Geometry of the Pfaff Lattices

Kodama, Yuji; Pierce, Virgil U.

doi:10.1093/imrn/rnm120

Cited by 11 publications

(36 citation statements)

References 18 publications

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“…By contrast, any t k flow in (B.10) is quite an intricate system, because the matrix L is a lower Hessenberg matrix and does not own concise structures. For more related work, please refer [3,7,34,35] etc..…”

Section: (B7b)mentioning

confidence: 99%

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

Chang

et al. 2018

Commun. Math. Phys.

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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 polynomials of the Bures random matrix ensemble studied in [22], we propose the concept of partial-skew-orthogonal polynomials (PSOPs) as a modification of the SOPs, and then the PSOPs with a variety of special skew-symmetric kernels and weight functions are addressed. By considering appropriate deformations of the weight functions, we derive nine integrable lattices in different dimensions. As a consequence, the tau-functions for these systems are shown to be expressed in terms of Pfaffians and the wave vectors PSOPs. In fact, the taufunctions also admit the representations of multiple integrals. Among these integrable lattices, some of them are known, while the others are novel to the best of our knowledge. In particular, one integrable lattice is related to the partition function of the Bures random matrix ensemble.Besides, we derive a discrete integrable lattice, which can be used to compute certain vector Padé approximants. This yields the first example regarding the connection between integrable lattices and vector Padé approximants, for which Hietarinta, Joshi and Nijhoff pointed out that " This field remains largely to be explored. " in the recent monograph [27, Section 4.4] .

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Section: (B7b)mentioning

confidence: 99%

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

Chang

et al. 2018

Commun. Math. Phys.

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“…We use this J because of its connection to the Pfaff lattice which is defined on lower Hessenberg matrices (see below). The Pfaff lattice can be viewed as an sp-version of the Toda lattice, it has the following form with sp-projection [1,16],…”

Section: (B) Paris Of Imaginary Eigenvalues (Z −Z)mentioning

confidence: 99%

“…As in the case of the Toda lattice hierarchy, one can also define the Pfaff lattice hierarchy (see [16], and also Section 3),…”

Section: (B) Paris Of Imaginary Eigenvalues (Z −Z)mentioning

confidence: 99%

“…The finite Pfaff lattice hierarchy is a set of Hamiltonian flows on sl(2n, R) with the Lie-Poisson structure induced by the splitting sl(2n, R) = k ⊕ sp(n), and it is defined as follows (see also [16]): For any X, Y ∈ sl(2n) we first define the R-bracket…”

Section: Pfaff Lattice Hierarchymentioning

confidence: 99%

“…This inner product may be obtained by the following inner product in the limit z 2k−1 → z 2k (see [16]), i.e.…”

Section: Remark 43mentioning

confidence: 99%

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The Pfaff lattice on symplectic matrices

Kodama

Pierce²

2010

J. Phys. A: Math. Theor.

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Abstract. 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 recent paper [Intern. Math. Res. Notices, (2007) rnm120], we studied the Pfaff lattice hierarchy for the case where the Lax matrix is defined to be a lower Hessenberg matrix. In this paper we deal with the case of a symplectic lower Hessenberg Lax matrix, this forces the Lax matrix to take a tridiagonal shape. We then show that the odd members of the Pfaff lattice hierarchy are trivial, while the even members are equivalent to the indefinite Toda lattice hierarchy defined in [Y. Kodama and J. Ye, Physica D, 91 (1996) 321-339]. This is analogous to the case of the Toda lattice hierarchy in the relation to the Kac-van Moerbeke system. In the case with initial matrix having only real or imaginary eigenvalues, the fixed points of the even flows are given by 2 × 2 block diagonal matrices with zero diagonals. We also consider a family of skew-orthogonal polynomials with symplectic recursion relation related to the Pfaff lattice, and find that they are succinctly expressed in terms of orthogonal polynomials appearing in the indefinite Toda lattice.

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Cyclic Olefin Copolymers

Fink¹

2010

Handbook of Engineering and Specialty Thermoplastics

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Geometry of the Pfaff Lattices

Cited by 11 publications

References 18 publications

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

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

The Pfaff lattice on symplectic matrices

Cyclic Olefin Copolymers

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