The Hilbert Series of Pfaffian Rings

The Hilbert Series of Pfaffian Rings


Sudhir R. Ghorpade 1

Department of Mathematics
Indian Institute of Technology, Bombay
Powai, Mumbai 400076, India
E-mail:

and

Christian Krattenthaler 2

Institut für Mathematik der Universität Wien
Strudlhofgasse 4, A-1090 Wien, Austria
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Dedicated to Professor Shreeram Abhyankar on his seventieth birthday


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.


Contents


1 Introduction 1
2 Definitions 3
3 The Hilbert series of a Pfaffian ring and nonintersecting lattice paths 4
4 The main results 8
A Appendix: Geometry of Degeneracy Loci and a Plethora of Multiplicity Formulae 14
References 18

Afternotes

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.

1 The first author was partially supported by a `Career Award' grant from AICTE, New Delhi and an IRCC grant from IIT Bombay.

2 The second author was partially supported by the Austrian Science Foundation FWF, grant P13190-MAT.

This paper is published in: Algebra, Arithmetic and Geometry with Applications (West Lafayette, 2000), C. Christensen, G. Sundaram, A. Sathaye and C. Bajaj Eds., Springer-Verlag, New York (2004), pp. 337-356.

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