On Supergroups and their Semisimplified Representation Categories

The representation category A=Rep(G,𝜖)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {A} = Rep(G,\epsilon )$\end{document} of a supergroup scheme G has a largest proper tensor ideal, the ideal N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {N}$\end{document} of negligible morphisms. If we divide A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {A}$\end{document} by N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {N}$\end{document} we get the semisimple representation category of a pro-reductive supergroup scheme Gred. We list some of its properties and determine Gred in the case GL(m|1).


Introduction
A fundamental fact about finite-dimensional algebraic representations of a reductive group over an algebraically closed field k of characteristic 0 is complete reducibility: Every representation decomposes into a direct sum of irreducible representations. This is no longer true if we consider representations of supergroups on super vector spaces. Indeed by a classical result of Djokovic-Hochschild [17] the representation category of a Lie superalgebra g is semisimple if and only if g is a semisimple Lie algebra or of the form osp(1|2n) for n ≥ 1. Correspondingly many standard techniques from Lie theory do not work for representations of supergroups. Although a lot of progress has been made on representations of special supergroups such as GL(m|n) and OSp(m|2n), many classical questions are still open, most notably the tensor product decompositon of two irreducible representations. The category of finite-dimensional super representations Rep(G) (or its full subcategory Rep(G, )), G a supergroup, is a tensor category. Every k-linear tensor category has a largest proper tensor ideal N , the tensor ideal of negligible morphisms [1,Proposition 7.1.4] [20]. By [1] the quotient category ω : Rep(G, ) → Rep(G, )/N is an abelian semisimple k-linear tensor category. This quotient is called the semisimplification of Rep (G, ).
Semisimplifications of other tensor categories have been studied in a variety of cases: A well-known example is the quotient of the category of tilting modules by the negligible modules (of quantum dimension 0) in the representation category of the Lusztig quantum group U q (g) where g is a semisimple Lie algebra over k [2,5]. In [28] Jannsen proved that the category of numerical motives as defined via algebraic correspondences modulo numerical equivalence is an abelian semisimple category. It was noted by André and Kahn [1] that taking numerical equivalence amounts to taking the quotient by the negligible morphisms. A generalization of these results was obtained in [38]. Recently Etingof and Ostrik [20] studied semisimplifications with an emphasis on finite tensor categories. This result follows immediately from a characterization of representation categories due to Deligne [16]. A natural question is to understand and possibly determine G red for given G. This is very difficult and not even possible in the general case. We assemble a few general results about these quotients and then focus on the GL(m|n)-case (m ≥ n).
We show that the classification of the irreducible representations of G red is a wild problem for n ≥ 3 in Theorem 4.4. Hence the question should be modified as follows: We should study the subcategory in Rep(G red , ) generated by the images ω(L(λ)) of the irreducible representations of G. To determine this subcategory would amount to determine the tensor product decomposition of irreducible representations up to superdimension 0 and would give a parametrization of the indecomposable summands of non-vanishing superdimension. We study this problem in [27] in the case of GL(n|n). The cases GL(m|1) and SL(m|1) are rather special since the blocks are of tame representation type and the indecomposable representations have been classified [21] and we can hope to determine the entire quotient category. From the classification it is easy to determine the irreducible objects of Rep(G red , ) in Lemma 5.2. We then compute their tensor product decomposition in Theorem 5.11. where Z/2Z acts by the parity automorphism on GL(m − 1) × GL(1) × GL (1).
In this statement we adopt the convention -as in the rest of the article -that any group scheme G is seen as a supergroup scheme with trivial odd part and Rep(G) is the category of super representations. In order to determine the tensor product decomposition we use two tools: The theory of mixed tensors [25] gives us the tensor product decomposition between the irreducible GL(m|1)-representations. We then use cohomological tensor functors DS : Rep(GL(m|1)) → Rep(GL(m − 1)) akin to those of [19,26] to reduce the tensor product decomposition between indecomposables to the irreducible case. The main point here is that DS(V ) is a Z-graded object for any V , hence DS could be interpreted as a functor to Z × Rep(GL(m − 1)).
To determine G red is probably in reach for the simple supergroups of maximal atypicality 1. To determine the subgroup of G red corresponding to the irreducible representations is already very difficult for GL(m|n) and even more so for the supergroups OSp(m|2n) and P (n).

Preliminaries
Super linear algebra. Throughout the article k is an algebraically closed field of characteristic 0. A super vector space is a finite-dimensional Z 2 -graded vector space V = V 0 ⊕ V 1 over k. Elements in V 0 respectively V 1 are called even respectively odd. An element is homogenuous if it is either even or odd. For a homogeneous element v write p(v) for the parity defined by We denote by H om(V , W ) the set of k-linear parity-preserving morphism between two super vector spaces V and W . The parity shift functor : svec → svec is defined by We refer the reader to [41,46] for some background on affine supergroup schemes. Recall that a functor from the category of commutative superalgebras G : salg → sets is called a supergroup scheme G if G is a representable group functor, i.e. there is some commutative Hopf superalgebra R such that G ∼ = Spec(R). It is called a supergroup if the representing superalgebra is finitely generated. A particular important example is the functor G = GL(m|n) from the category of commutative superalgebras to the category of groups which sends A = A 0 ⊕ A 1 to the invertible (m + n) × (m + n) matrices of the form where a is an (m × m)-matrix with entries in A 0 , b is an (m × n)-matrix with entries in A 1 , c is an (n × m)-matrix with entries in A 1 and d is an (n × n)-matrix with entries in A 0 .
Representations. We denote the category of finite-dimensional representations of G on super vector spaces with parity preserving morphisms by Rep(G). Let G be a supergroup scheme and let be an element of G(k) of order dividing 2 such that the automorphism int ( ) of G is the parity automorphism defined by x → (−1) p(x) x for homogeneous x. Then let Rep(G, ) be the category of (finite-dimensional) representations V = (V , ρ) such that ρ( ) is the parity automorphism of V .
Example: [16, Example 0.4 (i)] If G is an affine group scheme, then is central. If is trivial, one recovers the ordinary representation category of G (non-super). If is nontrivial, for every representation (V , ρ) of G the involution ρ( ) defines a Z 2 -graduation on V and the commutativity isomorphism of the tensor product is given by the Koszul rule. The category Rep(G, ) identifies itself with Rep(G, 1) as a k-linear monoidal category, but they are not equivalent as symmetric monoidal categories.
Example: [16,Example 0.4 (ii)] If G is an affine supergroup scheme, let μ 2 act on G by the parity automorphism. Then Rep(μ 2 G, = (−1, e)) is the category of super representations of G.
For the supergroup GL(m|n) and = diag(E m , −E n ) we put Rep(GL(m|n), ) = R m|n . For the whole category Rep(GL(m|n)) we also write T m|n . Then T m|n = R m|n ⊕ R m|n [9,Corollary 4.44]. The categories T m|n and R m|n are examples of super tannakian categories. For background on tensor categories we refer to [18]. We denote the unit object of a tensor category by 1. For a partition λ of n let S λ (−) be the associated Schur functor [16]. We put Sym n (X) = S (n) (X) (the n-th symmetric power) and n (X) = S (1,...,1) (X) (the n-th alternating power). An object X of A is called Schur-finite if there exists an integer n and a partition λ of n such that S λ (X) = 0. (1) The endomorphism ring of any indecomposable object is a local ring.
(2) Every object can be written as a direct sum of indecomposable objects.
(3) Every module has a unique projective cover An example is given by the category R m|n [21,41].

Negligible Morphisms
An additive category A is a Krull-Schmidt category if every object has a decomposition in a finite direct sum of elements with local endomorphism rings. An ideal in a k-linear category is for any two objects X, Y the specification of a k-submodule T (X, Y ) of H om A (X, Y ), such that for all pairs of morphisms f ∈ H om A (X, X ), g ∈ H om A (Y, Y ) the inclusion gT (X , Y )f ⊆ T (X, Y ) holds. Let T be an ideal in A. Then A/T is the category with the same objects as A and with H om A/T (X, Y ) = H om A (X, Y )/T (X, Y ). It is again a Krull-Schmidt category [34], [33,Lemma 2.1]. Suppose that A is abelian and that every object has finite length and let X be an indecomposable element and φ an endomorphism of X. By Fitting's lemma φ is either invertible or nilpotent. An element X is indecomposable if and only if its endomorphism ring is a local ring. We assume in the following that A is a super tannakian category or a pseudoabelian full tensor subcategory. Then all the above conditions hold.
An ideal in a tensor category is a tensor ideal if it is stable under id C ⊗ − and − ⊗ id C for all C ∈ A. The ideal is then stable under tensor products from left or right with arbitrary morphisms. Let T r be the trace.  The quotient A/N will be called the universal semisimple quotient of A. For completeness sake we assemble a few elementary lemmas about this quotient.

Lemma 3.4 An indecomposable object X of A maps to zero in A/N if and only if id X belongs to N (X, X).
The collection of these objects -called negligible objects -is denoted by N . The dimension of an object X in a tensor category is defined T r(id X ) ∈ End (1).

Lemma 3.5 An indecomposable object is in
Proof If X ∈ N we have T r(g) = 0 for all g ∈ End(X), in particular for g = id X . Let sdim(X) = 0. We have to show: id X ∈ N(X, X), ie. T r(g) = 0 for all g ∈ End(X). Since X is indecomposable g is either nilpotent or an isomorphism. If g is nilpotent T r(g) = 0 [8, Lemme 1.4.3]. Let g be an isomorphism. Since X is indecomposable g has a unique eigenvalue λ and T r(g) = λsdim(X), hence T r(g) = 0.

Lemma 3.6 ([8, Section 1.4]) The functor A → A/N induces a bijection between the isomorphism classes of indecomposable elements not in N and the isomorphism classes of irreducible elements in
Since A and A/N are abelian and every object has finite lenght, an object X is indecomposable if and only if End(X) is a local ring.

The Pro-Reductive Envelope
Since the quotient A/N is again a super-tannakian category, this defines a reductive supergroup scheme G red with A/N Rep(G red , ) with : μ 2 → G such that the operation of μ 2 gives the Z 2 -graduation of the representations. We call G red the pro-reductive envelope of G (following [1]). If G is an algebraic group, the pro-reductive envelope has been extensively studied by Andre and Kahn. Their proofs do not apply to the supergroup case. In the tannakian case N = R is equal to the radical ideal. In particular no indecomposable objects map to zero. Even in the tannakian case the pro-reductive cover will not be of finite type in general.

Theorem 3.8 [1], theorem C.5. The proreductive envelope of an affine k-group G is of finite type over k if and only if G is of finite type over k and the prounipotent radical of
Example: a) If G = G a , then G red = SL (2). b) If G = G a ×G a , then G red is no longer of finite type. In fact, the determination of G → G red is unsolvable since it would include a classification of the indecomposable representations of G which is a wild problem [1, 19.7].
More generally Rep(G red , ) is of finite or tame type if Rep(G) is of finite or tame type. The converse is not so obvious: If Rep(G) is of wild type, the problem of classifying indecomposable modules of non-vanishing superdimension will often be wild as well. However for the supergroup Q(n) corresponding to the simple Lie superalgebra q(n) every nontrivial irreducible representation has superdimension 0 [13, Theorem 1.2]. While the classification of the indecomposable representations is of Q(n) is wild, this might not be true for the ones of non-vanishing superdimension.
If the problem of classifying indecomposable modules of non-vanishing superdimension is wild, we should not try to determine G red in this case, but ask the following weaker questions. Given any object V ∈ Rep(G) or Rep(G, ), consider its image in A/N . The tensor category generated by it is a semisimple algebraic tensor category (since A/N is semisimple). The semisimple algebraic tensor categories in characteristic zero were classified in [44,Theorem 6]: H of a reductive algebraic k-group H with a product G = r≥1 Spo(1|2r) n r of simple supergroups of BC-type, where the semidirect product is defined by an abstract group homomorphism p : π 0 (H ) → r≥1 S n r . Now consider an irreducible object V ∈ Rep(G, ) and consider the tensor category generated by ω(V ) in Rep(G, )/N . This tensor subcategory corresponds to an algebraic supergroup G V → G red as in 3.9. Then this group is of finite type since it has a tensor generator. If G V is reductive (i.e. not containing OSp(1|2n)) we have the following weak a priori estimate. Lemma 3.10 Let T be a maximal torus in G V and X * (T ) its character group. Let R be the subgroup generated by the roots of G V . Then the center of G V has cyclic character group X/R and (G V ) 0 der has cyclic center.
Proof ω(V ) is a tensor generator of Rep(G V ) for the reductive group G V and likewise ω(V ) is a tensor generator of Rep(G V ) 0 der . Now use that a reductive group has a faithful irreducible representation if and only if X/R is cyclic and a semisimple group has a faithful irreducible representation if and only if its center is cyclic [36].

The Basic Classical Cases
Let G be basic classical [41] with underlying basic classical Lie superalgebra g [29]. Duflo and Serganova [19] and [40] constructed for certain elements x ∈ g 1 with [x, x] = 0, where g 1 denotes the odd part of g, tensor functors V → V x : Rep(g) → Rep(g x ) where g x is a classical Lie algebra or osp(1|2n). These functors are not full, hence need not factorize over the quotient Rep(G)/N . However it should be expected that G red contains groups G x with Lie superalgebra g x . For instance the superdimension of any irreducible representation in Rep(G) equals the superdimension of some representation in Rep(G red ) and in Rep(g x ).
Note that this representation in Rep(g x ) might not be irreducible.
For gl(m|n) we have g x = gl(|m − n|) and for osp(m|2n), m = 2l or 2l + 1, we have g x = osp(m − 2min(l, n), 2n − 2min(l, n)). For the exceptional Lie superalgebras the functor of Duflo-Serganov gives representations of the following Lie algebras: Hence G red should contain GL (1) or SL (2) or SL(3) as a subgroup respectively. We determine Rep(GL(m|1))/N in this article. For the GL(n|n)-case see [27]. The OSp(2|2n)-case can be treated similar to the GL(m|1)-case. In this case we obtain Remark: Analogs of DS can also be defined in the P (n) and Q(n)-case. For P (n) it is currently not known which irreducible representations have non-vanishing superdimension. In the Q(n)-case every nontrivial irreducible representation has superdimension 0. This does not mean that the group G red is trivial since we can still have many indecomposable representations with non-vanishing superdimension.

How to Determine the Semisimplification
In general it is quite hard to determine the semisimplification A/N or even the group G V corresponding to an indecomposable representation V . Here are some comments on possible approaches and their difficulties.
(1) The functor ω is not exact, hence it does not induce a homomorphism between the Grothendieck rings K 0 (A) and K 0 (A/N ). This would have allowed to compare the transcendence degrees of the Grothendieck rings and obtain estimates on the rank of the group G red . (2) Taking the quotient modulo negligible objects is not compatible with restriction to subgroups. If H ⊂ G is a subgroup, the corresponding restriction functor Res preserves the superdimension of an indecomposable object X of superdimension 0, Res(X) will in general be a direct sum of indecomposable summands of positive and negative superdimensions whose superdimensions add up to zero. (3) There is a vast literature on determining connected semisimple groups and a given representation from some discrete data. Examples are [35] (dimension data), Larsen's conjecture [23] and the classification of small representations [4]. All this requires the condition that the group is connected which we do not know for the pro-reductive envelope (see number 5). For the purpose of determining G V the most useful one might be [32] which often determines V and G V if Sym 2 (V ) and 2 (V ) are either irreducible or irreducible plus trivial. However even this requires the computation of 2 (V ) and Sym 2 (V ) which is in general difficult for an indecomposable representation of a supergroup.
(4) The functor DS, that is the crucial tool in the GL(m|1)-case, is not as powerful in more complicated cases. While DS(L(λ)) is known by [26,Theorem 16.1] in the T m|n -case, DS is not exact and it is not known how DS(X) behaves for indecomposable objects X of lenght 2 in T m|n . Note also that DS does not necessarily preserve negligible objects: the Kac module K(1) ∈ T n|n of superdimension zero decomposes under DS into a direct sum of irreducible representations of non-vanishing superdimension [26, Section 10]. (5) The most fundamental problem is that the groups G V might not be connected. This is not a problem in the tannakian case considered in [1]: If an algebraic group G is connected, G red is connected as well [1,Lemme 19.4.1]. No such statement is known in the super case. This means that there could be torsion up to negligible objects in Rep(G), e.g. there could be indecomposable objects X of superdimension 1 with X ⊗r ∼ = 1 ⊕ negligible (coming from π 0 (G red ) or a subgroup). Since there is typically no condition or restriction on the negligible part, such objects are hard to exclude. If we do not know that the group is connected, this makes the criteria in number 3 rather useless.

Preliminaries
Recall from Section 2 that T m|n denotes the category of finite dimensional algebraic representations of GL(m|n) with parity preserving morphisms. We always assume m ≥ n. The subcategory R m|n is stable under the dualities ∨ and * (the twisted dual [9, Section 4]). The irreducible representations in R m|n are parametrized by their highest weight with respect to the Borel subgroup of upper triangular matrices. A weight λ = (λ 1 , ..., λ m | λ m+1 , · · · , λ m+n ) ∈ X + of an irreducible representation in R m|n satisfies λ 1 ≥ . . . λ m , λ m+1 ≥ . . . λ m+n with integer entries [9, Section 4]. The Berezin determinant defines a one dimensional representation Ber. Its weight is is given by λ i = 1 for i = 1, . . . , m and λ m+i = −1 for i = 1, .., n.
For each weight λ ∈ X + we have the irreducible representation L(λ), its projective cover P (λ) and the Kac module (or standard objects) K(λ), the universal highest weight module in R m|n . The Kac module K(λ) has irreducible top and socle, and the top is given by the irreducible representation L(λ) [29,Proposition 2.4]. We say that a module is a Kac object if it has a filtration whose subquotients are Kac modules. The full subcategory of these modules is denoted C + . Similarly we have the category C − of objects which have a filtration by the twisted duals K(λ) * (costandard objects). By [9, Theorem 4.37] C + ∩ C − = P roj (every tilting module is projective).
The irreducible representations in T m|n are given by the {L(λ), L(λ) | λ ∈ X + } where denotes the parity shift.
Remark: Since T m|n = R m|n ⊕ R m|n and R m|n is closed under tensor products, we can often work in R m|n . This is however not true in Section 5 since the Duflo-Serganova functor defines a tensor functor DS : R m|n → T m|n .

Weight Diagrams
To each highest weight λ ∈ X + we associate, following [12, Section 1], two subsets of cardinality m respectively n of the numberline Z The integers in I × (λ)∩I • (λ) are labeled by ∨, the remaining ones in I × (λ) resp. I • (λ) are labeled by × resp. •. All other integers are labeled by a ∧. This labeling of the numberline Z uniquely characterizes the weight λ. If the label ∨ occurs r times in the labeling, then r is called the degree of atypicality of λ. Notice that 0 ≤ r ≤ n, and λ is called maximal atypical if r = n. Examples are the trivial module 1 and the standard representation V of highest weight λ = (1, . . . , 0|0, . . . , 0) for m = n. Another example is the Berezin determinant Ber = L(1, . . . , 1 | − 1, . . . , −1)

Wildness
If A is a category as in Lemma 2.4 we can associate to it its Ext-quiver. The vertex set X + is given by the set of isomorphism classes of simple modules and the number of arrows from λ to μ is given by Ext 1 A (L(λ), L(μ)).  A block of X + is a connected component of the Ext-quiver. Let A be the full subcategory of objects of A such that all composition factors are in (also called a block). This gives a decomposition A = A of full abelian subcategories. Every indecomposable module lies in a unique A and all its simple submodules belong to . Two irreducible representations L(λ) and L(μ) are in the same block if and only if the weights λ and μ define labelings with the same position of the labels × and •. The degree of atypicality is a block invariant, and the blocks of atypicality r are in 1-1 correspondence with pairs of disjoint subsets of Z of cardinality m − r resp. n − r.

Theorem 4.3 Assume m, n ≥ 2. Then GL(m|n) red is not of finite type.
Proof This follows from the description of the Tannaka group generated by the irreducible elements in [27, lemma 11.4].
The statement also follows from the following lemma. This lemma should of course also hold for m, n ≥ 2, but would require a more difficult argument. Let Q denote the Ext-quiver of R mn . Then there exists a system of relations R on Q such that R mn kQ/R − mod.

Lemma 4.4 Assume m, n ≥ 3. Then the problem of classifying indecomposable representations of non-vanishing superdimension is wild.
Proof We show that that the classication is wild for every maximally atypical block for n ≥ 3. Any such block is equivalent to the maximal atypical block of GL(n|n) [39, Theorem 3.6] [12]. Hence we show that the problem is wild in . By [10,Corollary 5.15], for any two irreducible modules L(λ), L(μ) ∈ R n dim(Ext 1 R n (L(λ), L(μ))) = p (1) λ,μ + p | | y y y y y y y y Since this subquiver has no path of length > 1, it embeds fully into k(Q)/R. The classification of indecomposable representations of the r-subspace quiver is wild for r ≥ 5 [ . This defines a bijection between the isomorphism classes of indecomposable representations of the (2n − 1)-subspace quiver with a subset of the indecomposable representations of the 2n-subspace quiver satisfying ( * ).
As explained above this means that we should not try to determine G red for general m, n. Instead we should determine the tensor category generated by the irreducible elements in A/N and the corresponding reductive supergroup. In the n = 1-case, where we have tame representation type, we determine the full quotient in this article (Theorem 5.12). We prove this by describing explicitly the image of an indecomposable representation I .

Germoni's Classification
Let be a singly atypical block of R m|1 . Then is equivalent to the category of nilpotent Z-graded finite-dimensional representations of A, the quotient of the free algebra on two generators d + and d − of respective degrees +1 and −1 by the relations (d + ) 2 = (d − ) 2 = 0 [21, Section 5]. Let˜ be the following quiver Then there is a bijection between the isomorphism classes of non-projective indecomposable representations of˜ and the finite connected subquivers I of˜ other than . These are also known as zigzag-modules [22,42] due to the following graphical interpretation of their Loewy structure. With I + [a, b] we denote the indecomposable module attached to [a, b] with L(b) in the top

Superdimensions
The superdimension of any Kac module is zero for any type I Lie superalgebra. The superdimension is additive in short exact sequences, hence we obtain sdimL(a) = −sdimL(a +1) for any atypical weight a.

Corollary 5.1 The superdimension of the indecomposable modules I + [a, b] and I − [a, b] is given by
The only other remaining indecomposable modules are the projective covers P (λ) of the atypical simple modules which have superdimension 0.

Corollary 5.2 The irreducible objects in A/N are up to isomorphism given by the
for all atypical blocks .

Mixed Tensors
We first determine the contribution from the irreducible modules. The subcategory T ⊂ R m|n (m ≥ n) of mixed tensors is the pseudoabelian full subcategory of objects, which are direct summands in a tensor product V ⊗r ⊗ (V ∨ ) ⊗s for some r, s. We have the equivalence Remark: The number s(λ) is described in the proof of [25,Theorem 8.2] in terms of the weight diagram of λ. Its precise description is not needed in the present article.
We now consider the tensor subcategory in T m|1 /N generated by the images of the irreducible modules. It is again a super tannakian category because it inherits the Schur finiteness of T m|1 /N (and then apply Theorem 2.3). It is clear from Theorem 5.3 that the reductive supergroup corresponding to this subcategory is GL(1) × GL(m − 1). Indeed the element defined in [14] [25, Section 10] and the determinant representation det.

Indecomposable Modules
We therefore understand the tensor product decomposition between the irreducible representations in T m|1 . In order to know the tensor product decomposition between any two irreducible representations of T m|1 /N , it remains to compute the tensor product decompo- a, b and a , b could be in different blocks) up to superdimension 0. This will be a complicated reduction to the case of irreducible representations.

Cohomological Tensor Functors
We now define cohomological tensors T m|1 → T m−1|0 . These were first defined in [19] and then later refined in [26]. We define a similar refined version in the T m|1 -case, but it can be easily extended to the general T m|n -case [24]. For any x ∈ X = {x ∈ g 1 | [x, x] = 0} and any representation (V , ρ), the operator ρ(x) defines a complex since ρ(x) • ρ(x) is zero, and we define V x = ker(ρ(x))/ im(ρ(x)). By [19,40] this defines a tensor functor and V x ∈ T m−1|0 . We fix the following element x ∈ X and denote the corresponding tensor functor V → V x by DS : T m|1 → T m−1|0 .

Parity Considerations
If V is in R m|1 , DS(V ) may not be in R m−1|0 . We need to study this more closely. We embed GL(m − 1|0) as an upper block matrix in GL(m|1). More precisely we fix the embedding Recall that we defined the categories R m|n in Section 2. We now fix the morphism : Then R m|1 is the full subcategory of objects V such that p V = ρ( m|1 ). If we use the embedding above, we Since ρ(x) is an odd morphism Hence ρ(x) induces the even morphism We use the notation ∂ for ρ(x).

Z-Grading
We equip DS(V ) with a Z-grading as in [26,Section 3]. Although DS(V ) is in T m−1|0 , it still has an action of the torus of diagonal matrices in GL(m|1). Let V = λ V λ be the weight decomposition and v = λ v λ in V . An easy calculation shows the following lemma.  form (1, . . . , 1, t −1 ) and denote by V l the eigenspace of V where H s acts by the eigenvalue t l . Another easy calculation shows the next lemma.
Since ∂V λ ⊂ V λ+μ we obtain a complex which we can write by the last lemma as We denote the cohomology of this complex by H l (V ). Then DS(V ) l = l (H l (V )) and we obtain a direct sum decomposition of DS(V ) into GL(m − 1)-modules This extra structure is very important since it carries a lot more information then just the Z 2 -graded version of DS(L) in T m−1|0 .

Lemma 5.6
A short exact sequence in R m|1 gives rise a long exact sequence for H l .
We denote by σ the automorphism of gl(m|1) defined by σ (x) = −x T where () T denotes the supertranspose. Then σ (x) is still a nilpotent element in g 1 . The corresponding tensor functor is denoted by DS σ : T m|1 → T m−1|0 . We can copy the arguments from above and endow DS σ (V ) with a Z-grading DS σ (V ) = l (H l σ (V )). DS and DS σ behave in the same way on irreducible objects, but differ on indecomposable elements, see Lemma 5.8.

Corollary 5.7 For any representation V , the objects DS(V ) and DS σ (V ) are Z-graded objects.
Another way of looking at this Z-grading is the following. The collection of cohomology functors H i : R m|1 → R m−1|0 for i ∈ Z defines a tensor functor to the category of Z-graded objects in R m−1|0 . Using the parity shift functor , this functor can be extended to a tensor functor As in [26,Theorem 4.1] we conclude from the support variety calculations in [6, Section 3] the following lemma Lemma 5. 8 The kernel of DS is C − and the kernel of DS σ is C + .

Cohomology Computations
We can now calculate DS(M) for any M of nonvanishing superdimension. By [19] DS(L(λ)) = mL core ⊕ m L core ∈ T m−1|0 for atypical L(λ) where L core ∈ R m|1 is the irreducible GL(m − 1)-representation obtained from L(λ) by replacing the single ∨ in the weight diagram by a ∧. We recall from [25, Section 10] that we have a commutative diagram where Rep(GL m−1 ) denotes Deligne's category to the parameter δ = m − 1. For this note that the image of any mixed tensor under DS is in R m−1|0 . We obtain the same diagram if we replace DS by DS σ since DS σ sends the standard to the standard representation.

Tensor Products up to Superdimension Zero
We now calculate the tensor product decomposition of I ± [a, b] ⊗ I ± [a , b ] up to superdimension zero using Lemma 5.10. Since DS sends every indecomposable module in the same block to the same irreducible GL(m − 1)-representation, it might be a bit surprising that we can use DS for this. It is crucial here that DS(I ± [a, b]) is a Z-graded object and that the intersection of the fibre of a Z-graded irreducible object in T m−1|0 under DS respectively DS σ consists of one indecomposable representation.
Notation. So far we worked within a fixed block so that the notation I + [a, b] was unambigious. In the following tensor product decomposition we need to specify the block of I + [a, b]. There is a natural bijection between atypical blocks of R m|1 and irreducible representations of GL(m − 1) given by → L core as in Section 5.  [39][7, Section 6.3] to fix this parametrization. We denote by L(0) the irreducible representation in λ corresponding to the trivial representation 1 ∈ R 1|1 under this equivalence. Then H a (L(a)) ∼ = L core and is zero in other degrees.

Lemma 5.11 Up to superdimension 0 summands we have the following decompositions
Proof We consider the tensor product I + [a, b] λ ⊗I + [a , b ] μ . Under DS the two modules map to two irreducible elements of Z × T m−1|0 , namely b × L (λ ) and b × L (μ ). Their tensor product is given by the Littlewood-Richardson-rule Note that not only DS : T m|1 → T m−1|0 is a tensor functor, but that also the induced functor DS : T m|1 → Z × T m−1|0 is compatible with the tensor product by the Künneth formula for the cohomology using that the cohomology of every indecomposable object is concentrated in one degree. Under the tensor functor DS σ the two indecomposable objects map again to two irreducible objects in Z × T m−1|0 . Now DS σ agrees with DS on irreducible representations, and on the indecomposable modules I + [a, b] λ and I + [a , b ] μ the two functors differ by a Z-shift: I + [a, b] λ maps toã × L (λ ) and I + [a , b ] μ maps toã × L (μ ). For the tensor product we obtain Hence the tensor products just differs by a Z-shift by (b −ã) − (ã −ã ). We look at the fibre of a summand (b + b ) × L (ν ) under DS and DS σ . The fibre under DS in the block ν consists of The only indecomposable representation in the intersection of the two fibres is I + [a + a , b+b ]. Hence this indecomposable module will appear as a direct summand in the tensor product decomposition.
The case I − [a, b] λ ⊗ I − [a , b ] μ can be reduced to the first case using (A ⊗ B) * ∼ = A * ⊗ B * for the twisted dual () * . The and under DS σ to Hence these two tensor products differ by a Z -shift by (b − a ) − (a − a ). The DS fibre of a summand (b + a ) × L (ν ) in the block ν consists of The DS σ fibre of the summand (a + b ) × L (ν ) in the block ν consists of Remark: In the sl(2|1)-case explicit formulas for the tensor products between any two indecomposable objects are already known [22].

The Pro-Reductive Envelope
In this section we prove the following theorem. Proof If I 1 , I 2 ∈ T m|1 are two indecomposable objects with positive superdimension, then every non-negligible summand in the decomposition I 1 ⊗ I 2 has positive superdimension. Indeed by Lemmas 5.9 and 5.10 using that that DS(L(λ)) ∼ = L core if p(λ) is even, each summand in DS(I 1 ) ⊗ DS(I 2 ) has positive superdimension. Now we consider the full subcategory in T m|1 of direct summands in iterated tensor products of indecomposable representations of superdimension ≥ 0. This category contains every indecomposable object up to a parity shift. It is closed under tensor products and duals and defines a pseudo-abelian tensor subcategory T + m|1 of T m|1 . Hence the quotient T + m|1 /N is defined and a super tannakian category by Proposition 3.2. By [15,Theorem 7.1] any tensor category A (in the sense of loc.cit) which satisfies dim A (X) > 0 for all objects X ∈ A is a tannakian category. Hence T + m|1 /N is the (ordinary) representation category of a reductive group G red,+ . Then T m|1 /N is the category of super representations of G red,+ and in particular G red = Z/2Z G red,+ .
By [18,Proposition 2.20] an algebraic group G is connected if and only if there exists an object X ∈ Rep(G) such that every object of Rep(G) is isomorphic to a subquotient of X n for some n ≥ 0. The tensor product decomposition in Lemma 5.11 shows then that there is no such X and G red,+ is therefore connected. We now use the preliminary considerations before the proof to define a homomorphism φ between the Grothendieck semirings K + 0 (G red,+ ) and K + 0 (GL(m − 1) × GL(1) × GL (1)). Clearly we should have for L(λ) ∼ = Ber s(λ) ⊗ R(λ L , λ R ) that We know by Theorem 5.3 that this is compatible with the ring structure for the classes of irreducible modules L(λ). It then follows from Lemma 5.11 and the preliminary considerations before the proof that this defines a ring homomorphism. By definition this sends (classes of) irreducible representations to irreducible representations and is clearly bijective. Since we know that G red,+ is a connected reductive group, we can use [30, Theorem 1.2] that every homomorphism of Grothendieck semirings φ : K 0 (H ) → K 0 (G) (for H, G connected reductive) which maps irreducible representations to irreducible, comes from a group homomorphism φ * : H → G. If φ is an isomorphism, so is φ * [30, Theorem 1.2]. Hence the result follows.
Remark: It might be tempting to define a tensor functor between the two representation categories directly instead of passing to K + 0 . While one may pass to the skeletal subcategories in order to define the functor on objects and not just on isomorphism classes, it is a priori not clear that this is compatible with the associativity constraints.
Remark: The reason for considering T + m|1 is to get rid of possible OSp(1|2n)-factors.
Remark: It is crucial here to consider the Grothendieck semiring, not the Grothendieck ring. Indeed if G is connected, semisimple, and simply connected, then the Grothendieck K 0 (G) is isomorphic to Z[x 1 , . . . , x r ] where r is the rank of G [30] and encodes only the rank.
Remark: By [43] the character formula for an atypical representation of GL(m|1) has the form of a character formula of GL(m − 1). Hence the superdimension of any irreducible representation equals up to a (−1) p(λ) factor the dimension of an irreducible GL(m − 1)representation. Our results give a conceptual explanation for this.