Determinantal ideals and the Capelli identities (Q1821150): Difference between revisions

Let k[X] denote the polynomial ring over a field k of characteristic zero, where \(X=(x_{ij})\) is a \(m\times n\) matrix of indeterminants. Let \(I_ r\) denote the ideal of k[X] generated by all \(r\times r\) minors and \(V_ r\) the corresponding subvariety of the variety of all \(m\times n\) matrices. The aim of this paper is to use the classical Capelli identities to give elementary proofs of the following known theorems. Theorem 1: If \(f\in k[X]\) and \(f(A)=0\) for all \(A\in V_ r\), then \(f\in I_ r.\) Theorem 2: If \(r=m\) then the p-th power of \(I_ r\) and the p-th symbolic power of \(I_ r\) are equal.