Hilbert Functions of Ladder Determinantal Varieties
# 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

#### September 15, 2000

## Abstract

We consider algebraic 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 a ladder shaped
region of the rectangular array.
Explicit formulae for the Hilbert function of such varieties are known in
(i) the rectangular case [Abhyankar, 1984], and (ii) the case of 2 ×2
minors in one-sided ladders [Kulkarni, 1985]. More recently, Krattenthaler
and Prohaska (1999) have proved a `remarkable formula', conjectured by
Conca and Herzog (1994) for the Hilbert series in the case of arbitrary sized
minors in one-sided ladders. We describe here an explicit, albeit
complicated,
formula for the Hilbert function and the Hilbert series in the case of
arbitrary sized minors in two-sided ladders. From a combinatorial viewpoint,
this is equivalent to the enumeration of certain sets of `indexed monomials'.

Contents

| Introduction | 1 |

1 | Preliminaries | 3 |

2 | Correspondence Between Radicals and Skeletons in a Biladder | 8 |

3 | Enumeration of Skeletons in a Biladder | 16 |

4 | Enumeration of Indexed Monomials | 23 |

5 | Applications | 30 |

| References | 34 |

^{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.

This paper is published in:
*Discrete
Mathematics*, Vol. 246 (2002), pp. 131-175.

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