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Let $S$ be the polynomial ring on the space of non-square generic matrices orthe space of odd-sized skew-symmetric matrices, and let $I$ be thedeterminantal ideal of maximal minors or $\operatorname{Pf}$ the ideal ofsub-maximal Pfaffians, respectively. Using desingularizations andrepresentation theory of the general linear group we expand upon work ofRaicu--Weyman--Witt to determine the $S$-module structures of$\operatorname{Ext}^j_S(S/I^t, S)$ and$\operatorname{Ext}^j_S(S/\operatorname{Pf}^t, S)$, from which we get thedegrees of generators of these $\operatorname{Ext}$ modules. As a consequence,via graded local duality we answer a question of Wenliang Zhang on the socledegrees of local cohomology modules of the form $H^j_\mathfrak{m}(S/I^t)$.

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