A Note of Hodge's Postulation Formula for Schubert Varieties
# A Note of Hodge's Postulation Formula for Schubert
Varieties^{1}

### Sudhir R. Ghorpade
^{2}

####
*Department of Mathematics *

Indian Institute of Technology, Bombay

Powai, Mumbai 400076 India

E-mail: srg@math.iitb.ernet.in

#### May 25, 2000

## Abstract

In this note we give a somewhat simpler description and an alternative proof
of the so called * postulation formula *, due to Hodge, which explicitly
gives the Hilbert function as well as the Hilbert polynomial of Schubert
varieties in Grassmannians. Our proof uses the combinatorics of
nonintersecting lattice paths and a mild generalization of a result of
Gessel and Viennot. The necessary background concerning such lattice paths
is also presented in a self-contained manner.

Contents

1 | Introduction | 1 |

2 | Schubert Varieties in Grassmannians | 2 |

3 | Nonintersecting Lattice Paths | 3 |

4 | Hodge's Postulation Formula | 5 |

| References | 8 |

^{1}
2000 * Mathematics Subject Classification. * Primary 14M15,
13D40, 05A15; Secondary 13P10, 14M12.

^{2}
Partially
supported by a `Career Award' grant from AICTE,
New Delhi and an IRCC grant from IIT Bombay.

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