A New Recursion in the Theory of Macdonald Polynomials

摘要

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 _(μv) occurring in the expansion ? _(p1)H? _μ[X; q, t]. In particular, it follows from this formula that the bigraded Hilbert series Fμ(q, t) of M _μ may be calculated from the recursion F _μ(q, t)=Σ _(v→μ)C _μF _v(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 from the fact that the c _(μv) 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].