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 $\mu$ of $n,$ one associates the matrix

${\Big(x_i^a\,y_i^b\Big)_{1\leq i\leq j,\ (a,b)\in\mu}}.$

Here, one considers $\mu$ as a Ferrers diagram (a subset of $\mathbb{N}\times \mathbb{N}$ in french notation), and $(i,j)\in\mu$ means that the cell $(i,j)$ lies in $\mu$. For example, the set of cells of the partition $32$ is

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

The $\mathbb{S}_n$-module $M_\mu$ is the smallest vector space over $\mathbb{Q}$ that contains the determinant $\Delta_\mu$ of the above matrix, and which is closed under partial derivatives with respect to any of the variables $x_i$ and $y_j,$ for $1\leq i,j\leq n$. Its dimension is $n!$. The symmetric group $\mathbb{S}_n$ acts by permutation of indices on both sets of variables, and this is called the diagonal action. The space $M_\mu$ is a submodule of the space of diagonal harmonic polynomials of $\mathbb{S}_n,$ whose dimension is $(n+1)^{n-1}$.

The bigraded Frobenius of the Garsia-Haiman module $M_\mu$ is equal to the combinatorial Macdonald polynomial $H_\mu$. This means that the multiplicity of an irreducible representation (of $\mathbb{S}_n$) corresponding to a partition $\nu,$ in the homogeneous component of bidegree $(i,j)$ of $M_\mu,$ is equal to the coefficient of $q^it^j\,S_\nu$ in $H_\mu$. For instance, with $\mu=21,$ this module is the span of

$\{\Delta_{21},\ \partial_{x_1}\Delta_{21},\ \partial_{x_2}\Delta_{21},\ \partial_{y_1}\Delta_{21},\ \partial_{y_2}\Delta_{21},\ 1\}$

where $\Delta_{21}=x_{{1}}y_{{2}}-x_{{1}}y_{{3}}-x_{{2}}y_{{1}}+x_{{2}}y_{{3}}+x_{{3}}y_{{1}}-x_{{3}}y_{{2}}$.

It is interesting to note that the $x$-free component (obtained by setting all variables $y_i$ equal to zero) of the above modules coincides with the Garsia-Procesi module $R_\mu$. The $y$-free component is the module $R_{\mu’}$.

For more about these modules, and their generalizations, see