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Rational solutions to the Pfaff lattice and Jack polynomials

Published online by Cambridge University Press:  02 October 2002

M. ADLER ,
V. B. KUZNETSOV  and
P. VAN MOERBEKE
M. ADLER
Affiliation:
Department of Mathematics, Brandeis University, Waltham, MA 02454, USA (e-mail: [email protected])
V. B. KUZNETSOV
Affiliation:
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK (e-mail: [email protected])
P. VAN MOERBEKE
Affiliation:
Department of Mathematics, Université de Louvain, 1348 Louvain-la-Neuve, Belgium and Brandeis University, Waltham, MA 02454, USA (e-mail: [email protected] and [email protected])
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Abstract

The finite Pfaff lattice is given by a commuting Lax pair 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 t_1,t_2,\dotsc, after conjugation by a diagonal matrix. The sequence of polynomial \tau-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 \tau-functions is given inductively by the action of a fixed vertex operator.

As an example, one such sequence is given by Jack polynomials for rectangular Young diagrams, while another chain starts with any two-column Jack polynomial.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 22 , Issue 5 , October 2002 , pp. 1365 - 1405
Copyright
© 2002 Cambridge University Press

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