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.
Sudhir R. Ghorpade