Lattice Diagram Polynomials and Extended Pieri Rules

Abstract

The lattice cell in thei+1 st row andj+1 st column of the positive quadrant of the plane is denoted (i,j). Ifμis a partition ofn+1, we denote byμ/ijthe diagram obtained by removing the cell (i,j) from the (French) Ferrers diagram ofμ. We setΔμ/ij=det‖xpjiyqji‖ni,j=1, where (p1,q1),…,(pn,qn) are the cells ofμ/ij, and letMμ/ijbe the linear span of the partial derivatives ofΔμ/ij. The bihomogeneity ofΔμ/ijand its alternating nature under the diagonal action ofSngivesMμ/ijthe structure of a bigradedSn-module. We conjecture thatMμ/ijis always a direct sum ofkleft regular representations ofSn, wherekis the number of cells that are weakly north and east of (i,j) inμ. We also make a number of conjectures describing the precise nature of the bivariate Frobenius characteristic ofMμ/ijin terms of the theory of Macdonald polynomials. On the validity of these conjectures, we derive a number of surprising identities. In particular, we obtain a representation theoretical interpretation of the coefficients appearing in some Macdonald Pieri Rules.