QUANTUM COHOMOLOGY OF Hilbn(C) AND THE WEIGHTED HOOK WALK ON YOUNG DIAGRAMS

Following the work of Okounkov-Pandharipande [OP1, OP2], Diaconescu [D], and the recent work [CDKM] studying the equivariant quantum cohomology QH∗ (C∗)2(Hilbn) of the Hilbert scheme and the relative Donaldson-Thomas theory of P ×C, we establish a connection between the J-function of the Hilbert scheme and a certain combinatorial identity in two variables. This identity is then generalized to a multivariate identity, which simultaneously generalizes the branching rule for the dimension of irreducible representations of the symmetric group in the staircase shape. We then establish this identity by a weighted generalization of the Greene-Nijenhuis-Wilf hook walk, which is of independent interest.