Let $V$ be a finite-dimensional (complex) vector space, and $Sym(V)$ be the symmetric algebra on this vector space. We can consider the multiplication map $Sym(V) \otimes V \to V$ as a complex of $GL(V)$-representations of length $2$. I this talk, I will describe how tensor powers of the above complex define interesting complexes of representations of the symmetric group $S_n$, which were studied by Deligne in the paper "La Categorie des Representations du Groupe Symetrique $S_t$, lorsque $t$ n’est pas un Entier Naturel". I will then explain how computing the cohomology of these complexes helps establish a relation between the Deligne categories and the representations of $S_{\infty}$, which are two natural settings for studying stabilization in the theory of finite-dimensional representations of the symmetric groups. This is joint work with D. Barter and Th. Heidersdorf