Multiple left regular representations generated by alternants

Let p(1) > ... p(n) greater than or equal to 0, and Delta(p) = det \\x(i)(p j)\\(n)(i, j=1). Let M-p be the linear span of the partial derivatives of D elta(p). Then M-p is a graded S-n-modlule. We prove that it is the direct s um of graded left regular representations of S-n. Specifically, set lambda( j) = p(j) - (n - j), and let Xi(lambda)(t) be the Hilbert polynomial of the span of all skew Schur functions s(lambda/mu) as mu varies in lambda. Then the graded Frobenius characteristic of M-p is Xi(lambda)(t) (H) over tilde (1n)(x; q, t), a multiple of a Macdonald polynomial. Corresponding results are also given for the span of partial derivatives of an alternant over any complex reflection group. Let (i, j) denote the lattice cell in the i+1st row and j+1st column of the positive quadrant of the plane. If L is a diagr am with lattice cells (p(1), q(1)), ..., (p(n), q(n)), we set Delta(L) = de t \\x(i)(pj)y(i)(qj)\\(n)(i, j=1), and let M-L be the linear span of the pa rtial derivatives of Delta(L). The bihomogeneity of Delta(L) and its altern ating nature under the diagonal action of S-n gives M-L the structure of a bigraded S-n-module. We give a family of examples and some general conjectu res about the bivariate Frobenius characteristic of M-L for two dimensional diagrams. (C) 2000 Academic Press.