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Two-Parameter Generalizations of Cauchy Bi-Orthogonal Polynomials and Integrable Lattices

Abstract

In this article, we consider the generalised two-parameter Cauchy two-matrix model and the corresponding integrable lattice equation. It is shown that with parameters chosen as \(1/k_i\), \(k_i\in {\mathbb {Z}}_{>0}\) (\(i=1,\,2\)), the average characteristic polynomials admit \((k_1+k_2+2)\)-term recurrence relations, which can be interpreted as spectral problems for integrable lattices. The tau function is then given by the partition function of the generalised Cauchy two-matrix model as well as Gram determinant. The simplest solvable example is given.

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Notes

  1. One can follow the procedure exhibited in Wang et al. (2010) and take the Gram determinant into the first equation of CKP equation (i.e. Wang et al. 2010, Eq. (3)). It is not difficult to see the asymmetric Gram determinant doesn’t satisfy the bilinear equation of CKP equation.

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Acknowledgements

The authors would also like to thank Profs. Xingbiao Hu and Dafeng Zuo for helpful discussions, and referees for helpful comments which improve the article a lot. X. Chang is partially supported by National Natural Science Foundation of China (Grant Nos. 11688101, 11731014 and 11701550) and the Youth Innovation Promotion Association CAS. S. Li was supported by the ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS). S. Tsujimoto is partially supported by JSPS KAKENHI (Grant Nos. 19H01792, 17K18725) and G. Yu is partially supported by National Natural Science Foundation of China (Grant No. 11871336).

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Correspondence to Guo-Fu Yu.

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Appendix A

Appendix A

This appendix is devoted to a statement about how the superpotential (1.3) relates to the extended affine group \({\tilde{W}}^{(k,k+1)}(A_\ell )\). To this end, firstly, we give some basic concepts about the extended affine Weyl group.

Let V be an \(\ell \)-dimensional Euclidean space with inner product \((\cdot ,\cdot )\) and \({\mathcal {R}}\) be an irreducible root system defined in V with simple roots \(\alpha _1,\cdots ,\alpha _\ell \) and coroots \(\alpha _1^\vee ,\cdots ,\alpha _\ell ^\vee \). Then, the Weyl group \(W({\mathcal {R}})\) can be generated by the reflection

$$\begin{aligned} x\mapsto x-(\alpha _j^\vee ,x)\alpha _j,\quad \forall x\in V,\quad j=1,\cdots ,\ell . \end{aligned}$$

The affine Weyl group \(W_a({\mathcal {R}})\) which acts on the Euclidean space V can be realised as the semi-product of \(W({\mathcal {R}})\) by the lattice of coroots via the map

$$\begin{aligned} {x}\mapsto w(x)+\sum _{j=1}^\ell m_j\alpha _j^\vee , \quad w\in W({\mathcal {R}}),\quad m_j\in {\mathbb {Z}}. \end{aligned}$$

Dubrovin and Zhang then proposed the extended affine Weyl group \({\tilde{W}}^{(k)}({\mathcal {R}})\) acting on \(V\oplus {\mathbb {R}}\) in the study of Frobenius manifold. It is generated by the transformations

$$\begin{aligned}&(x,x_{\ell +1})\mapsto (w(x)+\sum _{j=1}^\ell m_j\alpha _j^\vee , x_{\ell +1}),\quad w\in W({\mathcal {R}}), \quad m_j\in {\mathbb {Z}},\\&(x,x_{\ell +1})\mapsto (x+\gamma w_k, x_{\ell +1}-\gamma ),\quad 1\le k\le \ell \end{aligned}$$

with \(w_k\) being the fundamental weights defined by the relations \((w_k,\alpha _j^\vee )=\delta _{k,j}\) for \(k,j=1,\cdots ,\ell \) and \(\gamma \) is a constant related the root system. With \({\mathcal {R}}\) chosen as \(A_\ell \), a construction of Frobenius manifold structure on the orbit of \({\tilde{W}}^{(k)}(A_\ell )\) was given in Dubrovin and Zhang (1998). It was shown that \({\tilde{W}}^{(k)}(A_\ell )\) describes the monodromy of roots of trigonometric polynomials admits the form

$$\begin{aligned} \lambda (\phi )=e^{ik\phi }+\mathbf {a_1}e^{i(k-1)\phi }+\cdots +a_{\ell +1} e^{i(k-\ell -1)\phi },\quad a_{\ell +1}\ne 0. \end{aligned}$$

If one sets \(e^{i\phi }\) as the shift operator \(\Lambda \), then the above polynomials will become

$$\begin{aligned} L=\Lambda ^k+\mathbf {a_1}\Lambda ^{k-1}+\cdots +a_{\ell +1}\Lambda ^{k-\ell -1},\quad a_{\ell +1}\ne 0 \end{aligned}$$

which is regarded as the spectral operator of the bigraded Toda hierarchy (c.f. Eq. (1.2)).

In the recent work (Zuo 2020), the author studied another extended affine Weyl group \({\tilde{W}}^{(k,k+1)}(A_\ell )\) acting on the space \(V\oplus {\mathbb {R}}^2\), generated by the transformation

$$\begin{aligned}&(x,x_{\ell +1},x_{\ell +2})\mapsto (w(x)+\sum _{j=1}^\ell m_j\alpha _j^\vee ,x_{\ell +1},x_{\ell +2}),\quad w\in W(A_\ell ),\quad m_j\in {\mathbb {Z}},\\&(x,x_{\ell +1},x_{\ell +2})\mapsto (x+\gamma w_k,x_{\ell +1}-\gamma ,x_{\ell +2}),\quad (x,x_{\ell +1},x_{\ell +2})\\&\quad \mapsto (x+\gamma w_{k+1},x_{\ell +1},x_{\ell +2}-\gamma ) \end{aligned}$$

for \(1\le k\le \ell -1\). It was proven by Zuo (2020) that the orbit space of the extended affine Weyl group is locally isomorphic to a simple Hurwitz space \({\mathbb {M}}_{k,\ell -k+1,1}\) which contains a natural structure of Frobenius manifold. Moreover, this space consists of trigonometric Laurent series admitting the form

$$\begin{aligned} \lambda (\phi )=(e^{i\phi }-a_{\ell +2})^{-1}(e^{i(k+1)\phi }+\mathbf {a_1}e^{ik\phi } +\cdots +a_{\ell +1} e^{i(k-\ell )}\phi ),\quad a_{\ell +1}a_{\ell +2}\ne 0. \end{aligned}$$

Similarly, by setting \(e^{i\phi }=\Lambda \), one gets the spectral operator (1.3)

$$\begin{aligned} L\!=\!(\Lambda \!-\!a_{\ell +2})^{-1}(\Lambda ^{k+1}\!+\!\mathbf {a_1}\Lambda ^{k}\!+\cdots +a_{\ell +1}\Lambda ^{k-\ell }), \quad 1\!\le \! k\!\le \! \ell -1,\, a_{\ell +1}a_{\ell +2}\ne 0. \end{aligned}$$

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Chang, XK., Li, SH., Tsujimoto, S. et al. Two-Parameter Generalizations of Cauchy Bi-Orthogonal Polynomials and Integrable Lattices. J Nonlinear Sci 31, 30 (2021). https://doi.org/10.1007/s00332-021-09690-9

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Keywords

  • Two-parameter Cauchy two-matrix model
  • Toda-type lattice
  • Gram determinant technique

Mathematics Subject Classification

  • 37K10
  • 37K20
  • 15A15