Algebras of minors

Let $X$ be an $n\times m$ matrix of indeterminates over a field $K$ (of sufficiently large characteristic) and $M_t$ the set of $m$-minors of $X$. We consider two objects: (1) the Ress algebra of the polynomial ring $K[X]$ with respect to the ideal $I_t$ generated by $M_t$, and (2) the $A_t$ subalgebra of $K[X]$ generated by $M_t$. Note that $A_t$ is tHE coordinate ring of a Grassmannian if $t=\min(m,n)$; also the cases $t=1$ and $t=m-1=n-1$ are easily understood, since $A_t$ is a polynomial ring over $K$ in these cases. For both objects we compute the divisor class group and the canonical class. In particular we determine the Gorenstein rings among the $A_t$. It turns out that $A_t$ is Gorenstein exactly in the cases listed above and when $t(m+n)=mn$. We use initial methods, based on the straightening law and KRS. They can be applied to other types of determinantal ideals, too. We do this explicitly for generic Hankel matrices.