Schur powers of the cokernel of a graded morphism

Let $\phi :F⟶G$ be a graded morphism between free $R$-modules of rank $t$ and $t+c-1$, respectively, and let $I_j(\phi )$ be the ideal generated by the $j\times j$ minors of a matrix representing $\phi$. In this short note: (1) We show that the canonical module of $R\mathrm{/}{I}_j(\phi )$ is up to twist equal to a suitable Schur power ${\mathrm{\Sigma }}^{I}M$ of $M=\text{coker}({\phi }^{\ast })$; thus equal to ${\wedge }^{t+1-j}M$ if $c=2$ in which case we find a minimal free $R$-resolution of ${\wedge }^{t+1-j}M$ for any $j$, (2) For $c=3$, we construct a free $R$-resolution of ${\wedge }^{2}M$ which starts almost minimally (i.e. the first three terms are minimal up to a precise summand), and (3) For $c\ge 4$, we construct under a certain depth condition the first three terms of a free $R$-resolution of ${\wedge }^{2}M$ which are minimal up to a precise summand. As a byproduct we answer the first open case of a question posed by Buchsbaum and Eisenbud.