Let $k$ be a field and $S_{r,s}=k[x_{i,j}]$, $1\leq i \leq r, 1\leq j \leq s$. Let $I_t$ be the ideal of $t\times t$-minors of the generic $r\times s$-matrix $(x_{i,j})$, and let $R_{r,s,t}=S_{r,s}/I_t$. It's known that $R_{r,s,t}$ is a normal Cohen--Macaulay ring, and that $R_{r,s,t}$ is Gorenstein exactly when $r=s$ or $t=1$.
More generally, is there a criteria for when $R_{r,s,t}$ will be $\mathbb Q$-Gorenstein, i.e., for when some reflexive power of $\mathcal O_{R_{r,s,t}}(K_{R_{r,s,t}})$ will be locally free? From experimentations in Macaulay2, this already seems to be rare; even for the $2\times 2$-minors of the generic $2\times 3$ matrix this seems to fail. Does anyone know when (if ever) this will hold, and a reference?