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**

, CMS Treatise in Mathematics, CRC Press,*Algebraic Combinatorics and Coinvariant Spaces***2009**. 221 pages. (see the CRC Press website) (see Table of contents and Introduction).,*Multivariate Diagonal Coinvariant Spaces for Complex Reflection Groups*239 (**Advances in Mathematics**,**2013**), 97–108. (arXiv:1105.4358) MathReview.- (with
**L.F. Préville-Ratelle**),*Higher Trivariate Diagonal Harmonics via generalized Tamari Posets*3 (**Journal of Combinatorics**,**2012**), no. 3, 317–341. (arXiv:1105.3738) MathReview. - Jim Haglund’ book:
**The**, with an Appendix on the Combinatorics of Macdonald Polynomials, University Lecture Series of the AMS,*q, t*-Catalan Numbers and the Space of Diagonal Harmonics**2008**. - (with
**S. Hamel**)**Intersection of Modules Related to Macdonald’s Polynomials**, Discrete Mathematics 217 (**2000**), 51-64. - (with
**A. M. Garsia**and**G. Tesler**)**Multiple Left Regular Representations Generated by Alternants**,, vol. 91 (**Journal of Combinatorial Theory Series A****2000**), 49–83. - (with
**N. Bergeron**,**A. M. Garsia**,**M. Haiman**and**G. Tesler**)**Lattice Diagram Polynomials and Extended Pieri Rules**,, vol. 142 Issue 2, (**Advances in Mathematics****1999**), 244–334. (arXiv:math/9809126) - (with
**A. M. Garsia**,**M. Haiman**and**G. Tesler**)**Identities and Positivity Conjectures for Some Remarkable Operators in the Theory of Symmetric Functions**,Volume 6 Number 3 (**Methods and Applications of Analysis,****1999**), 363–420. - (with
**A. M. Garsia**)**Science Fiction and Macdonald’s Polynomials**, in Algebraic Methods and*q*-Special Functions, J.P. Van Dejen, L.Vinet (eds.),, AMS,**CRM Proceedings & Lecture Notes****1999**, 1–52. (arXiv:math/9809128) - J.C. Aval Thesis from Université de Bordeaux,
**Conjecture n! et généralisations**,**2001**.