Socle degrees for local cohomology modules of thickenings of maximal minors and sub-maximal Pfaffians

Let $S$ be the polynomial ring on the space of non-square generic matrices or the space of odd-sized skew-symmetric matrices, and let $I$ be the determinantal ideal of maximal minors or $\mathrm{Pf}$ the ideal of sub-maximal Pfaffians, respectively. Using desingularizations and representation theory of the general linear group we expand upon work of Raicu–Weyman–Witt to determine the $S$-module structures of ${\mathrm{Ext}}_S^j(S\mathrm{/}{I}^t,S)$ and ${\mathrm{Ext}}_S^j(S\mathrm{/}{\mathrm{Pf}}^t,S)$, from which we get the degrees of generators of these $\mathrm{Ext}$ modules. As a consequence, via graded local duality we answer a question of Wenliang Zhang on the socle degrees of local cohomology modules of the form $H_{\mathfrak{m}}^j(S\mathrm{/}{I}^t)$.