This material has been published in "Algebra, Arithmetic and Geometry with Applications," C. Christensen, G. Sundaram, A. Sathaye and C. Bajaj, eds., Springer-Verlag, New York, 2004, pp. 337-356, the only definitive repository of the content that has been certified and accepted after peer review. Copyright and all rights therein are retained by Springer-Verlag. This material may not be copied or reposted without explicit permission.

Sudhir R. Ghorpade and Christian Krattenthaler

The Hilbert series of Pfaffian rings

(20 pages)

Abstract. We give three determinantal expressions for the Hilbert series as well as the Hilbert function of a Pfaffian ring, and a closed form product formula for its multiplicity. An appendix outlining some basic facts about degeneracy loci and applications to multiplicity formulae for Pfaffian rings is also included.

The following versions are available:


In Footnote 7 of this paper, we remarked that most concepts and results discussed in the appendix on Degeneracy Loci would extend readily from the complex case to that of an arbitrary ground field, at least in characteristic zero case, if instead of cohomology rings, one works in the Chow ring of algebraic cycles modulo rational equivalence. Further, we stated that it is not clear to us how the proof of the squaring principle in the paper of Harris and Tu [Topology 23 (1984), 71-84] would go through in the general case.

In this context, Professor Fulton has kindly informed us that the `squaring principle' is, in fact, false in positive characteristic. See his e-mail for details.

Back to Christian Krattenthaler's home page.