$S_n$-Equivariant Sheaves and Kozsul Cohomology

We give a new interpretation of Kozsul cohomology, which is equivalent under the Bridgeland-King-Reid equivalence to Voisin’s Hilbert scheme interpretation in dimensions 1 and 2, but is different in higher dimensions. As an application, we prove that the dimension $K_{p,q}(B,L)$ is a polynomial in $d$ for $L=dA+P$ with $A$ ample and $d$ large enough.