents in wal le d Br auer algebr as ar e absolutely primitive. Pr o of. This follows from Theorem 4.5.1 along with Corollary 4.4.3.  4.6. Classiﬁcation of indecomp osable ob jects in Rep( GL δ ) . Giv en a biparti- tion λ , let L ( λ ) denote the image of e λ in Rep( GL δ ). Since the primitive idemp oten t e λ is only deﬁned up to conjugation, L ( λ ) is an indecomp osable ob ject in Rep( GL δ ) whic h is deﬁned up to isomorphism. The follo wing proposition concerning L ( λ ) will b e used to prov e our upcoming classiﬁcation of indecomp osable ob jects. Since the result will b e used later in the pap er, w e record it separately here: Prop osition 4.6.1. If λ and µ ar e bip artitions with Hom( L ( λ ) , L ( µ )) 6 = 0 , then | λ • | + | µ ◦ | = | λ ◦ | + | µ • | . Pr o of. Hom( L ( λ ) , L ( µ )) = Hom(im e λ , im e µ ) ⊂ Hom( w | λ • | , | λ ◦ | , w | µ • | , | µ ◦ | ), hence the prop osition follo ws from Remark 3.1.2.  No w we are ready to classify indecomposable ob jects in Rep( GL δ ). Theorem 4.6.2. The assignment λ 7→ L ( λ ) induc es a bije ction  bip artitions of arbitr ary size  bij. − →  nonzer o inde c omp osable obje cts in Rep( GL δ ) , up to isomorphism  Pr o of. By Prop osition 2.7.1(1) ev ery indecomp osable ob ject in Rep( GL δ ) is isomor- phic to the image of a primitive idemp oten t endomorphism in Rep 0 ( GL δ ). Since ev ery ob ject in Rep 0 ( GL δ ) is isomorphic to w r,s for some r , s ≥ 0, it follows that eac h indecomposable ob ject in Rep( GL δ ) is isomorphic to the image of a primitive idemp oten t in B r,s for some r, s ≥ 0. Hence, by Theorem 4.5.1 and Prop osition 4.4.2 the assignment λ 7→ L ( λ ) is surjective. No w supp ose λ ` ( r, s ) and µ ` ( r 0 , s 0 ) are tw o bipartitions with L ( λ ) ∼ = L ( µ ). F or con v enience, assume r ≥ r 0 so that ( r, s ) = ( r 0 + i, s 0 + i ) for some integer i ≥ 0 (Proposition 4.6.1). If µ = ( ∅ , ∅ ), then the existence of an isomorphism L (( ∅ , ∅ )) → L ( λ ) implies that the composition map (15) Hom( w r,s , 1 ) × Hom( 1 , w r,s ) → B 0 , 0 22 JONA THAN COMES AND BENJAMIN WILSON is nonzero. If δ = 0, then the map (15) is necessarily zero unless r = s = 0. Hence, if µ = ( ∅ , ∅ ) and δ = 0, then λ = ( ∅ , ∅ ) to o. No w assume µ 6 = ( ∅ , ∅ ) or δ 6 = 0 so that e ( i ) µ is deﬁned. By Prop osition 4.4.2, im e ( i ) µ ∼ = im e µ ∼ = im e λ , which implies e ( i ) µ and e λ are conjugate idemp oten ts in B r,s (see Prop osition 2.7.1(3)). By Theorem 4.5.1, λ = µ . Th us the assignment λ 7→ L ( λ ) is injectiv e.  Remark 4.6.3. Instead of relying on [CDDM], one can pro v e Theorem 4.6.2 with straigh tforward modiﬁcations of the pro of of [CO, Theorem 3.7]. W e end this subsection with a couple prop ositions concerning L ( λ ) which will b e useful later. Prop osition 4.6.4. Given a bip artition λ ` ( r, s ) , L (( λ • , ∅ )) ⊗ L (( ∅ , λ ◦ )) = im z λ = L ( λ ) ⊕ L ( µ (1) ) ⊕ · · · ⊕ L ( µ ( k ) ) for some bip artitions µ (1) , . . . , µ ( k ) which have the pr op erty µ ( j ) ` ( r − i j , s − i j ) with 0 < i j ≤ min( r , s ) for al l j = 1 , . . . , k . Pr o of. First, using Example 4.3.1(1) w e hav e e ( λ • , ∅ ) ⊗ e ( ∅ ,λ ◦ ) = z • λ • ⊗ z ◦ λ w hite = z λ whic h implies the left equalit y . F or the right equality , notice that by the deﬁnition of e λ w e can write z λ = e λ + e 1 + · · · + e k where e λ , e 1 , . . . , e k are mutually orthogonal primitiv e idemp oten ts in B r,s . Moreov er, π ( e j ) = 0 for all j = 1 , . . . , k . By Theorem 4.5.1, there exists a bipartition µ ( j ) ` ( r − i j , s − i j ) for some 0 ≤ i j ≤ min( r, s ) suc h that e j is conjugate to e ( i j ) µ ( j ) for all j = 1 , . . . , k . It follows that π ( e ( i j ) µ ( j ) ) = 0, whic h implies i j 6 = 0 for eac h j = 1 , . . . , k . Finally , by Propositions 2.7.1(2) and 4.4.2 we are done.  The following example illustrates Prop osition 4.6.4. Example 4.6.5. (1) Assume δ 6 = 0. Then by Examples 4.3.1(2) and 4.4.1(1) we ha ve the following orthogonal decomp osition of z ( 2 , 2 ) = id •◦ in to primitive idem- p oten ts: z ( 2 , 2 ) = e ( 2 , 2 ) + e (1) ( ∅ , ∅ ) . Hence •◦ = L (( 2 , 2 )) ⊕ L (( ∅ , ∅ )) in Re p( GL δ ). (2) When δ = 0, z ( 2 , 2 ) = e ( 2 , 2 ) , by Example 4.3.1(2). Hence •◦ = L (( 2 , 2 )) in Rep( GL 0 ). Prop osition 4.6.6. L ( λ ) ∗ = L ( λ ∗ ) . Pr o of. Giv en σ ∈ Σ r , it is easy to c hec k that the dual morphisms ( σ • ) ∗ = ( σ ◦ ) − 1 and ( σ ◦ ) ∗ = ( σ • ) − 1 . Given a partition α ` r , the idemp oten t z α ∈ K Σ r is inv ariant under the operation K Σ r → K Σ r , σ 7→ σ − 1 . Hence, z ∗ λ = ( z • λ • ⊗ z ◦ λ ◦ ) ∗ = z ◦ λ • ⊗ z • λ ◦ . Hence, up to isomorphism we ha v e (im z λ ) ∗ = im( z ∗ λ ) = im( z ◦ λ • ) ⊗ im( z • λ ◦ ) = im( z • λ ◦ ) ⊗ im( z ◦ λ • ) = im( z λ ∗ ) . The result now follo ws from Prop osition 4.6.4 after inducting on | λ | .  4.7. Indecomp osable summands of mixed tensor p o wers. Let V denote a δ -dimensional ob ject of a tensor category T , let V ∗ denote a dual for V in T , and write T ( r , s ) = T V ,V ∗ ( r , s ) = V ⊗ r ⊗ V ∗ ⊗ s for the mixe d tensor p ower of V . The following theorem giv es a useful criterion for the fullness of the functor F : Rep( GL δ ) → T , F ( • ) 7→ V DELIGNE’S Rep( GL δ ) AND REPRESENT A TIONS OF gl ( m | n ) 23 deﬁned b y Prop osition 3.5.1, and moreov er sho ws that an y indecomp osable sum- mand of a T ( r, s ) is isomorphic to the image under F of an indecomp osable ob ject from Rep( GL δ ). This will b e later applied in § 5 and § 8, where V will denote the nat- ural representations of the general linear group and the general linear sup ergroup, resp ectiv ely . Theorem 4.7.1. Supp ose that the K -algebr a maps K Σ p → End T ( V ⊗ p ) deﬁne d by the symmetric br aiding of T ar e surje ctive, and Hom T ( T ( r , s ) , T ( r 0 , s 0 )) = 0 whenever r + s 0 6 = r 0 + s . Then: (1) F is ful l. (2) Λ = { F ( L ( λ )) | λ is a bip artition, F ( L ( λ )) 6 = 0 } is a c omplete set of inde c omp osable summands of the mixe d tensor p owers T ( r, s ) . Mor e over, the memb ers of Λ ar e p airwise non-isomorphic. Pr o of. T o pro v e the ﬁrst part, it suﬃces to sho w the restriction of F to Rep 0 ( GL δ ) is full. Since every ob ject in Rep 0 ( GL δ ) is isomorphic (by braidings) to a word of the form w r,s for some r, s , and the restriction of F to Rep 0 ( GL δ ) is strict (hence it preserves braidings) it suﬃces to show (16) F : Hom Rep 0 ( GL δ ) ( w r,s , w r 0 ,s 0 ) → Hom T ( T ( r , s ) , T ( r 0 , s 0 )) is surjective for every r, r 0 , s, s 0 . By h yp othesis, we can assume r + s 0 = r 0 + s . Consider the diagram (17) Hom Rep 0 ( GL δ ) ( w r,s , w r 0 ,s 0 ) / / F   End Rep 0 ( GL δ ) ( • ⊗ r + s 0 ) F   Hom T ( T ( r , s ) , T ( r 0 , s 0 )) / / End T ( V ⊗ r + s 0 ) where the horizontal maps are given b y (18) f 7→ (id ⊗ r 0 • ⊗ ev w s 0 , 0 ⊗ id ⊗ s • )( f ⊗ c w s, 0 ,w s 0 , 0 )(id ⊗ r • ⊗ co ev w 0 ,s ⊗ id ⊗ s 0 • ) and (19) f 7→ (id ⊗ r 0 V ⊗ ev T ( s 0 , 0) ⊗ id ⊗ s V )( f ⊗ c T ( s, 0) ,T ( s 0 , 0) )(id ⊗ r V ⊗ co ev T (0 ,s ) ⊗ id ⊗ s 0 V ) . Since the restriction of F to Rep 0 ( GL δ ) is a strict tensor functor, the diagram (17) comm utes. Moreov er, the maps (18) and (19) are K -v ector space isomorphisms with inv erses f 7→ (id ⊗ r 0 • ⊗ ev w 0 ,s ⊗ id ⊗ s 0 ◦ )( f ⊗ c w 0 ,s 0 ,w 0 ,s )(id ⊗ r • ⊗ co ev w s 0 , 0 ⊗ id ⊗ s ◦ ) and f 7→ (id ⊗ r 0 V ⊗ ev T (0 ,s ) ⊗ id ⊗ s V ∗ )( f ⊗ c T (0 ,s 0 ) ,T (0 ,s ) )(id ⊗ r V ⊗ co ev T ( s 0 , 0) ⊗ id ⊗ s V ∗ ) resp ectiv ely (this is easily v eriﬁed using diagram calculus for tensor categories [Sel]). Since the rightmost v ertical map is surjectiv e, by h ypothesis, w e are done. W e now prov e the second part. A summand W of T ( r, s ) is the image of an idemp oten t in End T ( T ( r , s )). By part (1), such an idempotent has a pre-image e ∈ B r,s under F . W rite im e = L ( λ (1) ) ⊕ · · · ⊕ L ( λ ( k ) ) in Rep( GL δ ) for some bipartitions λ ( i ) (see Theorem 4.6.2). Then W = F ( L ( λ (1) )) ⊕ · · · ⊕ F ( L ( λ ( k ) )) in T , and by Proposition 2.7.4 the F ( L ( λ ( i ) )) are indecomp osable. Th us if W is indecomp osable, then W = F ( L ( λ ( i ) )) for some i .  24 JONA THAN COMES AND BENJAMIN WILSON 4.8. Generic semisimplicity of Rep( GL δ ) . W e close this section with the fol- lo wing theorem which tells us exactly when the category Rep( GL δ ) is semisimple 6 Theorem 4.8.1. Rep( GL δ ) is semisimple if and only if δ is not an inte ger. Pr o of. The semisimplicit y of the walled Brauer algebras is completely determined in [CDDM, Theorem 6.3]. In particular, if δ ∈ Z then B r,s ( δ ) is not semisimple for some r and s . In this case, using Theorem 4.5.1, there exist distinct bipartitions λ ` ( r − i, s − i ) and µ ` ( r − j, s − j ) for some i, j ≥ 0 with e ( j ) µ B r,s e ( i ) λ 6 = 0. Since e ( j ) µ B r,s e ( i ) λ = Hom(im e ( i ) λ , im e ( j ) µ ) = Hom( L ( λ ) , L ( µ )), we are done in case δ ∈ Z . No w assume δ 6∈ Z , and let λ and µ b e bipartitions with Hom( L ( λ ) , L ( µ )) 6 = 0. If λ ` ( r, s ), then µ ` ( r − i, s − i ) for some i (Prop osition 4.6.1). W e pro ceed under the assumption i ≥ 0, the case i ≤ 0 b eing dual, b y Prop osition 4.6.6. Then (20) 0 6 = Hom( L ( λ ) , L ( µ )) = e ( i ) µ B r,s e λ . Since δ 6∈ Z , B r,s is a semisimple algebra, hence (20) can only b e true if e λ and e ( i ) µ are conjugate, whic h implies λ = µ (Theorem 4.5.1). Moreov er, the semisimplicity of B r,s implies End( L ( λ )) = e λ B r,s e λ is indeed a division algebra.  5. Connection to represent a tions of the general linear group Fix a nonnegative in teger d and consider the category Rep( GL d ) of ﬁnite dimen- sional representations of the general linear group GL d o ver K . In this short section w e describ e ho w the categories Rep( GL δ ) and Rep( GL d ) are related. One can view this section as a preview of § 8 where w e show how Rep( GL δ ) is related to repre- sen tations of general linear sup ergroups. Ho w ev er, we will see in § 7 that form ulas for decomposing tensor products in Rep( GL δ ) can b e obtained by interpolating decomp osition formulas in Rep( GL d ); hence this preview of § 8 is also imp ortan t to the structure of the pap er. 5.1. The category Rep( GL d ) . Let GL d = GL ( d, K ) denote the general linear group, that is, the group of inv ertible d × d -matrices with entries from K , and write V for the d -dimensional natural mo dule. Let Rep( GL d ) denote the category of ﬁnite-dimensional GL d -mo dules. The tensor pro duct of GL d -mo dules deﬁned in the usual w ay , and 1 denoting the trivial one-dimensional mo dule, Rep( GL d ) b ecomes a monoidal category as p er example 2.1.1. Note that, in particular, we consider the monoidal category Rep( GL d ) to b e strict. F or any ob ject U of Rep( GL d ), let U ∗ , ev U , co ev U b e deﬁned as p er example 2.2.1. Recall that U ∗ is a GL d -mo dule via (21) ( x · φ )( u ) = φ ( x − 1 · u ) , ( x ∈ GL d , u ∈ U, φ ∈ U ∗ ) Then ev U , coev U are GL d -mo dule maps, and so ( U ∗ , ev U , coev U ) is a dual for U in Rep( GL d ). Th us Rep( GL d ) is a tensor category . It is w ell kno wn that Rep( GL d ) is semisimple (see, e.g. [FH]). In particular, an ob ject is simple if and only if it is indecomp osable. 6 Recall that a K -linear Krull Schmidt category C is semisimple if and only if End C ( L ) is a ﬁnite K -dimensional division algebra for all indecomp osable ob jects L and Hom C ( L, L 0 ) = 0 for all non-isomorphic indecomposable ob jects L, L 0 . DELIGNE’S Rep( GL δ ) AND REPRESENT A TIONS OF gl ( m | n ) 25 Let ε i denote the function that takes any matrix to the its ( i, i ) entry for i = 1 , . . . d , and let Γ denote the set of weigh ts. F or k ∈ Z , letting Γ k = { γ = d X i =1 γ i ε i | γ i ∈ Z , γ 1 > · · · > γ d , X i γ i = k } , w e ha v e Γ = u k ∈ Z Γ k . W rite γ i for the co eﬃcien ts of γ ∈ Γ with resp ect to the ε i . F or γ ∈ Γ, let V ( γ ) denote the ﬁnite-dimensional highest-weigh t GL d -mo dule of highest-w eight γ . The highest-w eigh t mo dules classify the indecomposable ob jects of Rep( GL d ) up to isomorphism. That is, any indecomp osable ob ject of Rep( GL d ) is isomorphic to V γ for some γ ∈ Γ, and if V γ ∼ = V γ 0 for γ , γ 0 ∈ Γ, then γ = γ 0 . The w eights Γ are in bijection with bipartitions λ with l ( λ ) ≤ d , via (22) w t ( λ ) = X i> 0 λ • i ε i − X j > 0 λ ◦ j ε d − j +1 . W riting V λ for V wt ( λ ) , the isomorphism classes of indecomp osable ob jects in Rep( GL d ) are thus parameterized b y such bipartitions. F or k ∈ Z , deﬁne a partial order 6 on Γ k b y declaring γ 6 γ 0 ⇔ γ 1 + · · · γ k 6 γ 0 1 + · · · γ 0 k , k = 1 , . . . , d. If γ ∈ Γ, then there exists k suc h that all weigh ts of the highest-weigh t mo dule V ( γ ) b elong to Γ k ( k is the rational degree of the mo dule). Th us the set of the w eights of V ( γ ) are partially ordered. If γ 0 is a weigh t of V ( γ ), then γ 0 ≤ γ . Prop osition 5.1.1. L et λ b e a bip artition with l ( λ ) ≤ d . Then ( V λ ) ∗ = V λ ∗ . Pr o of. Dualit y deﬁnes an endofunctor of Rep( GL d ). Th us ( V λ ) ∗ is indecomposable, so ( V λ ) ∗ ∼ = V µ for some bipartition µ with l ( µ ) ≤ d . It follo ws immediately from (21) that the dual of a weigh t space of V λ is a weigh t space of ( V λ ) ∗ with the weigh t negated. Negation in v erts the partial order on w eigh ts, so w t ( µ ) is the low est w eigh t of V λ . Recall that the symmetric group Σ d em b eds in GL d as the p erm utation matrices. Th us Σ d acts on V λ and hence on its set of weigh ts. If σ denotes the longest elemen t of Σ d , and γ is a weigh t, then ( γ σ ) i = γ d − i +1 for i = 1 , . . . d , and furthermore, w t ( ν ∗ ) = w t ( ν ) σ for an y bipartition ν . Finally , σ is an an ti-in v olution of the p oset of weigh ts, and so w t ( λ ) σ = w t ( µ ) is the low est w eigh t of V λ , and so µ = λ ∗ .  Theorem 5.1.2. (Comp ar e with [Koi, Theorem 2.4] ) Fix bip artitions λ ` ( r , s ) and µ ` ( r 0 , s 0 ) such that l ( λ ) , l ( µ ) ≤ d . F or e ach bip artition ν with l ( ν ) ≤ d let Γ ν λ,µ b e such that V λ ⊗ V µ = M ν V ⊕ Γ ν λ,µ ν . Then Γ ν λ,µ = 0 unless the | ν | ≤ ( r + r 0 , s + s 0 ) . Mor e over, if l ( λ ) + l ( µ ) ≤ d then (23) Γ ν λ,µ = X α,β ,η,θ ∈P X κ ∈P LR λ • κ,α LR µ ◦ κ,β !   X γ ∈P LR λ ◦ γ ,η LR µ • γ ,θ   LR ν • α,θ LR ν ◦ β ,η wher e LR α β ,γ ’s ar e the Littlewo o d Richar dson c o eﬃcients. 26 JONA THAN COMES AND BENJAMIN WILSON 5.2. The functor F d : Rep( GL d ) → Rep( GL d ) . W rite F d : Rep( GL d ) → Rep( GL d ) for the tensor functor which sends • 7→ V deﬁned by Proposition 3.5.1 . By classi- cal Sch ur-W eyl duality (see [W ey]), the K -algebra map K Σ p → End Rep( GL d ) ( V ⊗ p ) deﬁned by the symmetric braiding is surjective, for any p ≥ 0. W rite T ( r, s ) for the mixed tensor p o wers of V as p er section § 4.7. F or any ζ ∈ K , the central element ζ · id ∈ GL d acts on T ( r, s ) b y the scalar ζ r − s , so Hom Rep( GL d ) ( T ( r , s ) , T ( r 0 , s 0 )) = 0 unless r + s 0 = r 0 + s . Thus, by Theorem 4.7.1, the functor F d is full (this is the so-called First F undamen tal Theorem of in v ariant theory) and any indecomp os- able summand of a mixed tensor p o w ers T ( r, s ) is isomorphic to F d ( L ( λ )) for some bipartition λ . Giv en a bipartition λ , we write W ( λ ) = F d ( L ( λ )). In this section we will com- pletely describ e W ( λ ). W e start b y assuming one of λ • , λ ◦ is ∅ . Prop osition 5.2.1. Assume λ ` ( r , s ) with r s = 0 . If l ( λ ) ≤ d , then W ( λ ) = V λ . If l ( λ ) > d , then W ( λ ) = 0 . Pr o of. First, Σ r acts on V ⊗ r b y p erm uting tensors. Since strict tensor functors preserv e symmetric braidings, this action coincides with F d : K Σ r → End( V ⊗ r ). Hence if r = 0 then, by Example 4.3.1(1), W ( λ ) is the image of the idemp oten t z λ • ∈ K Σ r acting on V ⊗ r . This is precisely Weyl’s c onstruction of V λ (see for example [FH]). The case s = 0 follo ws from the case r = 0 using Prop ositions 4.6.6 and 5.1.1.  In fact, Prop osition 5.2.1 holds without the assumption that r s = 0. Theorem 5.2.2. Supp ose λ is an arbitr ary bip artition. Then W ( λ ) =  V λ if l ( λ ) ≤ d, 0 if l ( λ ) > d. Pr o of. W e induct on the size of λ . The base case λ = ( ∅ , ∅ ) is clear, so assume λ ` ( r, s ) with r s 6 = 0. By Example 4.3.1(1) w e ha v e z λ = e ( λ • , ∅ ) ⊗ e ( ∅ ,λ ◦ ) from whic h it follows (24) F d (im z λ ) = W (( λ • , ∅ )) ⊗ W (( ∅ , λ ◦ )) . Also, by Proposition 4.6.4 (25) F d (im z λ ) = W ( λ ) ⊕ W ( µ (1) ) ⊕ · · · ⊕ W ( µ ( k ) ) where µ (1) , . . . , µ ( k ) are bipartitions whose sizes are strictly smaller than ( r, s ). Assume l ( λ ) ≤ d . Then by (24) and Prop osition 5.2.1, F d (im z λ ) has a highest w eight v ector with weigh t w t ( λ ). On the other hand, since l ( λ ) ≤ d there is no cancelation in (22). Hence, b y induction, W ( µ ( j ) ) do es not hav e a highest weigh t v ector of w eight w t ( λ ) for an y j = 1 , . . . , k . Thus, by (25), W ( λ ) has a highest w eight vector with w eigh t w t ( λ ). By Prop osition 2.7.4, W ( λ ) is simple and we are done. No w assume l ( λ ) > d . If either l ( λ • ) or l ( λ ◦ ) is greater than d we are done b y Prop osition 5.2.1, hence w e can assume l ( λ • ) , l ( λ ◦ ) ≤ d . By Theorem 5.1.2, the highest weigh ts in (24) are all of the form w t ( ν ) with | ν |  ( r , s ). Thus, by induction, F d ( z λ ) decomp oses as a direct sum of W ( ν )’s with | ν |  ( r, s ). Hence, b y (25) and Prop osition 2.7.4, W ( λ ) = 0.  DELIGNE’S Rep( GL δ ) AND REPRESENT A TIONS OF gl ( m | n ) 27 6. The lifting map As b efore, w e ﬁx δ ∈ K . Let t b e an indeterminate. In this section we con- struct a ring isomorphism (called the lifting map ) b et ween the Grothendieck rings of Rep( GL δ ) and Rep( GL t ). The deﬁnition of the lifting map is not an explicit one, ho w ev er we show in § 6.4 that v alues of the lifting map can b e computed us- ing combinatorics of certain diagrams in tro duced b y Brundan and Stropp el. W e will see later ( § 7) that the lifting map will play a crucial role in our ability to de- comp ose tensor pro ducts in Rep( GL δ ). W e b egin by ﬁxing some notation for the Grothendiec k rings mentioned ab o ve. 6.1. The rings R δ and R t . W rite K ( t ) , K [[ t − δ ]], and K (( t − δ )) for the ﬁeld of fractions in t , ring of p o wer series in t − δ , and the ﬁeld of Laurent series 7 in t − δ respectively . Then we ha v e the following categories with their corresp onding additiv e Grothendieck rings (see § 2.8): category additiv e Grothendiec k ring bilinear form Rep( GL δ ) ov er K Rep( GL t ) o ver K ( t ) Rep( GL t ) o ver K (( t − δ )) R δ R t R t,δ ( − , − ) δ ( − , − ) t ( − , − ) t,δ By Theorem 4.6.2 w e can identify the elemen ts of R δ , R t , and R t,δ with formal Z -linear combinations of bipartitions, and w e will do so for the rest of the pap er. In particular, the rings R δ , R t , and R t,δ are clearly isomorphic as ab elian groups. Ho wev er, the multiplication in these rings dep ends on the parameter: Example 6.1.1. It is alw a ys true that e ( 2 , ∅ ) = id • and e ( ∅ , 2 ) = id ◦ , which implies L (( 2 , ∅ )) = • and L (( ∅ , 2 )) = ◦ . Hence L (( 2 , ∅ )) ⊗ L (( ∅ , 2 )) = •◦ alwa ys. Thus, it follows from example 4.3.1(2) that ( 2 , ∅ )( ∅ , 2 ) = ( 2 , 2 ) ∈ R 0 , whereas ( 2 , ∅ )( ∅ , 2 ) = ( 2 , 2 ) + ( ∅ , ∅ ) ∈ R t . The next prop osition sho ws that although the multiplication of bipartitions in R δ and R t ma y diﬀer, the rings R t and R t,δ can b e iden tiﬁed regardless of δ . Prop osition 6.1.2. (1) The Z -line ar map R t → R t,δ with λ 7→ λ for e ach bip arti- tion λ is a ring isomorphism. (2) ( λ, µ ) t = ( λ, µ ) t,δ =  1 if λ = µ, 0 if λ 6 = µ. Pr o of. (1) Supp ose (26) e λ ⊗ e µ = e 1 + · · · + e k is a decomp osition of e λ ⊗ e µ in to mutually orthogonal primitive idempotents o v er K ( t ). Then λµ = P ν a ν ν ∈ R t where a ν is the num b er of summands in (26) corresp ond to the bipartition ν . By Corollary 4.5.2, viewing (26) ov er the larger ﬁeld K (( t − δ )) ⊃ K ( t ) still gives an orthogonal decomposition of e λ ⊗ e µ in to primitiv e idemp oten ts, hence λµ = P ν a ν ν ∈ R t,δ to o. (2) By Corollary 4.5.2, we can w ork o v er the algebraic closure of K ( t ) (resp. K (( t − δ )) to compute ( λ, µ ) t (resp. ( λ, µ ) t,δ ). The result now follows from the fact that Rep( GL t ) is semisimple ov er any ﬁeld containing the indeterminate t (see Theorem 4.8.1).  7 In other words, K (( t − δ )) is the ﬁeld of fractions of K [[ t − δ ]]. 28 JONA THAN COMES AND BENJAMIN WILSON With Prop osition 6.1.2 in mind, for the rest of the pap er we will identify R t,δ with R t for every δ and write R t for b oth. 6.2. The ring map lift δ : R δ → R t . Fix a bipartition λ ` ( r, s ) and consider the idemp oten t e λ ∈ K B r,s ( δ ). W e can lift e λ to an idemp oten t ˜ e ∈ K (( t − δ )) B r,s ( t ), i.e. ˜ e is of the form ˜ e = P X a X X with a X ∈ K [[ t − δ ]] for all ( w r,s , w r,s )-diagrams X , and ˜ e | t = δ = e (see [CO, Theorem A.2]). Now, given another bipartition µ , let D λ,µ = D λ,µ ( δ ) denote the n um ber of times L ( µ ) o ccurs in a decomposition of im( ˜ e ) into a direct sum of indecomp osables in Rep( GL t ) ov er K (( t − δ )). One can sho w that D λ,µ do es not dep end on the choice of represen tativ e for e λ or on the c hoice of ˜ e (compare with [CO, Theorem 3.9]). Now, let lift δ : R δ → R t b e the Z -linear map deﬁned on bipartitions by lift δ ( λ ) = X µ D λ,µ µ. Example 6.2.1. If λ ` ( r, 0) then, by Example 4.3.1(1), e λ = z • λ • ∈ K B r, 0 ( δ ). Since z • λ • do es not dep end on δ , it can b e lifted to z • λ • = e λ ∈ K (( t )) B r, 0 ( t ). Hence, lift δ ( λ ) = λ for all λ ` ( r, 0), δ ∈ K . Similarly , lift δ ( λ ) = λ whenever λ ` (0 , s ). Example 6.2.2. (1) Assume δ = 0. By Example 4.3.1(2), e ( 2 , 2 ) = id •◦ ∈ K B 1 , 1 (0) whic h lifts to id •◦ ∈ K (( t )) B 1 , 1 ( t ). By Example 4.6.5(1), •◦ = L (( 2 , 2 )) ⊕ L (( ∅ , ∅ )) in Rep( GL t ). Th us lift 0 (( 2 , 2 )) = ( 2 , 2 ) + ( ∅ , ∅ ). (2) Assume δ 6 = 0. An explicit expression for e ( 2 , 2 ) ∈ K B 1 , 1 ( δ ) is given in Example 4.3.1(2). Since 1 t = P ∞ n =0 ( − 1) n δ n +1 ( t − δ ) n ∈ K [[ t − δ ]], a lift of e ( 2 , 2 ) is obtained by replacing δ with t in that expression. Hence, lift δ (( 2 , 2 )) = ( 2 , 2 ). The following theorem lists prop erties of lift δ whic h are very useful for this pap er. Theorem 6.2.3. (1) lift δ : R δ → R t is a ring isomorphism for every δ ∈ K . (2) D λ,λ = 1 for al l λ . Mor e over, D λ,µ = 0 unless µ = λ or µ ` ( | λ • | − i, | λ ◦ | − i ) for some i > 0 . (3) Fix a bip artition λ . lift δ ( λ ) = λ for al l but ﬁnitely many δ ∈ K . (4) (lift δ ( x ) , lift δ ( y )) t = ( x, y ) δ for al l x, y ∈ R δ . F or a pro of of Theorem 6.2.3 we refer the reader to [CO, Prop osition 3.12] where the analogous statements are prov ed for Rep( S t ). An imp ortan t consequence of Theorem 6.2.3 is the following: Corollary 6.2.4. ( λ, µ ) δ = P ν D λ,ν D µ,ν for al l bip artitions λ and µ . Pr o of. ( λ, µ ) δ = ( P ν D λ,ν ν , P ν 0 D µ,ν 0 ν 0 ) t (Theorem 6.2.3(4)) = P ν,ν 0 D λ,ν D µ,ν 0 ( ν, ν 0 ) t = P ν D λ,ν D µ,ν (Prop osition 6.1.2(2))  6.3. The diagrams of Brundan and Stropp el. W e will soon sho w that lift δ ( λ ) can b e computed explicitly using certain diagrams in tro duced b y Brundan and Stropp el [BS2-5]. In this subsection we introduce these diagrams and give a few examples. DELIGNE’S Rep( GL δ ) AND REPRESENT A TIONS OF gl ( m | n ) 29 As usual, we ﬁx δ ∈ K . Given a bipartition λ , set I ∧ ( λ ) = { λ • 1 , λ • 2 − 1 , λ • 3 − 2 , . . . } , I ∨ ( λ, δ ) = { 1 − δ − λ ◦ 1 , 2 − δ − λ ◦ 2 , 3 − δ − λ ◦ 3 , . . . } . No w, let x λ = x λ ( δ ) be the diagram obtained by lab eling the in teger vertices on the num b er line according to the following rule: lab el the the i th vertex b y         if i 6∈ I ∧ ( λ ) ∪ I ∨ ( λ, δ ) , ∧ if i ∈ I ∧ ( λ ) \ I ∨ ( λ, δ ) , ∨ if i ∈ I ∨ ( λ, δ ) \ I ∧ ( λ ) , × if i ∈ I ∧ ( λ ) ∩ I ∨ ( λ, δ ) . F or example, , , , . Remark 6.3.1. Notice that the integer i in x λ is lab elled ∧ for i  0. Moreo v er, if δ ∈ Z (resp. δ 6∈ Z ) then i is labelled by ∨ (resp.  ) for i  0. In fact, it is not diﬃcult to show that when δ ∈ Z there is a bijection b et w een the set of all bipartitions and the set of all diagrams with (1) i lab elled ∧ for i  0; (2) i lab elled ∨ for i  0; and (3) the num b er of × ’s min us the num ber of  ’s equal to δ . Next, w e construct the c ap diagr am c λ = c λ ( δ ) in the following recursive manner: Step 0: Start with x λ . Step n : Draw a cap connecting v ertices i and j on the num ber line whenever (i) i < j ; (ii) i is lab elled by ∨ and j is lab elled by ∧ in x λ ; and (iii) eac h in teger b et ween i and j in x λ is either lab elled by  , labelled by × , or already part of a cap from an earlier step. It follows from Remark 6.3.1 that no new caps will b e added after a ﬁnite num b er of steps, leaving us with the cap diagram c λ . Example 6.3.2. If δ = 1 and λ = ((5 2 , 4 2 , 3 2 ) , (5 3 , 4 , 3 , 2)) then Giv en in tegers i < j , w e sa y ( i, j ) is a ∨∧ -p air in x λ if there is a cap from j to i in c λ . F or instance, in Example 6.3.2 there are four ∨∧ -pairs: ( − 5 , 5) , ( − 4 , 2) , ( − 3 , − 2), and (3 , 4). Next, giv en bipartitions µ and λ , w e say that µ is linke d to λ if there 30 JONA THAN COMES AND BENJAMIN WILSON exists an in teger k ≥ 0 and bipartitions ν ( n ) for 0 ≤ n ≤ k such that (i) ν (0) = λ , (ii) ν ( k ) = µ , and (iii) x ν ( n ) is obtained from x ν ( n − 1) b y swapping the lab els of some ∨∧ -pair in x λ whenev er 0 < n ≤ k . Finally , set D 0 λ,µ = D 0 λ,µ ( δ ) =  1 if µ is link ed to λ, 0 otherwise . Remark 6.3.3. It is shown in [CD] that D 0 λ,µ giv e decomposition n um b ers for w alled Brauer algebras. This is easy to see when δ 6∈ Z . Indeed, when δ 6∈ Z there are no ∨ lab els on x λ , so there are no ∨∧ -pairs; hence D 0 λ,µ 6 = 0 if and only if λ = µ . Example 6.3.4. Fix δ = − 1. In this example w e will compute the n um b ers D 0 λ,µ where λ = ((3 , 2) , (3 , 1)) and µ is arbitrary . No w we swap lab els on ∨∧ -pairs in x λ to determine whic h bipartitions µ are linked to λ . The follo wing table lists our results: ∨ ∧ -pairs sw app ed x µ µ none ((3 , 2) , (3 , 1)) ( − 1 , 1) ((3) , (1 2 )) (2 , 3) ((2 2 ) , (3)) b oth ((2) , (1)) Hence D 0 λ,µ = 1 when µ is one of the four bipartitions listed in the table ab o ve, and D 0 λ,µ = 0 for all other µ . Example 6.3.5. In this example we compute D 0 ( 2 , 2 ) ,µ ( δ ) for all bipartitions µ and all δ ∈ K . Since I ∧ (( 2 , 2 )) = { 1 , − 1 , − 2 , . . . } , the diagram x ( 2 , 2 ) ( δ ) has a ∨∧ -pair if and only if 0 ∈ I ∨ (( 2 , 2 ) , δ ) and 1 6∈ I ∨ (( 2 , 2 ) , δ ), which o ccurs if and only if δ = 0 since I ∨ (( 2 , 2 ) , δ ) = {− δ, 2 − δ, 3 − δ, . . . } . Hence, when δ 6 = 0 we hav e D 0 ( 2 , 2 ) ,µ ( δ ) =  1 if µ = ( 2 , 2 ) , 0 otherwise . On the other hand, DELIGNE’S Rep( GL δ ) AND REPRESENT A TIONS OF gl ( m | n ) 31 Sw apping the ∨∧ -pair (0 , 1) in x ( 2 , 2 ) (0) gives x ( ∅ , ∅ ) (0). Hence D 0 ( 2 , 2 ) ,µ (0) =  1 if µ = ( 2 , 2 ) or µ = ( ∅ , ∅ ) , 0 otherwise . In particular, D 0 ( 2 , 2 ) ,µ = D ( 2 , 2 ) ,µ for all µ regardless of δ (see Example 6.2.2). In § 6.4 we will sho w D 0 λ,µ = D λ,µ alw ays (see Corollary 6.4.2). The follo wing prop osition describ es ho w swapping the lab els on ∨∧ -pairs aﬀects the size of the corresp onding bipartitions. Prop osition 6.3.6. Supp ose λ and µ ar e bip artitions and ( i, j ) is a ∨∧ -p air in x λ . If x µ is obtaine d fr om x λ by swapping the lab els of i and j , then | µ | = ( | λ • | + i − j, | λ ◦ | + i − j ) . Pr o of. Set a k = λ • k − k + 1 and b k = k − δ − λ ◦ k for each k > 0 so that I ∧ ( λ ) = { a 1 , a 2 , a 3 , . . . } , I ∨ ( λ, δ ) = { b 1 , b 2 , b 3 , . . . } . Then | λ • | = P k> 0 ( a k + k − 1) and | λ ◦ | = P k> 0 ( k − δ − b k ). Now let L, M ∈ Z b e suc h that a M = j and b L = i . Then sw apping the lab els of i and j in x λ results in x µ with I ∧ ( µ ) = { a 1 , . . . , a M − 1 , a M +1 , . . . , a N , b L , a N +1 , a N +2 , . . . } , I ∨ ( µ, δ ) = { b 1 , . . . , b L − 1 , b L +1 , . . . , b N 0 , a M , b N 0 +1 , b N 0 +2 , . . . } for some N , N 0 . Hence | µ • | = X 0 N ( a k + k − 1) + X M 0 ( a k + k − 1) − ( a M + M − 1) − ( N − M ) + ( b L + N − 1) = | λ • | + i − j. Similarly , | µ ◦ | = X 0 N 0 ( k − δ − b k ) + X L 0 ( k − δ − b k ) − ( L − δ − b L ) − ( N 0 − L ) + ( N 0 − δ − a M ) = | λ ◦ | + i − j.  The next corollary , an immediate consequence of Prop osition 6.3.6, will be useful later. Corollary 6.3.7. D 0 λ,λ = 1 for al l λ . Mor e over, D 0 λ,µ = 0 unless µ = λ or | µ | = ( | λ • | − i, | λ ◦ | − i ) for some i > 0 . 32 JONA THAN COMES AND BENJAMIN WILSON 6.4. Computing lift δ ( λ ) . The follo wing theorem will allow us to explicitly com- pute v alues of the lifting map using the combinatorics developed in § 6.3 (see Corol- lary 6.4.2). Our pro of of this theorem relies heavily on the results in [CD]. Theorem 6.4.1. P ν D 0 λ,ν D 0 µ,ν = ( λ, µ ) δ for al l bip artitions λ, µ and al l δ ∈ K . Pr o of. It follo ws from Corollary 6.2.4 that the statement of theorem is symmetric in λ and µ , hence we may assume | µ | 6 > | λ | . Supp ose λ ` ( r , s ) and µ ` ( r 0 , s 0 ). ( λ, µ ) δ is zero unless r + s 0 = r 0 + s (Prop osition 4.6.1). Hence, by Corollary 6.3.7, it suﬃces to consider the case µ ` ( r − i, s − i ) for some i ≥ 0. Assume δ 6 = 0. Then B r,s is a quasi-hereditary (hence cellular) algebra (see [CDDM, Corollary 2.8]) with decomp osition n umbers given b y D 0 λ,ν (see [CD, The- orem 4.10] 8 ). In particular, this implies that the B r,s -mo dule homomorphisms b et ween the pro jective mo dules e λ B r,s and e ( i ) µ B r,s satisfy the following: (27) X ν D 0 λ,ν D 0 µ,ν = dim K Hom B r,s ( e λ B r,s , e ( i ) µ B r,s ) (see, for instance [GL, Theorem 3.7(iii)]). Since (28) Hom B r,s ( e λ B r,s , e ( i ) µ B r,s ) = e ( i ) µ B r,s e λ = Hom Rep( GL δ ) ( L ( λ ) , L ( µ )) , it follows that ( λ, µ ) δ agrees with (27). If δ = 0, the algebra B r,s is no longer quasi-hereditary , but it is still cellular (see [CDDM, Theorem 2.7]). The decomp osition n umbers are still giv en by D 0 λ,ν , ho wev er there is no PIM lab elled by ( ∅ , ∅ ) in this case, hence we must require λ 6 = ( ∅ , ∅ ). Hence, if neither λ nor µ is ( ∅ , ∅ ), (27) and (28) still hold w e are done as b efore. Since w e are assuming | µ | 6 > | λ | , to complete the pro of of the theorem w e only need to pro v e the case µ = ( ∅ , ∅ ), λ ` ( r, r ) and δ = 0. Since D 0 ( ∅ , ∅ ) ,ν = 0 whenev er ν 6 = ( ∅ , ∅ ) and D 0 ( ∅ , ∅ ) , ( ∅ , ∅ ) = 1, w e ha v e P ν D 0 λ,ν D 0 ( ∅ , ∅ ) ,ν = D 0 λ, ( ∅ , ∅ ) . The decomp osition num b er D 0 λ, ( ∅ , ∅ ) is the composition m ultiplicit y of the simple B r,r -mo dule lab elled by λ in Hom( w r,r , 1 ), the standard B r,r -mo dule lab elled by ( ∅ , ∅ ). Hence, (29) D 0 λ, ( ∅ , ∅ ) = dim K Hom B r,r ( e λ B r,r , Hom( w r,r , 1 )) . Since Hom B r,r ( e λ B r,r , Hom( w r,r , 1 )) = Hom( w r,r , 1 ) e λ = Hom Rep( GL 0 ) ( L ( λ ) , L (( ∅ , ∅ ))) , ( λ, ( ∅ , ∅ )) 0 agrees with (29).  Corollary 6.4.2. D λ,µ ( δ ) = D 0 λ,µ ( δ ) for al l bip artitions λ, µ and al l δ ∈ K . Pr o of. First, put the following partial order on pairs of bipartitions: ( λ, µ ) > ( λ 0 , µ 0 ) means either | λ | > | λ 0 | , or λ = λ 0 and | µ | > | µ 0 | . W e pro ve the corollary by inducting on this partial order. First notice that D ( ∅ , ∅ ) , ( ∅ , ∅ ) = 1 = D 0 ( ∅ , ∅ ) , ( ∅ , ∅ ) . No w assume ( λ, µ ) 6 = (( ∅ , ∅ ) , ( ∅ , ∅ )). By Theorem 6.2.3(2) and Corollary 6.3.7 we 8 The results in [CD] are only prov ed for K = C , how ever, using Corollary 4.4.3 it can b e sho wn that their results hold ov er arbitrary ﬁelds of characteristic zero. DELIGNE’S Rep( GL δ ) AND REPRESENT A TIONS OF gl ( m | n ) 33 ma y assume | λ | > | µ | . Th us D λ,µ = ( λ, µ ) δ − P ν | ν | < | µ | D λ,ν D µ,ν (Corollary 6.2.4 and Theorem 6.2.3(2)) = ( λ, µ ) δ − P ν | ν | < | µ | D 0 λ,ν D 0 µ,ν (Induction) = D 0 λ,µ (Theorem 6.4.1 and Corollary 6.3.7) .  Example 6.4.3. Using Corollary 6.4.2 and Example 6.3.4 we ha v e lift − 1 (((3 , 2) , (3 , 1))) = ((3 , 2) , (3 , 1)) + ((3) , (1 2 )) + ((2 2 ) , (3)) + ((2) , (1)) . 7. Decomposing tensor products in Rep ( GL δ ) In this section w e give a generic decomp osition formula for decomp osing tensor pro ducts of indecomp osable ob jects in Rep( GL t ). W e then show how this generic decomp osition formula along with the lifting map from the previous section can b e used to decomp ose arbitrary tensor products in Rep( GL δ ). Throughout this section we will w ork in the Grothendiec k rings R δ and R t (see § 6.1). 7.1. The generic case. The following theorem explains ho w to decompose the tensor pro duct of tw o indecomposable ob jects in Rep( GL t ). Theorem 7.1.1. Given bip artitions λ, µ , and ν , let Γ ν λ,µ b e as in (23). Then λµ = P ν Γ ν λ,µ ν in R t . Pr o of. Fix bipartitions λ and µ and let ν (1) , . . . , ν ( k ) b e bipartitions suc h that (30) λµ = ν (1) + · · · + ν ( k ) in R t . By Theorem 6.2.3(3) there exists a p ositiv e integer d whic h sim ultaneously satisﬁes (i) d ≥ l ( λ ) + l ( µ ); (ii) d ≥ l ( ν ( i ) ) for each i = 1 , . . . , k ; and (iii) lift d ﬁxes λ, µ, ν (1) , . . . , ν ( k ) . Now, lift d is a ring isomorphism (Theorem 6.2.3(1)), hence (30) holds in R d b y assumption (iii). Since F d is a tensor functor, b y Theorem 5.2.2 along with assumptions (i) and (ii) we ha ve V λ ⊗ V µ = V ν (1) ⊕ · · · ⊕ V ν ( k ) in Rep( GL d ). The result no w follows from Theorem 5.1.2 and assumption (i).  The following corollary lists sp ecial cases of Theorem 7.1.1, which are easy to pro ve using basic prop erties of Littlew o od Richardson co eﬃcien ts: Corollary 7.1.2. The fol lowing e quations hold in R t . (31) ( λ • , ∅ )( µ • , ∅ ) = X α ∈P LR α λ • ,µ • ( α, ∅ ) , (32) ( ∅ , λ ◦ )( ∅ , µ ◦ ) = X α ∈P LR α λ ◦ ,µ ◦ ( ∅ , α ) , (33) ( λ • , ∅ )( ∅ , µ ◦ ) = X ν X κ ∈P LR λ • κ,ν • LR µ ◦ κ,ν ◦ ν, 34 JONA THAN COMES AND BENJAMIN WILSON (34) λ ( 2 , ∅ ) = X λ • + ( λ • + , λ ◦ ) + X λ ◦− ( λ • , λ ◦− ) , (35) λ ( ∅ , 2 ) = X λ ◦ + ( λ • , λ ◦ + ) + X λ •− ( λ •− , λ ◦ ) , wher e the sums in (34) ar e taken over al l p artitions λ • + (r esp. λ ◦− ) obtaine d fr om the Y oung diagr am λ • (r esp, λ ◦ ) by adding one b ox (r esp. r emoving one b ox). Similarly for (35). Example 7.1.3. In this example we compute ((2) , ∅ )( 2 , 2 ) ∈ R t . By Example 6.1.1 (or using (33), (34), or (35)) we hav e ( 2 , ∅ )( ∅ , 2 ) = ( 2 , 2 ) + ( ∅ , ∅ ) ∈ R t . Hence, using (34) and (35) we ha v e the following in R t : ((2) , ∅ )( 2 , 2 ) = ((2) , ∅ )(( 2 , ∅ )( ∅ , 2 ) − ( ∅ , ∅ )) = ((2) , ∅ )( 2 , ∅ )( ∅ , 2 ) − ((2) , ∅ ) = (((2 , 1) , ∅ ) + ((3) , ∅ ))( ∅ , 2 ) − ((2) , ∅ ) = ((2 , 1) , 2 ) + ((3) , 2 ) + ((1 2 ) , ∅ ) + ((2) , ∅ ) . 7.2. Decomp osing arbitrary tensor products. T o compute the product of tw o bipartitions λ, µ ∈ R δ for arbitrary δ ∈ K (i.e. to decomp ose tensor pro ducts in Rep( GL δ )) we (1) determine the co eﬃcients in lift δ ( λµ ) = P ν,ν 0 D λ,ν D µ,ν 0 ν ν 0 using Corollary 6.4.2, (2) use the results in § 7.1 to expand P ν,ν 0 D λ,ν D µ,ν 0 ν ν 0 = ν (1) + · · · + ν ( k ) ∈ R t , (3) determine lift − 1 δ ( ν (1) + · · · + ν ( k ) ) = λµ , which b y Theorem 6.2.3(2) consists of a sum of a subset of the bipartitions ν (1) , . . . , ν ( k ) . The following examples illustrate the pro cess describ ed abov e: Example 7.2.1. Consider ((2 2 ) , (3 , 1))( 2 , ∅ ) ∈ R − 1 . Since and b y Corollary 6.4.2 we ha ve lift − 1 (((2 2 ) , (3 , 1))) = ((2 2 ) , (3 , 1)) + ((2) , (1 2 )). More- o ver, by Example 6.2.1, lift − 1 (( 2 , ∅ )) = ( 2 , ∅ ). Since lift − 1 is a ring map (Theorem 6.2.3(1)) it follows that lift − 1 (((2 2 ) , (3 , 1))( 2 , ∅ )) = ((2 2 ) , (3 , 1))( 2 , ∅ ) + ((2) , (1 2 ))( 2 , ∅ ) , whic h, by (34), is equal to ((3 , 2) , (3 , 1)) +((2 2 , 1) , (3 , 1)) +((2 2 ) , (3)) +((2 2 ) , (2 , 1))+((2 , 1) , (1 2 ))+((3) , (1 2 ))+((2) , 2 ) . No w, by Example 6.3.4 along with Corollary 6.4.2 lift − 1 (((3 , 2) , (3 , 1))) = ((3 , 2) , (3 , 1)) + ((3) , (1 2 )) + ((2 2 ) , (3)) + ((2) , 2 ) . Similarly , one can show lift − 1 (((2 2 , 1) , (3 , 1))) = ((2 2 , 1) , (3 , 1)) and lift − 1 (((2 2 ) , (2 , 1))) = ((2 2 ) , (2 , 1)) + ((2 , 1) , (1 2 )) . DELIGNE’S Rep( GL δ ) AND REPRESENT A TIONS OF gl ( m | n ) 35 Hence, lift − 1 (((2 2 ) , (3 , 1))( 2 , ∅ )) = lift − 1 (((3 , 2) , (3 , 1)) + ((2 2 , 1) , (3 , 1)) + ((2 2 ) , (2 , 1))) . It follows from Theorem 6.2.3(1) that ((2 2 ) , (3 , 1))( 2 , ∅ ) = ((3 , 2) , (3 , 1)) + ((2 2 , 1) , (3 , 1)) + ((2 2 ) , (2 , 1)) ∈ R − 1 . Example 7.2.2. In this example we compute ((2) , ∅ )( 2 , 2 ) ∈ R δ for arbitrary δ ∈ K . First, since lift δ is a ring map (Theorem 6.2.3(1)), b y Examples 6.2.1 and 6.2.2 we ha v e lift δ (((2) , ∅ )( 2 , 2 )) =  ((2) , ∅ )( 2 , 2 ) + ((2) , ∅ ) if δ = 0 , ((2) , ∅ )( 2 , 2 ) if δ 6 = 0 . Hence, by Example 7.1.3 we ha v e lift δ (((2) , ∅ )( 2 , 2 )) =  ((2 , 1) , 2 ) + ((3) , 2 ) + ((1 2 ) , ∅ ) + 2((2) , ∅ ) if δ = 0 , ((2 , 1) , 2 ) + ((3) , 2 ) + ((1 2 ) , ∅ ) + ((2) , ∅ ) if δ 6 = 0 . No w, lift δ (((1 2 ) , ∅ )) = ((1 2 ) , ∅ ) and lift δ (((2) , ∅ )) = ((2) , ∅ ) for all δ (Example 6.2.1). Moreo ver, using Corollary 6.4.2, w e compute lift δ (((2 , 1) , 2 )) =    ((2 , 1) , 2 ) + ((1 2 ) , ∅ ) if δ = − 1 , ((2 , 1) , 2 ) + ((2) , ∅ ) if δ = 1 , ((2 , 1) , 2 ) if δ 6 = ± 1 . lift δ (((3) , 2 )) =  ((3) , 2 ) + ((2) , ∅ ) if δ = − 2 , ((3) , 2 ) if δ 6 = − 2 . Since lift δ is an isomorphism for all δ , we ha v e the following in R δ : ((2) , ∅ )( 2 , 2 ) =        ((2 , 1) , 2 ) + ((3) , 2 ) + ((1 2 ) , ∅ ) + 2((2) , ∅ ) if δ = 0 , ((2 , 1) , 2 ) + ((3) , 2 ) + ((2) , ∅ ) if δ = − 1 , ((2 , 1) , 2 ) + ((3) , 2 ) + ((1 2 ) , ∅ ) if δ ∈ { 1 , − 2 } , ((2 , 1) , 2 ) + ((3) , 2 ) + ((1 2 ) , ∅ ) + ((2) , ∅ ) otherwise . 8. Represent a tions of the general linear supergr oup Fix m, n ≥ 0 and consider the algebraic sup ergroup GL ( m | n ) o v er K . In this pa- p er we will only deal with ﬁnite dimensional representations of GL ( m | n ), which can b e identiﬁed with in tegrable represen tations of the corresponding Lie sup eralgebra gl ( m | n ). W e prefer to exploit this identiﬁcation and work with gl ( m | n ) rather than GL ( m | n ), and we will do so for the rest of the pap er. W e b egin b y ﬁxing notation and conv entions for representations of gl ( m | n ). 8.1. The category Rep( gl ( m | n )) . Let V = V ¯ 0 ⊕ V ¯ 1 denote a superspace ov er K with dim K V ¯ 0 = m , dim K V ¯ 1 = n . W rite gl ( m | n ) for the associated gener al line ar Lie sup er algebr a , that is, for the Lie superalgebra of endomorphisms of the sup erspace V , considered as ( m + n ) × ( m + n ) matrices. Then V is called the natur al mo dule for gl ( m | n ). Let Rep( gl ( m | n )) denote the category of ﬁnite-dimensional gl ( m | n )-mo dules. Giv en gl ( m | n )-mo dules U, U 0 , the tensor pro duct of sup erspaces U ⊗ U 0 is again a gl ( m | n )-mo dule with action (36) x · ( u ⊗ u 0 ) = ( x · u ) ⊗ u 0 + ( − 1) ¯ x ¯ u u ⊗ ( x · u 0 ) , for x ∈ gl ( m | n ) , u ∈ U, u 0 ∈ U 0 . Th us, as p er example 2.1.2, Rep( gl ( m | n )) is a monoidal category , where the one-dimensional purely-even mo dule 1 carries 36 JONA THAN COMES AND BENJAMIN WILSON the trivial action. Note that, in particular, we consider the tensor pro duct of Rep( gl ( m | n )) to b e strict. If U is a gl ( m | n )-mo dule, then the dual sup erspace U ∗ (cf. example 2.2.2) is again a gl ( m | n )-mo dule with action (37) ( x · φ )( u ) = − ( − 1) ¯ x ¯ φ φ ( x · u ) , for x ∈ gl ( m | n ) , φ ∈ U ∗ , u ∈ U . The maps ev U , coev U are maps of gl ( m | n )-mo dules, and so Rep( gl ( m | n )) is a tensor category . It is well known that the category Rep( gl ( m | n )) is not semisimple when n > 0 or m > 0 (see e.g. [Ser1]). 8.2. Characters. Let h ⊂ gl ( m | n ) denote the subalgebra of diagonal matrices. F or an y ob ject U of Rep( gl ( m | n )), let c h U = X µ ∈ h ∗ dim K U µ e µ , denote the char acter of U , where dim K U µ denotes the dimension of the µ -weigh t space of U as a v ector space ov er K , and e µ denotes the formal exponential. Addition and multiplication of characters of ob jects in Rep( gl ( m | n )) are deﬁned comp onen t-wise and by con v olution, resp ectiv ely . Since h is purely ev en, it follo ws that from the deﬁnition of the bipro duct and equation (36) that c h( U ⊕ U 0 ) = c h U + ch U 0 , c h( U ⊗ U 0 ) = c h U · ch U 0 for any ob jects U, U 0 of Rep( gl ( m | n )). F or an y r, s ≥ 0, write T ( r , s ) = V ⊗ r ⊗ ( V ∗ ) ⊗ s , as per section § 4.7. Let { ε i } 1 ≤ i ≤ m + n denote the diagonal co ordinate functions of gl ( m | n ), so that ε i tak es an y matrix to its ( i, i )-entry . Then for an y r, s ≥ 0, all w eigh ts of T ( r, s ) are in tegral linear combinations of the ε i (hence also of an y submo dule of T ( r , s )). Let x i = e ε i , 1 ≤ i ≤ m, y j = e ε m + j , 1 ≤ j ≤ n. Then if U is a submo dule of T ( r, s ), then ch U = c h U ( x | y ) is a Lauren t poly- nomial in the v ariables x = { x i } and y = { y j } . W rite ¯ x = { x − 1 i } 1 ≤ i ≤ m and ¯ y = { y − 1 i } 1 ≤ i ≤ n . Then it follo ws from equation (37) that c h ( U ∗ )( x | y ) = c h U ( ¯ x | ¯ y ) . Example 8.2.1. One has that ch 1 = 1, while c h V = x 1 + · · · + x m + y 1 + · · · + y n , c h V ∗ = x − 1 1 + · · · + x − 1 m + y − 1 1 + · · · + y − 1 n . 8.3. The functor F m | n : Rep( GL m − n ) → Rep( gl ( m | n )) . Let F m | n : Rep( GL m − n ) → Rep( gl ( m | n )) denote the tensor functor whic h sends • 7→ V deﬁned b y Prop osition 3.5.1 . F or an y ζ ∈ K , the central elemen t ζ · id m + n ∈ gl ( m | n ) acts on the mixed tensor p o w er T ( r , s ) b y the scalar ζ r − s , and so Hom Rep( gl ( m | n )) ( T ( r , s ) , T ( r 0 , s 0 ) = 0 unless r + s 0 = r 0 + s . Moreo ver, it is well-kno wn that the K -algebra map K Σ p → End Rep( gl ( m | n )) ( V ⊗ p ) deﬁned b y the symmetric braiding is surjective, for any p ≥ 0 (see [BR, Remark 4.15] or [Ser2]). Thus, b y Theorem 4.7.1, the functor F m | n is full and an y indecomp osable summand of a mixed tensor p o w ers T ( r , s ) is isomorphic to F m | n ( L ( λ )) for some bipartition λ . Generalizing our notation from § 5, w e set W ( λ ) = F m | n ( L ( λ )). The rest of this section is dev oted to describing W ( λ ) for arbitrary λ . More precisely , DELIGNE’S Rep( GL δ ) AND REPRESENT A TIONS OF gl ( m | n ) 37 w e will giv e a form ula for computing the character of W ( λ ) in § 8.5 and giv e a criterion for the v anishing of W ( λ ) in § 8.7. By Theorem 4.7.1, these results give a classiﬁcation of indecomp osable summands of mixed tensor space in Rep( gl ( m | n )). 8.4. Comp osite supersymmetric Sch ur p olynomials. In § 8.5 we give a for- m ula for computing the character of W ( λ ) in Rep( gl ( m | n )) (see Theorem 8.5.2). In the case λ ◦ = ∅ (resp. λ • = ∅ ), the c haracter of W ( λ ) was computed in [BR] and [Ser2] and is called a c ovariant (r esp. c ontr avariant) sup ersymmetric Schur p olynomial . Our formula for ch W ( λ ) for arbitrary λ is in terms of the n umbers D λ,µ ( § 6) and the so-called c omp osite sup ersymmetric Schur p olynomials 9 (see for instance [MV2]). There are man y equiv alent deﬁnitions of composite sup ersymmet- ric Sc h ur p olynomials; we will use the determinan tal formula found, for instance, in [MV2, (38)]. In order to state this formula we need a few preliminary deﬁni- tions. As in § 8.1, we work with the v ariables x = { x i } 1 ≤ i ≤ m and y = { y i } 1 ≤ i ≤ n , and write ¯ x = { x − 1 i } 1 ≤ i ≤ m and ¯ y = { y − 1 i } 1 ≤ i ≤ n . No w, w e deﬁne the c omplete sup ersymmetric p olynomials by h k = h k ( x | y ) = k X i =0 h k − i ( x ) e i ( y ) where h k ( x ) and e k ( y ) are the complete and elemen tary symmetric p olynomials resp ectiv ely (see for instance [Mac1, § I.2]). In particular, h 0 = 1 and h k = 0 whenev er k < 0. Next, we write ¯ h k = h k ( ¯ x | ¯ y ). Now, given a bipartition λ , w e deﬁne the c omp osite sup ersymmetric Schur p olynomial s λ = s λ ( x | y ) as the follo wing determinate (compare with [MV2, (38)] 10 ): s λ = det                      ¯ h λ ◦ q . . . ¯ h λ ◦ q − 1 . . . ¯ h λ ◦ 2 +1 . . . . . . ¯ h λ ◦ 2 ¯ h λ ◦ 1 +1 . . . ¯ h λ ◦ 2 − 1 ¯ h λ ◦ 1 h λ • 1 − 1 . . . . . . ¯ h λ ◦ 1 − 1 h λ • 1 h λ • 2 − 1 . . . h λ • 1 +1 h λ • 2 . . . . . . h λ • 2 +1 . . . h λ • p − 1 . . . h λ • p                      where p (resp. q ) is any in teger greater than or equal to l ( λ • ) (resp. l ( λ ◦ )). 9 Also known as comp osite sup ersymmetric S-p olynomials. 10 In the literature s λ ( x | y ) is sometimes denoted s λ ◦ ; λ • ( x/y ) or even { λ ◦ ; λ • } . 38 JONA THAN COMES AND BENJAMIN WILSON Example 8.4.1. Fix m = 1 and n = 2. Then s ((1) , (2)) = det  ¯ h 2 h 0 ¯ h 1 h 1  = det 1 x 2 1 + 1 x 1 y 1 + 1 x 1 y 2 + 1 y 1 y 2 1 1 x 1 + 1 y 1 + 1 y 2 x 1 + y 1 + y 2 ! = x 1 y 1 y 2 + y 1 x 1 y 2 + y 2 x 1 y 1 + y 1 x 2 1 + y 2 x 2 1 + 2 1 x 1 + 1 y 1 + 1 y 2 , whereas s ((1 2 ) , (3)) = det   ¯ h 3 h 0 h − 1 ¯ h 2 h 1 h 0 ¯ h 1 h 2 h 1   = det    1 x 3 1 + 1 x 2 1 y 1 + 1 x 2 1 y 2 + 1 x 1 y 1 y 2 1 0 1 x 2 1 + 1 x 1 y 1 + 1 x 1 y 2 + 1 y 1 y 2 x 1 + y 1 + y 2 1 1 x 1 + 1 y 1 + 1 y 2 x 2 1 + x 1 y 1 + x 1 y 2 + y 1 y 2 x 1 + y 1 + y 2    = y 2 1 x 3 1 + y 1 y 2 x 3 1 + y 2 2 x 3 1 + y 2 1 x 2 1 y 2 + y 2 2 x 2 1 y 1 + 2 y 1 x 2 1 + 2 y 2 x 2 1 + 1 x 1 + y 1 x 1 y 2 + y 2 x 1 y 1 − x 1 y 1 y 2 . 8.5. The c haracter of W ( λ ) . The following prop osition lists some of the w ell- kno wn prop erties of s λ whic h will b e useful for this pap er. Prop osition 8.5.1. (1) ch W ( λ ) = s λ whenever λ • = ∅ or λ ◦ = ∅ . (2) s ( λ • , ∅ ) s ( ∅ ,µ ◦ ) = P ν P κ ∈P LR λ • κ,ν • LR µ ◦ κ,ν ◦ s ν . Pr o of. (1) follo ws from the corresp onding determinantal formula for the (non- comp osite) sup ersymmetric Sch ur p olynomials (see [MV1, (6)] and references therein). F or (2) see [CK, (3.2)].  W e are no w ready to pro v e our formula for computing the c haracter of W ( λ ). Theorem 8.5.2. c h W ( λ ) = P µ D λ,µ ( m − n ) s µ for any bip artition λ . Pr o of. W e induct on the size of λ . The case λ = ( ∅ , ∅ ) is easy to c hec k. Now, by Prop osition 4.6.4 w e can write (38) λ = ( λ • , ∅ )( ∅ , λ ◦ ) − X ν | ν | < | λ | a λ,ν ν ∈ R m − n for some a λ,ν ∈ Z . W rite D λ,µ = D λ,µ ( m − n ). Applying lift m − n to (38) and using Example 6.2.1 we ha v e (39) X µ D λ,µ µ = ( λ • , ∅ )( ∅ , λ ◦ ) − X ν | ν | < | λ | a λ,ν X µ D ν,µ µ ∈ R t . Using formula (33) w e compute the coeﬃcient of µ in (39) to b e (40) D λ,µ = X κ ∈P LR λ • κ,µ • LR λ ◦ κ,µ ◦ − X ν | ν | < | λ | a λ,ν D ν,µ . On the other hand, since F m | n is a tensor functor, it follows from (38) that c h W ( λ ) = c h W (( λ • , ∅ )) ch W (( ∅ , λ ◦ )) − X ν | ν | < | λ | a λ,ν c h W ( ν ) . DELIGNE’S Rep( GL δ ) AND REPRESENT A TIONS OF gl ( m | n ) 39 Hence, by Proposition 8.5.1(1) along with induction, w e hav e c h W ( λ ) = s ( λ • , ∅ ) s ( ∅ ,λ ◦ ) − X ν | ν | < | λ | a λ,ν X µ D ν,µ s µ . Th us, by Proposition 8.5.1(2), c h W ( λ ) = X µ X κ ∈P LR λ • κ,µ • LR λ ◦ κ,µ ◦ s µ − X ν | ν | < | λ | a λ,ν X µ D ν,µ s µ , and we are done by (40).  Example 8.5.3. Let m = 1, n = 2 and consider the bipartition ((1 2 ) , (3)). Since and b y Corollary 6.4.2 we hav e lift − 1 (((1 2 ) , (3))) = ((1 2 ) , (3)) + ((1) , (2)). Hence, b y Theorem 8.5.2 and Example 8.4.1 we ha v e c h W (((1 2 ) , (3))) = s ((1 2 ) , (3)) + s ((1) , (2)) = y 2 1 x 3 1 + y 1 y 2 x 3 1 + y 2 2 x 3 1 + y 2 1 x 2 1 y 2 + y 2 2 x 2 1 y 1 + 3 y 1 x 2 1 + 3 y 2 x 2 1 + 2 y 1 x 1 y 2 + 2 y 2 x 1 y 1 + 3 1 x 1 + 1 y 1 + 1 y 2 . F or any Lauren t p olynomial f = f ( z 1 , . . . , z k ), write edeg f = max { deg ( f | z i := z ε i i , 1 ≤ i ≤ k ) | ε i ∈ {± 1 } , 1 ≤ i ≤ k } for the extr emal de gr e e of f , where deg deﬁnes the familiar total degree. Then one has edeg ( f g ) ≤ edeg f + edeg g . F or an y bipartition λ , w e consider s λ and ch W ( λ ) as Laurent polynomials in the v ariables x = { x i } and y = { y j } . Then w e hav e the follo wing corollary to Theorem 8.5.2: Corollary 8.5.4. L et λ b e a bip artition. Then edeg c h W ( λ ) ≤ | λ | . Pr o of. That edeg s µ for an y bipartition µ follo ws from e.g. [MV2, (32)]. The claim then follows from Theorems 6.2.3 (2) and 8.5.2.  8.6. Dimensions. In this subsection we derive a form ula for the K -dimension of W ( λ ) in Rep( gl ( m | n )). T o do so, let d k = d k ( m | n ) denote the result of setting x i = y j = 1 for all 1 ≤ i ≤ m , 1 ≤ j ≤ n in h k . F rom the deﬁnition of h k w e hav e (41) d k = X 0 ≤ i ≤ k  m + k − i − 1 m − 1  n i  whenev er m > 0, and d k =  n k  when m = 0. Now, given a bipartition λ w e let d λ = d λ ( m | n ) denote the result of se tting x i = y j = 1 for all 1 ≤ i ≤ m , 1 ≤ j ≤ n 40 JONA THAN COMES AND BENJAMIN WILSON in s λ , so that (42) d λ = det                      d λ ◦ q . . . d λ ◦ q − 1 . . . d λ ◦ 2 +1 . . . . . . d λ ◦ 2 d λ ◦ 1 +1 . . . d λ ◦ 2 − 1 d λ ◦ 1 d λ • 1 − 1 . . . . . . d λ ◦ 1 − 1 d λ • 1 d λ • 2 − 1 . . . d λ • 1 +1 d λ • 2 . . . . . . d λ • 2 +1 . . . d λ • p − 1 . . . d λ • p                      where p (resp. q ) is any integer greater than or equal to l ( λ • ) (resp. l ( λ ◦ )). It follo ws immediately from Theorem 8.5.2 that (43) dim K W ( λ ) = X µ D λ,µ d µ in Rep( gl ( m | n )). The following technical lemma concerning d k will be useful later: Lemma 8.6.1. Assume m > 0 and let g k ( u ) = Q 1 ≤ j 0. Assume λ is almost ( m | n )-cross so l ( λ • ) , l ( λ ◦ ) ≤ m +1 and λ • i = n + 1 − λ ◦ m − i +2 for all 1 ≤ i ≤ m + 1. Then, by (42), d λ is the determinan t of the (2 m + 2) × (2 m + 2) blo c k matrix ( A B ) where A =      d λ ◦ m +1 d λ ◦ m − 1 · · · d λ ◦ 1 + m d λ ◦ m +1 − 1 d λ ◦ m · · · d λ ◦ 1 + m − 1 . . . . . . . . . d λ ◦ m +1 − 2 m − 1 d λ ◦ m − 2 m · · · d λ ◦ 1 − m − 1      , 44 JONA THAN COMES AND BENJAMIN WILSON B =      d n − m − λ ◦ m +1 d n − m +1 − λ ◦ m · · · d n − 2 m − λ ◦ 1 d n − m +1 − λ ◦ m +1 d n − m − λ ◦ m · · · d n − 2 m +1 − λ ◦ 1 . . . . . . . . . d n + m +1 − λ ◦ m +1 d n + m − λ ◦ m · · · d n +1 − λ ◦ 1      . Hence, if we let C denote the (2 m + 2) × ( m + 1) matrix whose j th column is giv en b y Col j ( C ) = Col j ( A ) + ( − 1) m − 1 Col j ( B ), then by Lemma 8.6.1 we ha v e C = 1 ( m − 1)! X 0 ≤ i ≤ n  n i       g λ ◦ m +1 ( i ) g λ ◦ m +1 ( i ) · · · g λ ◦ 1 + m ( i ) g λ ◦ m +1 − 1 ( i ) g λ ◦ m ( i ) · · · g λ ◦ 1 + m − 1 ( i ) . . . . . . . . . g λ ◦ m +1 − 2 m − 1 ( i ) g λ ◦ m − 2 m ( i ) · · · g λ ◦ 1 − m − 1 ( i )      where g k ( u ) is the p olynomial deﬁned in Lemma 8.6.1. T o sho w d λ = 0 we will v erify that the columns of C (and hence of ( A B )) are linearly dep enden t. T o do so, notice g k − l ( u ) = g k ( u + l ) for any k, l ∈ Z . Hence, the j th column of C is (46) Col j ( C ) = 1 ( m − 1)! X 0 ≤ i ≤ n  n i       g λ ◦ m +2 − j + j − 1 ( i ) g λ ◦ m +2 − j + j − 1 ( i + 1) . . . g λ ◦ m +2 − j + j − 1 ( i + 2 m + 1)      . The m + 1 polynomials g λ ◦ m +2 − j + j − 1 ( u ) for 1 ≤ j ≤ m + 1 are each of degree m − 1 in the v ariable u . Hence, there exist a 1 , . . . , a m +1 ∈ K (not all zero) with P m +1 j =1 a j g λ ◦ m +2 − j + j − 1 ( u ) = 0. It follo ws from (46) that P m +1 j =1 a j Col j ( C ) = 0.  Finally , we are in p osition to prov e our criterion for the v anishing of W ( λ ). Theorem 8.7.6. W ( λ ) 6 = 0 in Rep( gl ( m | n )) if and only if λ is ( m | n ) -cr oss. Pr o of. One direction is Lemma 8.7.2. T o pro v e the other, assume λ is not ( m | n )- cross. W e will pro ceed by inducting on | λ | = ( r, s ). The non-( m | n )-cross bipar- titions of minimal size are exactly the almost ( m | n )-cross bipartitions, hence the base case of our induction is Lemma 8.7.5. If λ is not almost ( m | n )-cross, then there exists a non-( m | n )-cross bipartition µ which is obtained from λ by removing a box. If µ ` ( r − 1 , s ) then b y (34) we hav e µ ( 2 , ∅ ) = λ + µ (1) + · · · + µ ( k ) ∈ R t for some bipartitions µ (1) , . . . , µ ( k ) with | µ ( i ) | < | λ | for all 1 ≤ i ≤ k . Hence, by Theorem 6.2.3(2), µ ( 2 , ∅ ) = λ + ν (1) + · · · + ν ( l ) ∈ R m − n for some bipartitions ν (1) , . . . , ν ( l ) , which implies W ( µ ) ⊗ W (( 2 , ∅ )) = W ( λ ) ⊕ W ( ν (1) ) ⊕ · · · ⊕ W ( ν ( l ) ) in Rep( gl ( m | n )). By induction, we know W ( µ ) = 0, hence W ( λ ) = 0 to o. F or the case µ ` ( r , s − 1), one argues similarly using (35).  Remark 8.7.7. As men tioned ab o v e, if λ ◦ = ∅ (resp. λ • = ∅ ), then λ b eing ( m | n )-cross is equiv alen t to λ • (resp. λ ◦ ) b eing ( m | n )-ho ok. Hence Theorem 8.7.6 is a generalization of Theorem 8.7.1. On the other hand, supp ose n = 0 and set m = d so that F m | n is iden tiﬁed with F d (see § 5.2). In this case λ is ( m | n )-cross if and only if l ( λ ) ≤ d . Hence, Theorem 8.7.6 also generalizes the v anishing criterion in Theorem 5.2.2. DELIGNE’S Rep( GL δ ) AND REPRESENT A TIONS OF gl ( m | n ) 45 8.8. Decomp osing W ( λ ) ⊗ W ( µ ) . Since F m | n is a tensor functor, Theorem 8.7.6 along with our metho d of decomp osing arbitrary tensor products in Rep( GL δ ) from § 7.2 immediately allow us to decomp ose tensor pro ducts of the form W ( λ ) ⊗ W ( µ ). Example 8.8.1. F rom Example 7.2.2 with δ = 0 we ha ve the follo wing decomp o- sition in Rep( gl ( m | m )): W ((2) , ∅ ) ⊗ W (( 2 , 2 )) ∼ = W (((2 , 1) , 2 )) ⊕ W (((3) , 2 )) ⊕ W (((1 2 ) , ∅ )) ⊕ W (((2) , ∅ )) ⊕ 2 . If m > 1, all the bipartitions in the decomp osition ab o ve are ( m | m )-cross. Hence b y Theorem 8.7.6 the summands in the decomp osition ab o v e are all nonzero. On the other hand, ((2 , 1) , 2 ) is the only bipartition ab o v e which is not (1 | 1)-cross. Hence by Theorem 8.7.6, we ha ve the following decomp osition into nonzero inde- comp osables in Rep( gl (1 | 1)): W ((2) , ∅ ) ⊗ W (( 2 , 2 )) ∼ = W (((3) , 2 )) ⊕ W (((1 2 ) , ∅ )) ⊕ W (((2) , ∅ )) ⊕ 2 . References [AF] F.W. Anderson and K.R. F uller. Ri ngs and c ate gories of modules . 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J.C.: Dep ar tment of Ma thema tics, University of Oregon, Eugene, OR 97403, USA E-mail address : jcomes@uoregon.edu B.W.: Dieffenbachstr., 27, Berlin 10967, Germany E-mail address : benjamin@asmusas.net

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