The n! conjecture of Garsia and Haiman states that certain explicitly defined vector spaces, now known as Garsia-Haiman modules, have dimension n!. It was originally motivated by the Macdonald positivity conjecture that Macdonald polynomials have can be expanded in terms of Schur functions using polynomials with positive integer coefficients. While both conjectures have been proven, the known proofs are indirect, and it remains an open problem to give an explicit basis for the Garsia-Haiman modules. This reading seminar will study papers on the combinatorics of the Macdonald polynomials and on approaches to finding a basis for the Garsia-Haiman modules, as well as papers on various related conjectures.

Students are invited to read and present on one or part of one of the papers listed in the bibliography. Please talk to me if you are interested. Those of you who have volunteered to present a paper (Thanks!) are welcome to ask me for help, if they want to practice, et c.

We will meet on Thursdays from 10-12 in 3106 MSB. Although we will be scheduled for two full hours, we will rarely use all that time.

• Thursday September 27, 10AM. Organizational Meeting. I gave a short overview lecture, including all statements of the representation theory and symmetric function theory we will assume (and then rarely use).
• October 4. I will give a more detailed presentation of the representation theory and symmetric function theory. Tentatively, Nicolas Thiery will give a software demonstration at 11:15.
• October 11. Chris Berg will speak on the Haglund-Haiman-Loehr paper.
• October 18. Eddie Kim will speak on the Garsia-Haiman paper.
• October 25. Jason Bandlow will speak on the Bergeron-Garsia paper.
• November 1. Sonya Berg will speak on the Haglund-Haiman-Loehr-Remmel-Ulyanov paper. (to be confirmed)
• November 8. Steven Pon will speak on the Assaf preprint.
• November 15. Qiang Wang will speak on the Haiman paper on the diagonal invariants.
• November 29. Jeff Ferreira will speak on the Allen paper on Procesi's conjecture.

• J. Haglund, M. Haiman, and N. Loehr, A Combinatorial Formula for Macdonald Polynomials, J. Amer. Math. Soc. 18 (2005), 735--761.
• S. Assaf, A combinatorial proof of LLT and Macdonald positivity, preprint available from her web page, www.math.upenn.edu/~sassaf/
• A. Garsia and M. Haiman, Some natural bigraded $S_n$-modules and $q,t$-Kostka coefficients, Electronic J. Combin. 3 (1996) no. 2, Research Paper 24.
• F. Bergeron and A. Garsia, Science Fiction and Macdonald Polynomials, in Algebraic Methods and $q$-special functions, CRM Proceedings and Lecture Notes, American Mathematical Society.
• F. Bergeron, N. Bergeron, A. Garsia, M. Haiman, and G. Tesler, Lattice diagram polynomials and extended Pieri rules, Adv. Math. 142 (1999) 244-334.
• A. Garsia and M. Haiman, A remarkable $q,t$-Catalan sequence and $q$-Lagrange inversion, J. Algebraic Combin. 5 (1996), 191--244.
• J. Haglund, M. Haiman, N. Loehr, J. Remmel, and A. Ulyanov, A combinatorial formula for the character of the diagonal coinvariants, Duke Math. J. 126 (2005, 195--232.
• I. Gordon, On the quotient ring by diagonal invariants, Invent. Math. 153 (2003) 503--518.

• A. Garsia and C. Procesi, On certain graded $S_n$-modules and the $q$-Kostka polynomials, Adv. Math. 94 (1992), 82--138.
• N. Bergeron and A. Garsia, On certain spaces of harmonic polynomials, in Hypergeometric functions on domains of positivity, Jack polynomials, and applications, Contemp. Math. 138 (1992), 51--86.
• E. Allen, A conjecture of Procesi and a new basis for the decomposition of the graded left regular representation of $S_n$, Adv. Math. 100 (1993), 262--292.
• S. Ariki, T. Terasoma, and H. Yamada, Higher Specht polynomials, Hiroshima Math. J. 27 (1997), 177-188.
• F. Bergeron, A. Garsia, and G. Tesler, Multiple left regular representations generated by alternants, J. Combin. Theory Ser. A, 81 (2000), 49--83.
• J.-C. Aval, F. Bergeron, and N. Bergeron, Spaces of lattice diagram polynomials in one set of variables, Adv. in Appl. Math. 28 (2002), 343--359.
• J.-C. Aval and N. Bergeron, Vanishing ideals of lattice diagram determinants, J. Combin. Theory Ser. A 99 (2002), 244-260.

• 7.1-7.18 of Stanley's Enumerative Combinatorics
• Macdonald's book
• Sections 1 and 3 of the lecture notes from Mark Haiman's PCMI lectures.