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… 
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64 Citations

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References

SHOWING 1-10 OF 30 REFERENCES
The Pfaff lattice, Matrix integrals and a map from Toda to Pfaff
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,…
Symmetric random matrices and the Pfaff lattice
Consider a symmetric (finite) matrix ensemble, with a certain probability distribution. What is the probability that the spectrum belongs to a certain interval or union of intervals on the real line?…
Multicritical points of unoriented random surfaces
Infinite dimensional Lie algebras: Frontmatter
Completely Integrable Systems, Euclidean Lie-algebras, and Curves
Matrix integrals, Toda symmetries, Virasoro constraints, and orthogonal polynomials
into the algebras of skew-symmetric As and lower triangular (including the diagonal) matrices Ab (Borel matrices). We show that this splitting plays a prominent role also in the construction of the…
The Spectrum of coupled random matrices
The study of the spectrum of coupled random matrices has received rather little attention. To the best of our knowledge, coupled random matrices have been studied, to some extent, by Mehta. In this…
Group factorization, moment matrices, and Toda lattices
∗appears in: International Mathematics Research notices, 12 (1997), received 4 september 1996. †Department of Mathematics, Brandeis University, Waltham, Mass 02254, USA (adler@math.brandeis.edu). The…
The geometry of spinors and the multicomponent BKP and DKP hierarchies
We develop a formalism of multicomponent BKP hierarchies using elementary geometry of spinors. The multicomponent KP and the modified KP hierarchy (hence all their reductions like KdV, NLS, AKNS or…
Random Matrices
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