Mixed Tensors of the General Linear Supergroup

We describe the image of the canonical tensor functor from Deligne’s interpolating category $Rep(G{L}_{m-n})$ to $Rep(GL(m\mathrm{\mid }n))$ attached to the standard representation. This implies explicit tensor product decompositions between any two projective modules and any two Kostant modules of $GL(m\mathrm{\mid }n)$, covering the decomposition between any two irreducible $GL(m\mathrm{\mid }1)$-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.