The University of Michigan Combinatorics Seminar
We use a special kind of directed networks in an annulus to study a cluster algebra structure on a space of rational functions with a pole at infinity and subject to some genericity conditions. Distinguished clusters in this cluster algebra are in natural correspondence with pairs of Coxeter elements of the permutation groups. We show that sequences of cluster transformations connecting two distinguished clusters are closely associated with Backlund-Darboux transformations of Toda flows on corresponding double Bruhat cells in GL(n). This is joint work with M. Shapiro and A. Vainshtein.