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
Here, one considers
The
The bigraded Frobenius of the Garsia-Haiman module
where
It is interesting to note that the
For more about these modules, and their generalizations, see
- Algebraic Combinatorics and Coinvariant Spaces, CMS Treatise in Mathematics, CRC Press, 2009. 221 pages. (see the CRC Press website) (see Table of contents and Introduction).
- Multivariate Diagonal Coinvariant Spaces for Complex Reflection Groups, Advances in Mathematics, 239 (2013), 97–108. (arXiv:1105.4358) MathReview.
- (with L.F. Préville-Ratelle) Higher Trivariate Diagonal Harmonics via generalized Tamari Posets, Journal of Combinatorics, 3 (2012), no. 3, 317–341. (arXiv:1105.3738) MathReview.
- Jim Haglund’ book: The q, t-Catalan Numbers and the Space of Diagonal Harmonics, with an Appendix on the Combinatorics of Macdonald Polynomials, University Lecture Series of the AMS, 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, Journal of Combinatorial Theory Series A, vol. 91 (2000), 49–83.
- (with N. Bergeron, A. M. Garsia, M. Haiman and G. Tesler) Lattice Diagram Polynomials and Extended Pieri Rules, Advances in Mathematics, vol. 142 Issue 2, (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, Methods and Applications of Analysis, Volume 6 Number 3 (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.), CRM Proceedings & Lecture Notes, AMS, 1999, 1–52. (arXiv:math/9809128)
- J.C. Aval Thesis from Université de Bordeaux, Conjecture n! et généralisations, 2001.