## Abstract

We present the third in the series of papers describing Poisson properties of planar directed networks in the disk or in the annulus. In this paper we concentrate on special networks N_{u,v} in the disk that correspond to the choice of a pair (u, v) of Coxeter elements in the symmetric group S_{n} and the corresponding networks, in the annulus. Boundary measurements for N_{u,v} represent elements of the Coxeter double Bruhat cell G^{u,v}⊂GL_{n}. The Cartan subgroup H acts on G^{u,v} by conjugation. The standard Poisson structure on the space of weights of N_{u,v} induces a Poisson structure on G^{u,v}, and hence on the quotient G^{u,v}/H, which makes the latter into the phase space for an appropriate Coxeter-Toda lattice. The boundary measurement for, is a rational function that coincides up to a non-zero factor with the Weyl function for the boundary measurement for N_{u,v}. The corresponding Poisson bracket on the space of weights of, induces a Poisson bracket on the certain space, of rational functions, which appeared previously in the context of Toda flows. Following the ideas developed in our previous papers, we introduce a cluster algebra, on, compatible with the obtained Poisson bracket. Generalized Bäcklund-Darboux transformations map solutions of one Coxeter-Toda lattice to solutions of another preserving the corresponding Weyl function. Using network representation, we construct generalized Bäcklund-Darboux transformations as appropriate sequences of cluster transformations in.

Original language | English |
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Pages (from-to) | 245-310 |

Number of pages | 66 |

Journal | Acta Mathematica |

Volume | 206 |

Issue number | 2 |

DOIs | |

State | Published - Jun 2011 |

### Bibliographical note

Funding Information:We wish to express gratitude to A. Berenstein, P. Di Francesco, R. Kedem and A. Zele-vinsky for useful comments. A. V. would like to thank the University of Michigan, where he spent a sabbatical term in Spring 2009 and where this paper was finished. He is grateful to Sergey Fomin for warm hospitality and stimulating working conditions. M. S. expresses his gratitude to the Stockholm University and the Royal Institute of Technology, where he worked on this manuscript in the Fall 2008 during his sabbatical leave. M. G. was supported in part by NSF Grant DMS #0801204. M. S. was supported in part by NSF Grants DMS #0800671 and PHY #0555346. A. V. was supported in part by ISF Grant #1032/08.

## ASJC Scopus subject areas

- Mathematics (all)