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


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'.


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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