Garsia-Haiman Modules

The following construction generalizes the (well known) construction of the space of (strongly) harmonic polynomials for the symmetric group. This space of  (strong) harmonic polynomial is particularly nice for any reflection group. All such polynomials may be obtained as derivatives of the jacobian determinant of the group considered (see this book). In the symmetric group case, this coincides with the Vandermonde determinant.

To each partition μ of n, one associates the matrix

(xiayib)1ij, (a,b)μ.

Here, one considers μ as a Ferrers diagram (a subset of N×N in french notation), and (i,j)μ means that the cell (i,j) lies in μ. For example, the set of cells of the partition 32 is

{(0,0),(1,0),(2,0),(0,1),(1,1)}.

The Sn-module Mμ is the smallest vector space over Q that contains the determinant Δμ of the above matrix, and which is closed under partial derivatives with respect to any of the variables xi and yj, for 1i,jn. Its dimension is n!. The symmetric group Sn acts by permutation of indices on both sets of variables, and this is called the diagonal action. The space Mμ is a submodule of the space of diagonal harmonic polynomials of Sn, whose dimension is (n+1)n1.

The bigraded Frobenius of the Garsia-Haiman module Mμ is equal to the combinatorial Macdonald polynomial Hμ. This means that the multiplicity of an irreducible representation (of Sn) corresponding to a partition ν, in the homogeneous component of bidegree (i,j) of Mμ, is equal to the coefficient of qitjSν in Hμ. For instance, with μ=21, this module is the span of

{Δ21, x1Δ21, x2Δ21, y1Δ21, y2Δ21, 1}

where Δ21=x1y2x1y3x2y1+x2y3+x3y1x3y2.

It is interesting to note that the x-free component (obtained by setting all variables yi equal to zero) of the above modules coincides with the Garsia-Procesi module Rμ. The y-free component is the module Rμ.

For more about these modules, and their generalizations, see