On the variety of projectors

Abstract

In this paper we are going to give an answer to the following two questions. Let K be a field and A4 the space of n x n square matrices with coefficients in K. The first question we want to investigate is what are the equations of the subvariety A’, of ikf, consisting of rank h matrices A such that A2 = A. By equations we mean of course a set of explicit generators of the prime ideal of x,. The second question is what are the equations of the varieties Yh, h <n/2, which are the closures of the nilpotent conjugacy classes in h4, E, consisting of matrices of rank h whose square is the matrix 0. In this case we shall also prove that these varieties Y, are normal and Cohen-Macaulay in arbitrary characteristic, thus generalizing a result of Hesselink [3] to arbitrary characteristic. Note that in characteristic 0 all closures of nilpotent conjugacy classes have been proved to be normal and Cohen-Macaulay by Kraft and Procesi [4], but our method is not suitable for generalizing their result. The method used in this paper is the one of Hodge algebras [ 11, namely we shall exhibit an explicit linear basis for the coordinate rings of the varieties X,, and Y,.