# The Pfaff lattice and skew-orthogonal polynomials

@article{Adler1999ThePL, title={The Pfaff lattice and skew-orthogonal polynomials}, author={Mark Adler and Emil Horozov and Pierre van Moerbeke}, journal={International Mathematics Research Notices}, year={1999}, volume={1999}, pages={569-588} }

Consider a semi-inflnite skew-symmetric moment matrix, m1 evolving according to the vector flelds @m=@tk = ⁄ k m + m⁄ >k ; where ⁄ is the shift matrix. Then The skew-Borel decomposition m1 := Q i1 JQ >i1 leads to the so-called Pfafi Lattice, which is integrable, by virtue of the AKS theorem, for a splitting involving the a‐ne symplectic algebra. The tau-functions for the system are shown to be pfa‐ans and the wave vectors skew-orthogonal polynomials; we give their explicit form in terms of…

## 64 Citations

### Partial-Skew-Orthogonal Polynomials and Related Integrable Lattices with Pfaffian Tau-Functions

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Skew-orthogonal polynomials (SOPs) arise in the study of the n-point distribution function for orthogonal and symplectic random matrix ensembles. Motivated by the average of characteristic…

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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…

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Pfaff lattice was introduced by Adler and van Moerbeke to describe the partition functions for the random matrix models of GOE and GSE type. The partition functions of those matrix models are given…

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Non-Hermitian random matrices with symplectic symmetry provide examples for Pfaffian point processes in the complex plane. These point processes are characterised by a matrix valued kernel of…

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We study the Pfaff lattice, introduced by us in the context of a Lie algebra splitting of gl(infinity) into sp(infinity) and lower-triangular matrices. We establish a set of bilinear identities,…

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The Pfaff lattice is an integrable system arising from the SR-group factorization in an analogous way to how the Toda lattice arises from the QR-group factorization. In our earlier paper (Kodama and…

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Skew orthogonal polynomials arise in the calculation of the n-point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry. In particular,…

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The eigenvalue probability density function for symplectic invariant random matrix ensembles can be generalized to discrete settings involving either a linear or an exponential lattice. The…

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