15 March 2002 Toda versus Pfaff lattice and related polynomials
M. Adler, P. van Moerbeke
Duke Math. J. 112(1): 1-58 (15 March 2002). DOI: 10.1215/S0012-9074-02-11211-3


The Pfaff lattice was introduced by us in the context of a Lie algebra splitting of ${\rm gl}(\infty)$ into ${\rm sp}(\infty)$ and an algebra of lower-triangular matrices. The Pfaff lattice is equivalent to a set of bilinear identities for the wave functions, which yield the existence of a sequence of "$\tau$-functions". The latter satisfy their own set of bilinear identities, which moreover characterize them.

In the semi-infinite case, the $\tau$-functions are Pfaffians, in the same way that for the Toda lattice the $\tau$-functions are Hänkel determinants; interesting examples occur in the theory of random matrices, where one considers symmetric and symplectic matrix integrals for the Pfaff lattice and Hermitian matrix integrals for the Toda lattice.

There is a striking parallel between the Pfaff lattice and the Toda lattice, and even more striking, there is a map from one to the other, mapping skew-orthogonal to orthogonal polynomials. In particular, we exhibit two maps, dual to each other, mapping Hermitian matrix integrals to either symmetric matrix integrals or symplectic matrix integrals.


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M. Adler. P. van Moerbeke. "Toda versus Pfaff lattice and related polynomials." Duke Math. J. 112 (1) 1 - 58, 15 March 2002. https://doi.org/10.1215/S0012-9074-02-11211-3


Published: 15 March 2002
First available in Project Euclid: 18 June 2004

zbMATH: 1067.37111
MathSciNet: MR1890646
Digital Object Identifier: 10.1215/S0012-9074-02-11211-3

Primary: 37K60
Secondary: 33C45 , 37N20 , 82C23

Rights: Copyright © 2002 Duke University Press


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Vol.112 • No. 1 • 15 March 2002
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