On classical tensor categories attached to the irreducible representations of the general linear supergroups $$GL(n\vert n)$$

Abstract

We study the quotient of $$\mathcal {T}_n = Rep(GL(n|n))$$ T n = R e p ( G L ( n | n ) ) by the tensor ideal of negligible morphisms. If we consider the full subcategory $$\mathcal {T}_n^+$$ T n + of $$\mathcal {T}_n$$ T n of indecomposable summands in iterated tensor products of irreducible representations up to parity shifts, its quotient is a semisimple tannakian category $$Rep(H_n)$$ R e p ( H n ) where $$H_n$$ H n is a pro-reductive algebraic group. We determine the $$H_n$$ H n and the groups $$H_{\lambda }$$ H λ corresponding to the tannakian subcategory in $$Rep(H_n)$$ R e p ( H n ) generated by an irreducible representation $$L(\lambda )$$ 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 $$\pi _0(H_n)$$ π 0 ( H n ) .