A New Recursion in the Theory of Macdonald Polynomials

Abstract

The bigraded Frobenius characteristic of the Garsia-Haiman module M _μ is known [7, 10] to be given by the modified Macdonald polynomial ${\tilde{H}_{\mu}[X; q, t]}$. It follows from this that, for ${\mu \vdash n}$ the symmetric polynomial ${{\partial_{p1}} \tilde{H}_{\mu}[X; q, t]}$ 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 _{μ v} occurring in the expansion ${{\partial_{p1}} \tilde{H}_{\mu}[X; q, t] = \sum_{v \to \mu}c_{\mu v} \tilde{H}_{v}[X; q, t]}$. 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 ${F_\mu (q, t) = \sum_{v \to \mu}c_{\mu v} F_v (q, t)}$. One of the frustrating problems of the theory of Macdonald polynomials has been to derive from this recursion that ${F\mu (q, t) \in \mathbf{N}[q, t]}$. This difficulty arises from the fact that the c _{μ v} 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 Yoo in [15].