Universal characters, integrable chains and the Painlevé equations

The universal character is a generalization of the Schur polynomial attached to a pair of partitions; see [8]. We prove that the universal character solves the Darboux chain. The N -periodic closing of the chain is equivalent to the Painlevé equation of type A N−1. Consequently we obtain an expression of rational solutions of the Painlevé equations in terms of the universal characters. Introduction The universal character S[λ,μ], defined by Koike [8], is a polynomial attached to a pair of partitions [λ, μ] and is a generalization of the Schur polynomial. The universal character is in fact the irreducible character of a rational representation of the general linear group GL(n,C), while the Schur polynomial that of a polynomial representation. The Darboux chain, given by (1.1) below, is a sequence of ordinary differential equations with quadratic nonlinearity. This is closely interconnected with the spectral theory of Schrödinger operators; in fact, governs a sequence of Schrödinger operators connected with the neighbours by the Darboux transformations; see [16, 17, 18, 24, 25]. The Painlevé equations can be derived from the Darboux chains with suitable boundary conditions; as is well known that the chains, (1.1), with periods of order three and four yield Painlevé equations PIV and PV respectively. In general, for an integer N ≥ 3, the N -periodic closing of the chain coincides with the (higher order) Painlevé equation of type ∗Current address: Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan. E-mail address: [email protected], [email protected]