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

We study the quotient of $\mathcal{T}_n = Rep(GL(n|n))$ by tensor ideal negligible morphisms. If we consider full subcategory $\mathcal{T}_n^+$ $\mathcal{T}_n$ indecomposable summands in iterated products irreducible representations up to parity shifts, its is a semisimple tannakian category $Rep(H_n)$ where $H_n$ pro-reductive algebraic group. determine connected derived subgroup $G_n \subset H_n$ and groups $G_{\lambda} (H_{\lambda}^0)_{der}$ corresponding generated an representation $L(\lambda)$. This gives structural information about $Rep(GL(n|n))$, including decomposition law product superdimension zero. Some results are conditional on hypothesis $2$-torsion $\pi_0(H_n)$.