Generalized Bäcklund–Darboux transformations for Coxeter–Toda flows from a cluster algebra perspective

Michael Gekhtman Department of Mathematics, University of Notre Dame Michael Shapiro Department of Mathematics, Michigan State University Alek Vainshtein Department of Mathematics and Department of Computer Science, University of Haifa

Quantum Algebra mathscidoc: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.
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@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.
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