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 Nu,v in the disk that correspond to the choice of a pair (u, v) of Coxeter elements in the symmetric group Sn and the corresponding networks, in the annulus. Boundary measurements for Nu,v represent elements of the Coxeter double Bruhat cell Gu,v⊂GLn. The Cartan subgroup H acts on Gu,v by conjugation. The standard Poisson structure on the space of weights of Nu,v induces a Poisson structure on Gu,v, and hence on the quotient Gu,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 Nu,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.
Bibliographical noteFunding 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)