Partial-skew-orthogonal polynomials and related integrable lattices with Pfaffian tau-functions

Xiang-Ke Chang Yi He Xing-Biao Hu Shi-Hao Li

Dynamical Systems mathscidoc:1912.43164

Communications in Mathematical Physics, 364, (3), 1069-1119, 2018.12
Skew-orthogonal polynomials (SOPs) arise in the study of the <i>n</i>-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 Forrester and Kieburg (Commun Math Phys 342(1):151187, 2016), we propose the concept of <i>partial-skew-orthogonal polynomials</i> (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 tau-functions also admit the multiple integral representations. Among these integrable lattices, some of them are
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@inproceedings{xiang-ke2018partial-skew-orthogonal,
  title={Partial-skew-orthogonal polynomials and related integrable lattices with Pfaffian tau-functions},
  author={Xiang-Ke Chang, Yi He, Xing-Biao Hu, and Shi-Hao Li},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221112659879157724},
  booktitle={Communications in Mathematical Physics},
  volume={364},
  number={3},
  pages={1069-1119},
  year={2018},
}
Xiang-Ke Chang, Yi He, Xing-Biao Hu, and Shi-Hao Li. Partial-skew-orthogonal polynomials and related integrable lattices with Pfaffian tau-functions. 2018. Vol. 364. In Communications in Mathematical Physics. pp.1069-1119. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221112659879157724.
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