The equations of conjugacy classes of nilpotent matrices. (English) Zbl 0717.20033

Weyman, J.

Invent. Math. 98, No. 2, 229-245 (1989).

Let X be the set of \(n\times n\) matrices over a field k of characteristic 0, and let O(u) be the set of nilpotent matrices in X with Jordan blocks of sizes \(u_ 1,...,u_ s\) where \(u=(u_ 1,u_ 2,...,u_ s)\) is a partition of n. For \(u=(n)\), \(O(u)\) is the set of all nilpotent matrices and an old result of B. Kostant proved in the fundamental paper [Am. J. Math. 85, 327-404 (1963; Zbl 0124.268)] 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 C. De Concini and C. Procesi [in Invent. Math. 64, 203-219 (1981; Zbl 0475.14041)] where the authors calculated the generators of ideals of schematic intersections \(O(u)\cap D\) (D is the set of diagonal matrices). De Concini and Procesi, and T. Tanisaki [Tôhoku Math. J., II. Ser. 34, 575-585 (1982; Zbl 0544.14030)] proposed different sets of generators of the ideals of O(u).
It follows from the main result of this paper that all their conjectures are true. Moreover, the author constructs minimal sets of generators for the ideals of O(u), and calculates the generators of ideals of “rank varieties” introduced by Eisenbud and Saltman.

Reviewer: K.Otsuka

MSC:

20G15	Linear algebraic groups over arbitrary fields
15A72	Vector and tensor algebra, theory of invariants
15A30	Algebraic systems of matrices
14L30	Group actions on varieties or schemes (quotients)

Keywords:

nilpotent matrices; Jordan blocks; GL(n)-invariants; coordinate ring; generators of ideals; schematic intersections; diagonal matrices

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