A New Recursion in the Theory of Macdonald

The bigraded Frobenius characteristic of the Garsia-Haiman module M μ is known [7] [10] to be given by the modified Macdonald polynomial˜H μ [X; q, t]. It follows from this that, for μ n the symmetric polynomial ∂ p1˜H μ [X; q, t] is the bigraded Frobenius characteristic of the restriction of M μ from S n to S n−1. The theory of Macdonald polynomials gives explicit formulas for the coefficients c μν occurring in the expansion ∂ p1˜H μ [X; q, t] = ν→μ c μν˜H ν [X; q, t]. In particular it follows from this formula that the bigraded Hilbert series F μ (q, t) of M μ may be calculated from the recursion F μ (q, t) = ν→μ c μν F ν (q, t). One of the frustrating problems of the theory of Macdonald polynomials has been to derive from this recursion that F μ (q, t) ∈ N[q, t]. This difficulty arises form the fact that the c μν have rather intricate expressions as rational functions in q, t. We give here a new recursion, from which a new combinatorial formula for F μ (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 M. Yoo in [15].