Rational solutions to the Pfaff lattice and Jack polynomials

The finite Pfaff lattice is given by commuting Lax pairs involving a finite matrix L (zero above the first subdiagonal) and a projection onto Sp(N). The lattice admits solutions such that the entries of the matrix L are rational in the time parameters t1, t2, . . . , after conjugation by a diagonal matrix. The sequence of polynomial τ -functions, solving the problem, belongs to an intriguing chain of subspaces of Schur polynomials, associated to Young diagrams, dual with respect to a finite chain of rectangles. Also, this sequence of τ -functions is given inductively by the action of a fixed vertex operator. As examples, one such sequence is given by Jack polynomials for rectangular Young diagrams, while another chain starts with any twocolumn Jack polynomial. Department of Mathematics, Brandeis University, Waltham, Mass 02454, USA. Email: [email protected]. The support of a National Science Foundation grant # DMS01-00782 is gratefully acknowledged. Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT. E-mail: [email protected]. The support of a FSR grant, Universite de Louvain, Belgium, and from the EPSRC, UK, is gratefully acknowledged. This work was done, while visiting the University of Louvain and Brandeis University. Department of Mathematics, Université de Louvain, 1348 Louvain-la-Neuve, Belgium and Brandeis University, Waltham, Mass 02454, USA. E-mail: [email protected] and @brandeis.edu. The support of a National Science Foundation grant # DMS-01-00782, a Nato, a FNRS and a Francqui Foundation grant is gratefully acknowledged.