Schur-Weyl Duality for Heisenberg Cosets

Let $V$ be a simple vertex operator algebra containing a rank $n$ Heisenberg vertex algebra $H$ and let $C=\text{Com}\left( {H}, {V}\right)$ be the coset of ${H}$ in ${V}$. Assuming that the representation categories of interest are vertex tensor categories in the sense of Huang, Lepowsky and Zhang, a Schur-Weyl type duality for both simple and indecomposable but reducible modules is proven. Families of vertex algebra extensions of ${C}$ are found and every simple ${C}$-module is shown to be contained in at least one ${V}$-module. A corollary of this is that if ${V}$ is rational and $C_2$-cofinite and CFT-type, and $\text{Com}\left( {C}, {V}\right)$ is a rational lattice vertex operator algebra, then so is ${C}$. These results are illustrated with many examples and the $C_1$-cofiniteness of certain interesting classes of modules is established.