The thought process that led me to this question is that the identity $$ \left(\prod_i \frac1{1-x_i}\right)\left(\prod_i {1-x_i}\right)=1$$ can be understood as expressing exactness of the Koszul complex. This identity is rewritten by taking $\left(\prod_i \frac1{1-x_i}\right)$ as the generating function for the complete symmetric functions $h_n$ and $\left(\prod_i {1+x_i}\right)$ as the generating function for the elementary symmetric functions $e_n$.

Next we have the Jacobi-Trudi identity which expresses a Schur function as the determinant of a matrix whose entries are complete (or elementary) symmetric functions. Also the Specht module is sometimes constructed as a quotient (or submodule) of the trivial representation of the Young subgroup induced to a representation. This suggests that this is the start of a BGG resolution.

I imagine that if this works then it is well-known. Could I have some references? and where does line of thought lead?