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We define and study cohomological tensor functors from the category T n of finite-dimensional representations of the supergroup Gl(n|n) into T n−r for 0 < r ≤ n. In the case DS : T n → T n−1 we prove a formula DS(L) = Π ni L i for the image of an arbitrary irreducible representation. In particular DS(L) is semisimple and multiplicity free. We derive a few applications of this theorem such as the degeneration of certain spectral sequences and a formula for the modified superdimension of an irreducible representation.

Abstract. We describe the image of the canonical tensor functor from Deligne's interpolating category Rep(GL m−n ) to Rep(GL(m|n)) attached to the standard representation. This implies explicit tensor product decompositions between any two projective modules and any two Kostant modules of GL(m|n), covering the decomposition between any two irreducible GL(m|1)-representations. We also obtain character and dimension formulas. For m > n we classify the mixed tensors with non-vanishing superdimension. For m = n we characterize the maximally atypical mixed tensors and show some applications regarding tensor products.

The representation category A = Rep(G, ǫ) of a supergroup scheme G has a largest proper tensor ideal, the ideal N of negligible morphisms. If we divide A by N we get the semisimple representation category of a pro-reductive supergroup scheme G red . We list some of its properties and determine G red in the case GL(m|1).