dorsal/arxiv
View SchemaThe Pfaff lattice and skew-orthogonal polynomials
| Authors | M. Adler, E. Horozov, P. van Moerbeke |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9903005 |
| URL | https://arxiv.org/abs/solv-int/9903005 |
| Journal | Intern. Math. Research Notices, 1999 |
Abstract
Consider a semi-infinite skew-symmetric moment matrix, $m_{\iy}$ evolving according to the vector fields $\pl m / \pl t_k=\Lb^k m+m \Lb^{\top k} ,$ where $\Lb$ is the shift matrix. Then the skew-Borel decomposition $ m_{\iy}:= Q^{-1} J Q^{\top -1} $ leads to the so-called Pfaff Lattice, which is integrable, by virtue of the AKS theorem, for a splitting involving the affine symplectic algebra. The tau-functions for the system are shown to be pfaffians and the wave vectors skew-orthogonal polynomials; we give their explicit form in terms of moments. This system plays an important role in symmetric and symplectic matrix models and in the theory of random matrices (beta=1 or 4).
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"abstract": "Consider a semi-infinite skew-symmetric moment matrix, $m_{\\iy}$ evolving\naccording to the vector fields $\\pl m / \\pl t_k=\\Lb^k m+m \\Lb^{\\top k} ,$ where\n$\\Lb$ is the shift matrix. Then the skew-Borel decomposition $ m_{\\iy}:= Q^{-1}\nJ Q^{\\top -1} $ leads to the so-called Pfaff Lattice, which is integrable, by\nvirtue of the AKS theorem, for a splitting involving the affine symplectic\nalgebra. The tau-functions for the system are shown to be pfaffians and the\nwave vectors skew-orthogonal polynomials; we give their explicit form in terms\nof moments. This system plays an important role in symmetric and symplectic\nmatrix models and in the theory of random matrices (beta=1 or 4).",
"arxiv_id": "solv-int/9903005",
"authors": [
"M. Adler",
"E. Horozov",
"P. van Moerbeke"
],
"categories": [
"solv-int",
"nlin.SI"
],
"journal_ref": "Intern. Math. Research Notices, 1999",
"title": "The Pfaff lattice and skew-orthogonal polynomials",
"url": "https://arxiv.org/abs/solv-int/9903005"
},
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