Schur Partial Derivative Operators

A lattice diagram is a finite list L=((p_1,q_1),…,(p_n,q_n) of lattice cells. The corresponding lattice diagram determinant is \Delta_L(X;Y)=\det | x_i^{p_j}y_i^{q_j} |. These lattice diagram determinants are crucial in the study of the so-called n! conjecture'' of A. Garsia and M. Haiman. The space M_L is the space spanned by all partial derivatives of \Delta_L(X;Y). The shift operators’', which are particular partial symmetric derivative operators are very useful in the comprehension of the structure of the M_L spaces. We describe here how a Schur function partial derivative operator acts on lattice diagrams with distinct cells in the positive quadrant.