Generalized Bäcklund–darboux Transformations for Coxeter–toda Flows from Cluster Algebra Perspective

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 S n and the corresponding networks N u,v 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 N u,v is a rational function that coincides up to a nonzero factor with the Weyl function for the boundary measurement for Nu,v. The corresponding Poisson bracket on the space of weights of N u,v induces a Poisson bracket on the certain space Rn 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 A on Rn 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 A.