Determinantal ideals and the Capelli identities
type: Item
instance of: scholarly article
title: Determinantal ideals and the Capelli identities
zbMATH Open document ID: 0616.13009
DOI: 10.1016/0024-3795(85)90081-3
author: I. M. Anderson
published in: Linear Algebra and its Applications
publication date: 1985-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13B25
Mathematics Subject Classification ID: 14M12
zbMATH DE Number: 3997951
zbMATH Keywords: determinantal variety
zbMATH Keywords: determinantal ideal
zbMATH Keywords: polynomial ring
zbMATH Keywords: Capelli identities
MaRDI profile type: Publication
full work available at URL: 0024-3795(85)90081-3
OpenAlex ID: W2004494466
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description: scientific article; zbMATH DE number 3997951