Symmetries in the fourth Painleve equation and Okamoto polynomials

Abstract

We propose a new representation of the fourth Painlev\'e equation in which the $A^{(1)}_2$-symmetries become clearly visible. By means of this representation, we clarify the internal relation between the fourth Painlev\'e equation and the modified KP hierarchy. We obtain in particular a complete description of the rational solutions of the fourth Painlev\'e equation in terms of Schur functions. This implies that the so-called Okamoto polynomials, which arise from the $\tau$-functions for rational solutions, are in fact expressible by the 3-reduced Schur functions.