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#### Quantum Algebramathscidoc:1701.29001

Acta Mathematica, 206, (2), 245-310, 2009.6
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 $N_{u,v}^\circ$ 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 $N_{u,v}^\circ$ 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 $N_{u,v}^\circ$ induces a Poisson bracket on the certain space ${\mathcal{R}_n}$ of rational functions, which appeared previously in the context of Toda flows.
@inproceedings{michael2009generalized,
title={Generalized Bäcklund–Darboux transformations for Coxeter–Toda flows from a cluster algebra perspective},
author={Michael Gekhtman, Michael Shapiro, and Alek Vainshtein},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203356558690739},
booktitle={Acta Mathematica},
volume={206},
number={2},
pages={245-310},
year={2009},
}

Michael Gekhtman, Michael Shapiro, and Alek Vainshtein. Generalized Bäcklund–Darboux transformations for Coxeter–Toda flows from a cluster algebra perspective. 2009. Vol. 206. In Acta Mathematica. pp.245-310. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203356558690739.