A New Recursion in the Theory of Macdonald Polynomials

The bigraded Frobenius characteristic of the Garsia-Haiman module M _μ is known 7, 10] to be given by the modified Macdonald polynomial (H)\tilde]_mX; q, t]{\tilde{H}_{\mu}X; q, t]}. It follows from this that, for m\vdash n{\mu \vdash n} the symmetric polynomial ?_p1 (H)\tilde]_mX; q, t]{{\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 ?_p1 (H)\tilde]_mX; q, t] = ?_{v ? m}c_mv (H)\tilde]_vX; q, t]{{\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_m (q, t) = ?_{v ? m}c_mv F_v (q, t){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 Fm(q, t) ? Nq, t]{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].