On certain spaces of lattice diagram determinants

The aim of this work is to study some lattice diagram determinants ∆L(X, Y ) as defined in [5] and to extend results of [3]. We recall that ML denotes the space of all partial derivatives of ∆L. In this paper, we want to study the space M i,j(X, Y ) which is defined as the sum of ML spaces where the lattice diagrams L are obtained by removing k cells from a given partition, these cells being in the “shadow” of a given cell (i, j) in a fixed Ferrers diagram. We obtain an upper bound for the dimension of the resulting space M i,j(X, Y ), that we conjecture to be optimal. This dimension is a multiple of n! and thus we obtain a generalization of the n! conjecture. Moreover, these upper bounds associated to nice properties of some special symmetric differential operators (the “shift” operators) allow us to construct explicit bases in the case of one set of variables, i.e. for the subspace M i,j(X) consisting of elements of 0 Y -degree.