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 Varieties1
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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