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