Pfaff - Brandeis University

Download PDF
3 downloads 0 Views 318KB Size Report
Comment
(e-mail: [email protected] and @math.brandeis.edu) ... ∗∗The support of Japanese ministry of education's grant-in-aid for scientific ... 1 Here “t, s ∈ C∞” is an informal way of saying that t and s are two sequences of independent ...... As in the last section, we use the notation A := Z≥0 or Z to treat both the semi ...
Mathematische Annalen

Math. Ann. (2001) Digital Object Identifier (DOI) 10.1007/s002080100153

Pfaff τ -functions M. Adler · T. Shiota · P. van Moerbeke Received: 20 September 1999 / Published online: ♣ – © Springer-Verlag 2001 Consider the two-dimensional Toda lattice, with certain skew-symmetric initial condition, which is preserved along the locus s = −t of the space of time variables. Restricting the solution to s = −t, we obtain another hierarchy called Pfaff lattice, which has its own tau function, being equal to the square root of the restriction of 2D-Toda tau function. We study its bilinear and Fay identities, W and Virasoro symmetries, relation to symmetric and symplectic matrix integrals and quasiperiodic solutions.

0. Introduction Consider the set of equations ∂m∞ = Λn m∞ , ∂tn

∂m∞ = −m∞ (Λ )n , ∂sn

n = 1, 2, . . . ,

(0.1)

on infinite matrices m∞ = m∞ (t, s) = (µi,j (t, s))0≤i,j