Quantum cohomology of Hilbn(C2) and the weighted hook walk on Young diagrams

Following the work of Okounkov and Pandharipande (2010) [OP1,OP2], Diaconescu [D], , and the recent work of I. Ciocan-Fontanine et al. (in preparation) [CDKM] studying the equivariant quantum cohomology of the Hilbert scheme and the relative Donaldson–Thomas theory of P1×C2, 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.