• go down to a brief Outline of these subjects

 
• go down to Publications from various sources concerning these subjects

 
• go down to some Tables of values for various objects

For more information on the  Garsia-Haiman conjecture (also known as the  n!  conjecture) stating that the bigraded Frobenius characteristic of the modules mentioned above is closely related to  Macdonald polynomials.

The study of similar questions for diagonal harmonics is linked to the study of the nabla operator, which can be used to define a (q,t)-analog of the Catalan polynomials.

Also available, tables of small values of the bigraded Frobenius characteristic of the modules mentioned above, as well as some of Macdonald q,t-Kostka coefficients appearing as coefficients of Macdonald polynomials when expanded in the Schur polynomial basis.
 

• J. Alfano, A basis for the Y2 subspace of Diagonal Harmonics, PHD. Thesis UCSD 1994.
• DVI (61K)
• PostScript (165K)

 
• E. Allen, A conjecture of Procesi and the Straightening Algorithm of Rota, Proc. Nat. Accad. V89 1992 3980-3984

 
• E. Allen, A conjecture of Procesi and a new Basis for the Standard Graded Left Regular Representation of S_n, Advances in Math. V100 #2 1993 262-292.

 
• E. Allen, The Descent Monomials and a Basis for the Diagonally Symmetric Polynomials, Journal of Algebraic Combinatorics, V3 1994 5-16.

 
• E. Allen, The Decomposition of a Bigraded Left Regular Representation of the Diagonal Action of S_n, Journal of Comb. Theory Series A, Series A Vol. 71, No. 1, (1995) 97-111

 
• E. Allen, New Bases for the Decomposition of the Graded Left Regular Representation of the Reflection Groups of types B_n and D_n, Journal of Algebra, 173, (1995) 122-143.

 
• E. Allen, Bitableau Bases for Diagonally Symmetric Polynomial Quotient Rings, (in preparation)

 
• F. Bergeron, Algèbre et combinatoire, Lecture notes in french, 123 pages.
• .ps.zip (412 K)
• .pdf

 
• F. Bergeron and S. Hamel, Intersection of Modules Related to Macdonald's Polynomials, Discrete Math., accepté (1999).
• Abstract
• .ps.gz (68 K)

 
• F. Bergeron and A. Garsia, Science Fiction and Macdonald Polynomials, Algebraic methods and q-special functions. R. Floreanini, L. Vinet (eds.), CRM Proceedings & Lecture Notes, American Mathematical Society,, Volume 22, (1999), 1-52.
• Abstract
• .ps.gz

 
• F. Bergeron, A. M. Garsia, M. Haiman and G. Tesler,  Identities and positivity conjectures for some remarkable operators in the theory of symmetric function, Methods of Analysis and Applications, Vol. 6, Accepted, 60 pages.
• Abstract
• .ps.gz (272 K)

 
• F. Bergeron, N. Bergeron, A. Garsia, M. Haiman and G. Tesler, Lattice Diagram Polynomials and Extended Pieri Rules, , Advances in Mathematics, Volume 142, (1999), 244-234.
• Abstract
• PostScript

 
• F. Bergeron, A. M. Garsia, and G. Tesler,  Multiple left regular representations associated with alternants of the symmetric Groups,
• Abstract included in Summaries

 
• N. Bergeron and A. M. Garsia, On Certain Spaces of Harmonic Polynomials, Hypergeometric Functions on domains of Positivity, Jack polynomials and Applications, Contemporary Mathematics, #138 (1992) 51-86.

 
• A. M. Garsia, Orthogonality of Milne's polynomials and raising operators, Discrete Mathematics #99 (1992) 247-264.

 
• A. M. Garsia, Recent Progress on the Macdonald q,t-Kostka conjecture, Actes du 4e Colloque sur les Series Formelles et Combinatoire Algébrique, U.Q.A.M., Pub. L.A.C.I.M. Montreal, (1992) 249-255.

 
• A. M. Garsia and M. Haiman, A graded representation module for Macdonald's polynomials, Proc. Natl. Acad. Sci. USA V 90 (1993) 3607-3610.

 
• A. Garsia and M. Haiman, Orbit Harmonics and Graded Representations, Research Monograph to appear as part of the Collection Published by the Lacim, edited by S. Brlek, U. du Québec a Montréal.
• PostScript: Chapters 1 & 2 (470K), Chapter 3 (2.1 megs), Chapter 4 (1 meg)

 
• A. M. Garsia and M. Haiman, Factorizations of Pieri rules for Macdonald polynomials, Discrete Mathematics 139 (1995) 219-256.

 
• A. Garsia and M. Haiman, A Remarkable q,t-Catalan Sequence and q-Lagrange inversion, J. of Alg. Comb. V. 5 (1996) pp. 191-244.
• PostScript (1.7 meg)

 
• A. Garsia and M. Haiman, Some bigraded Sn-modules and the Macdonald q,t-Kostka coefficients, Electronic Journal of Alg. Comb. V. 3 #2 (1996) pp. 561-620.
• All versions

 
• A. Garsia and M. Haiman, A random q,t-hook walk and a Sum of Pieri Coefficients, J. Combin. Theory Ser. A #82 (1998), no. 1, 74--111.
• PostScript (2.1 megs)

 
• A. Garsia, M. Haiman and G. Tesler, Explicit Plethystic Formulas for the Macdonald q,t-Kostka Coefficients, Séminaire Lotharingien de Combinatoire, B42m (1999), 45 pp.
• Abstract and versions

 
• A. M. Garsia and C. Procesi, On certain graded S_n-modules and the q-Kostka polynomials, Advances in Mathematics, #94 (1992) 82-138.

 
• A. M. Garsia and J. Remmel, Plethystic Formulas and positivity for q,t-Kostka Coefficients, Mathematical Essays in Honor of Gian-Carlo Rota, B. Sagan & R. Stanley. Ed., Progress in Mathematics V 161, Birkhauser (1998) pp. 245-262

 
• A. M. Garsia and G. Tesler, Plethystic Formulas for the Macdonald q,t-Kostka coefficients, Advances in Mathematics, V. 123 #2 (1996) pp. 144-222.
• PostScript (630 K)

 
• M. Haiman, Conjectures on the quotient ring by diagonal invariants, Journal of Algebraic Combinatorics #3 (1994) 17-76.
• PostScript (705K)

 
• A. Lascoux, Polynômes harmoniques,
• Postscript
• Pdf

 
• A. Lascoux, L. Lapointe, and J. Morse, Determinantal Expressions of Macdonald polynomials, submitted.
• Postscript

 
• L. Lapointe and J. Morse, Tableaux statistics for two part Macdonald polynomials, submitted.
• Postscript

 
• I.G. Macdonald, A new class of symmetric functions, Séminaire Lotharingien de Combinatoire, B20a (1988), 41 pp.
• Abstract and versions

 
• J. Morse, Some classical expansions for Sahi-Knop and Macdonald polynomials, Séminaire Lotharingien de Combinatoire, B41a (1998), 29 pp.
• Abstract and versions
• Abstract included in  Summaries

 
• J. Morse, Sahi-Knop polynomials and basic hypergeometric series, Proceedings of Formal Power Series and Algebraic Combinatorics, #3, 437-446 (1997).

 
• J. Morse, Recursions and Explicit formulas for particular n-variable Sahi-Knop and Macdonald polynomials, Journal of Combinatorial Theory Series A, 16 pages, November (1997).
• Abstract included in  Summaries

 
• J. Morse, Sahi-Knop and Macdonald polynomials Related to q-Ultraspherical functions and basic hypergeometric series, Discrete Mathematics, 10 pages, September (1997).
• Abstract included in  Summaries

 
• E. Reiner, Some Applications of the Theory of Orbit Harmonics, Thesis UCSD 1993

 
• E. Reiner, A Proof of the n! Conjecture for Generalized Hooks, Journal of Comb. Theory Series A (accepted).

 
• R. Stanley, Positivity problems and conjectures in algebraic combinatorics, To appear in Mathematics: Frontiers and Perspectives, published by the International Mathematics Union to celebrate the turn of the century.
• Postscript.gz (116 K)

 
• G. Tesler, Isotypic Decompositions of Lattice Determinants, Journal of Combinatorial Theory, Series A, 85 (1999), 208-227.

 
• M. Zabrocki, On the action of the Hall-Littlewood vertex operator, Doctoral Thesis, UCSD (1998)
• Abstract included in Summaries

 
• M. Zabrocki, Vertex operators for standard bases of the symmetric function,
• Postscript

 
• M. Zabrocki, Positivity for special cases of  (q,t)-Kostka coefficients and standard tableaux statistics,
• Postscript

• (q,t)-Catalan polynomials    (as matrices in .ps.gz 12 K)  (.pdf)

 
• (q,t)-Kostka polynomials   (as matrices in .ps.gz)  (.pdf)

 
• Frobenius of  M_m  for partitions of n < 6 (html)      (Other more extensive tables)

 
• Frobenius of Diagonal Harmonics Module (html)

 
• Matrices of operator Nabla (maple in .gz)