Algebras of minors

Abstract

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.

17 pages