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Rational solutions to the Pfaff lattice and Jack polynomials

Mark Adler·Vadim B. Kuznetsov·Pierre van Moerbeke
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Abstract

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 t_1,t_2,..., 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 examples, one such sequence is given by Jack polynomials for rectangular Young diagrams, while another chain starts with any two-column Jack polynomial.

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Mark Adler, Vadim B. Kuznetsov, Pierre van Moerbeke. 2002-02-18. Rational solutions to the Pfaff lattice and Jack polynomials. https://arxiv.org/abs/nlin/0202037

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