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A New Recursion in the Theory of Macdonald Polynomials
A.M. Garsia1 and J. Haglund2
11Department of Mathematics, University of California, San Diego (UCSD), 9500 Gilman Drive # 0112, La Jolla, CA 92093-0112, USA
garsia@math.ucsd.edu
2Department of Mathematics, University of Pennsylvania, 209 South 33rd St., David Rittenhouse Laboratory, Philadelphia, PA 19104-6395, USA
jhaglund@math.upenn.edu
Annals of Combinatorics 16 (1) pp.77-106 March, 2012
AMS Subject Classification: 05E10, 05E05
Abstract:
The bigraded Frobenius characteristic of the Garsia-Haiman module Mμ is known [7, 10] to be given by the modified Macdonald polynomial Hμ [X; q, t]. It follows from this that, for μ n the symmetric polynomial ∂p1Hμ [X; q, t] is the bigraded Frobenius characteristic of the restriction of Mμ from Sn to Sn−1. The theory of Macdonald polynomials gives explicit formulas for the coefficients cμν occurring in the expansion ∂p1Hμ [X; q, t]= Σν→μ cμνHv [X; q, t]. In particular, it follows from this formula that the bigraded Hilbert series Fμ (q, t) of Mμ may be calculated from the recursion Fμ (q, t) = Σν→μ cμν F v [X; q, t]. One of the frustrating problems of the theory of Macdonald polynomials has been to derive from this recursion that Fμ (q, t) ∈ N[q, t]. This difficulty arises from the fact that the cμν have rather intricate expressions as rational functions in q, t. We give here a new recursion, from which a new combinatorial formula for Fμ (q, t) can be derived when μ is a two-column partition. The proof suggests a method for deriving an analogous formula in the general case. The method was successfully carried out for the hook case by Yoo in [15].
Keywords: Hilbert series, Frobenius series, Garsia-Haiman modules, science fiction, Macdonald polynomials

References:

1. Bergeron, F., et al.: Lattice diagram polynomials and extended pieri Rules. Adv. Math. 142(2), 244–334 (1999)

2. Bergeron, F., Garsia, A.M.: Science fiction and Macdonald polynomials. In: Van Diejen, J.P., Vinet, L. (eds.) Algebraic Methods and q-Special Functions, pp. 1–52. Amer. Math. Soc., Providence, RI (1999)

3. Bergeron, N., Garsia, A.M.: On certain spaces of harmonic polynomials. In: Richards, D.St.P. (ed.) Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications, Contemp. Math., Vol. 138, pp. 51–86. Amer. Math. Soc., Providence, RI (1992)

4. Garsia, A.M., Haiman, M.: Orbit Harmonics and Graded Representations. Laboratoire de combinatoire et d’informatique mathématique, Université du Québec à Montréal. In preparation.

5. Garsia, A.M., Haiman, M.: A graded representation model for Macdonald’s polynomials. Proc. Natl. Acad. Sci. USA 90(8), 3607–3610 (1993)

6. Garsia, A.M., Haiman, M.: Factorizations of Pieri rules for Macdonald polynomials. Discrete Math. 139(1-3), 219–256 (1995)

7. Garsia, A.M., Haiman, M.: Some natural bigraded Sn-modules and q, t-Kostka coefficients. Electron. J. Combin. 3(2), #R24 (1996)

8. Garsia, A.M., Procesi, C.: On certain graded Sn-modules and the q-Kostka polynomials. Adv. Math. 94(1), 82–138 (1992)

9. Haglund, J., Haiman, M., Loehr, N.: A combinatorial formula for Macdonald polynomials. J. Amer. Math. Soc. 18(3), 735–761 (2005)

10 . Haiman, M.: Hilbert Schemes, polygraphs, and the Macdonald positivity conjecture. J. Amer. Math. Soc. 14(4), 941–1006 (2001)

11. Lapointe, L., Lascoux, A., Morse, J.: Tableau atoms and a new Macdonald positivity conjecture. Duke Math. J. 116(1), 103–146 (2003)

12. Macdonald, I.G.: A new class of symmetric functions. Publ. I.R.M.A. Strasbourg 372/S- 20, 131–171 (1988)

13. Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd Ed. Oxford University Press, New York (1995)

14. Stembridge, J.: Some particular entries of the two-parameter Kostka matrix. Proc. Amer. Math. Soc. 121(2), 367–373 (1994)

15. Yoo, M.: Combinatorial Formulas Connected to Diagonal Harmonics andMcdonald Polynomials. PhD Thesis, University of Pennsylvania, Philadelphia (2009)