A New Recursion in the Theory of Macdonald Polynomials

作者：Garsia A M^*; Haglund J

来源：Annals of Combinatorics, 2012, 16(1): 77-106.

DOI：10.1007/s00026-011-0122-5

• Hilbert series
• Frobenius series
• Garsia-Haiman modules
• science fiction
• Macdonald polynomials

The bigraded Frobenius characteristic of the Garsia-Haiman module M-mu is known [7, 10] to be given by the modified Macdonald polynomial (H) over tilde(mu) [X; q,t]. It follows from this that, for mu (sic) n the symmetric polynomial. partial derivative(p1) (H)over tilde(mu) [X; q, t] is the bigraded Frobenius characteristic of the restriction of M-mu from S-n to Sn-1. The theory of Macdonald polynomials gives explicit formulas for the coefficients c(mu nu). occurring in the expansion. partial derivative(p1) (H)over tilde(mu) [X; q, t] = Sigma(v ->mu)c(mu nu)(H) over tilde(nu) [X; q, t]. In particular, it follows from this formula that the bigraded Hilbert series F-mu (q, t) of M mu may be calculated from the recursion F-mu (q, t) = Sigma(v ->mu)c(mu nu) (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). N[q, t]. This difficulty arises from the fact that the c(mu nu). have rather intricate expressions as rational functions in q, t. We give here a new recursion, from which a new combinatorial formula for F-mu (q, t) can be derived when mu 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].

• 出版日期2012-3