Universal characters, integrable chains and the Painleve equations

The universal character is a generalization of the Schur polynomial attached to a pair of partitions; see (Adv. Math. 74 (1989) 57). We prove that the universal character solves the Darboux chain. The N-periodic closing of the chain is equivalent to the Painleve equation of type A(N-1)((1)) Consequently we obtain an expression of rational solutions of the Painleve equations in terms of the universal characters. (c) 2004 Elsevier Inc. All rights reserved.