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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
2Department of Mathematics, University of Pennsylvania, 209 South 33rd St., David Rittenhouse Laboratory, Philadelphia, PA 19104-6395, USA
Annals of Combinatorics 16 (1) pp.77-106 March, 2012
AMS Subject Classification: 05E10, 05E05
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


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