Equivariant resolutions over Veronese rings

Working in a polynomial ring S=k[x1, horizontal ellipsis ,xn]$S={\mathbf {k}}[x_1,\ldots ,x_n]$, where k${\mathbf {k}}$ is an arbitrary commutative ring with 1, we consider the d$d$th Veronese subalgebras R=S(d)$R={S<^>{(d)}}$, as well as natural R$R$-submodules M=S(> r,d)$M={S<^>{({\geqslant r},{d})}}$ inside S$S$. We develop and use characteristic-free theory of Schur functors associated to ribbon skew diagrams as a tool to construct simple GLn(k)$GL_n({\mathbf {k}})$-equivariant minimal free R$R$-resolutions for the quotient ring k=R/R+${\mathbf {k}}=R/R_+$ and for these modules M$M$. These also lead to elegant descriptions of ToriR(M,M ')$\operatorname{Tor}<^>R_i(M,M<^>{\prime})$ for all i$i$ and HomR(M,M ')$\operatorname{Hom}_R(M,M<^>{\prime})$ for any pair of these modules M,M '$M,M<^>{\prime}$.