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

E-mail:

*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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