From Toda Hierarchy to KP Hierarchy

Using the matrix-resolvent method and a formula of the second-named author on the $n$-point function for a KP tau-function, we show that the tau-function of an arbitrary solution to the Toda lattice hierarchy is a KP tau-function. We then generalize this result to tau-functions for the extended Toda hierarchy (ETH) by developing the matrix-resolvent method for the ETH. As an example the partition function of Gromov–Witten invariants of the complex projective line is a KP tau-function, and an application on irreducible representations of the symmetric group is obtained.