Hilbert functions of ladder determinantal varieties

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 obtained in (i) the rectangular case by Abhyankar (Rend. Sem. Mat. Univers. Politecn. Torino 42 (1984) 65), and (ii) the case of 2×2 minors in one-sided ladders by Kulkarni (Semigroup of ordinary multiple point, analysis of straightening formula and counting monomials, Ph.D. Thesis, Purdue University, West Lafayette, USA, 1985). More recently, Krattenthaler and Prohaska (Trans. Amer. Math. Soc. 351 (1999) 1015) have proved a ‘remarkable formula’, conjectured by Conca and Herzog (Adv. Math. 132 (1997) 120) 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’.