On classical tensor categories attached to the irreducible representations of the General Linear Supergroups $GL(n\vert n)$

Abstract

We study the quotient of Tn = Rep(GL(n|n)) by the tensor ideal of negligible morphisms. If we consider the full subcategory T + n of Tn of indecomposable summands in iterated tensor products of irreducible representations up to parity shifts, its quotient is a semisimple tannakian category Rep(Hn) where Hn is a pro-reductive algebraic group. We determine the connected derived subgroup Gn ⊂ Hn and the groups G λ = (H λ ) 0 der corresponding to the tannakian subcategory in Rep(Hn) generated by an irreducible representation L(λ). This gives structural information about the tensor category Rep(GL(n|n)), including the decomposition law of a tensor product of irreducible representations up to summands of superdimension zero. Some results are conditional on a hypothesis on 2-torsion in π0(Hn). Contents 1. Introduction 2. The superlinear groups 3. Weight and cup diagrams 4. Cohomological tensor functors. 5. Tannakian arguments 6. The structure of the derived connected groups G n 7. Proof of the structure theorem: Overview 8. Small representations 9. The cases n = 2, 3 and the S i -case 10. Tannakian induction: Proof of the structure theorem 11. A conjectural structure theorem 12. The Picard group of T n 13. The Picard group of T n and the group H n Appendix A. Equivalences and derivatives

Semisimplicity of the $DS$ functor for the orthosymplectic Lie superalgebra

“…Theorems A,B allow us to compute the superdimension of an irreducible osp(m|2n)module (since DS is symmetric monoidal it preserves the superdimensions); see [HW] and [EAS] for analogous results in the gl(m|n) and p(n)-case. Our main theorem also allows to reduce certain questions about tensor products to lower rank similarly to [HW2], [H].…”