Symmetries in the fourth Painlevé equation and Okamoto polynomials

Abstract

The fourth Painlev\'e equation $P_{IV}$ is known to have symmetry of the affine Weyl group of type $A^{(1)}_2$ with respect to the Bäcklund transformations. We introduce a new representation of $P_{IV}$, called the symmetric form , by taking the three fundamental invariant divisors as the dependent variables. A complete description of the symmetry of $P_{IV}$ is given in terms of this representation. Through the symmetric form, it turns out that $P_{IV}$ is obtained as a similarity reduction of the 3-reduced modified KP hierarchy. It is proved in particular that the special polynomials for rational solutions $P_{IV}$, called Okamoto polynomials , are expressible in terms of the 3-reduced Schur functions.