On the Enumeration of Indexed Monomials and the Computation of Hilbert
Functions of Ladder Determinantal Varieties
#
On the Enumeration of Indexed Monomials and the Computation of Hilbert
Functions of Ladder Determinantal 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

#### October 1, 1999

## Abstract

We outline the computation of an explicit formula for the Hilbert function
of the ladder determinantal varieties defined by the vanishing of all minors
of a fixed size of a rectangular matrix with indeterminate entries such that
the indeterminates in these minors are restricted to lie in some ladder shaped
region of the rectangular array.
Finding such a formula is equivalent to enumerating the set of monomials of a
fixed degree such that the support of these monomials is a subset of a `ladder'
and satisfies a certain ``index condition''.
We also describe applications of this formula for estimating the dimension
of ladder determinantal varieties.

Contents

1 | Introduction | 1 |

2 | Preliminaries | 3 |

3 | Radicals and Skeletons | 5 |

4 | General Case | 6 |

5 | Degree Computations | 8 |

| References | 11 |

^{1}
1991 * Mathematics Subject Classification. * Primary 05A15, 13C40, 13D40, 14M12; Secondary 05A19, 05E10, 14M15.

^{2} A part of this work was supported by
research grant No. 93-106/RG/MATHS/AS from the
Third World Academy of Sciences, Italy. Currently, the
author is partially
supported by a `Career Award' grant from AICTE,
New Delhi and an IRCC grant from IIT Bombay.

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