Abstract

Let X be the set of n• n matrices over a field k of characteristic 0. For a partition u=(nl, u2..... us) of n we denote by O (u) the set of nilpotent matrices in X with Jordan blocks of sizes ul,..., us. We are interested in equations of the closure of O (u) in Xi. e. in the generators of the ideal of polynomial functions on Xvanishing on O (u). For u=(n), O (u) is the set of all nilpotent matrices and an old result of Kostant proved in the fundamental paper [K] says that the equations are the GL (n)-invariants in the coordinate ring of X (GL (n) acts on X by conjugation). The problem of calculating the equations of O (u) in general was proposed by DeConcini and Procesi in [DP] where the authors calculated the generators of ideals of schematic intersections O (u) c~ D (D is the set of diagonal matrices). DeConcini and Procesi, Tanisaki [T] and Eisenbud and Saltman [ES] proposed different sets of generators of the ideals of O (u).