GROMOV-WITTEN INVARIANTS OF SymdPr

We give a graph-sum algorithm that expresses any genus-g Gromov-Witten invariant of the symmetric product orbifold Symd Pr := [(Pr)d/Sd]in terms of "Hurwitz-Hodge integrals" -integrals over (compacti-fied) Hurwitz spaces. We apply the algorithm to prove a mirror-type theorem for Symd Pr in genus zero. The theorem states that a generating function of Gromov-Witten invariants of Symd Pr is equal to an explicit power series ISymd Pr , conditional upon a conjectural combinatorial identity. This is a first step in the direction of proving Ruan's Crepant Resolution Conjecture for the resolution Hilb(d)(P2) of the coarse moduli space of Symd P2.