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are rational functions in the edge weights and λ, see Corollary 9.3 and study how

they depend on the choice of the cut. Besides, we provide a cohomological descrip-

tion of the space of face and trail weights, which is a higher analog of the space

of edge weights used in the previous chapter. Poisson properties of the obtained

boundary measurement map from the space of edge weights to the space of rational

k × m matrix functions in one variable are treated in Section 9.2. We prove an

analog of Theorem 8.6, saying that for fixed sets of sources and sinks, the induced

Poisson structures on the space of matrix functions form a 2-parametric family

that does not depend on the internal structure of the network, see Theorem 9.4.

Explicit expressions for this family are much more complicated than those for the

disk, see Proposition 9.6. The proof itself differs substantially from the proof of

Theorem 8.6. It relies on the fact that any rational k × m matrix function belongs

to the image of the boundary measurement map for an appropriated perfect net-

work, see Theorem 9.10. The section is concluded with Theorem 9.15 claiming that

for a specific choice of sinks and sources one can recover the Sklyanin bracket cor-

responding to the trigonometric R-matrix. In Section 9.3 we extend these results

to the Grassmannian boundary measurement map from the space of edge weights

to the space of Grassmannian loops. We define the path reversal map and prove

that this map commutes with the Grassmannian boundary measurement map, see

Theorem 9.17. Further, we prove Theorem 9.22, which is a natural extension of

Theorem 8.12; once again, the proof is very different and is based on path reversal

techniques and the use of face weights.

In the concluding Chapter 10 we apply techniques developed in the previous

chapter to providing a cluster interpretation of generalized B¨ acklund–Darboux trans-

formations for Coxeter–Toda lattices. Section 10.1 gives an overview of the chapter

and contains brief introductory information on Toda lattices, Weyl functions of the

corresponding Lax operators and generalized B¨ acklund–Darboux transformations

between phase spaces of different lattices preserving the Weyl function. A Coxeter

double Bruhat cell in GLn is defined by a pair of Coxeter elements in the symmetric

group Sn. In Section 10.2 we have collected and proved all the necessary technical

facts about such cells and the representation of their elements via perfect networks

in a disk. Section 10.3 treats the inverse problem of restoring factorization param-

eters of an element of a Coxeter double Bruhat cell from its Weyl function. We

provide an explicit solution for this problem involving Hankel determinants in the

coeﬃcients of the Laurent expansion for the Weyl function, see Theorem 10.9. In

Section 10.4 we build and investigate a cluster algebra on a certain space Rn of

rational functions related to the space of Weyl functions corresponding to Coxeter

double Bruhat cells. We start from defining a perfect network in an annulus corre-

sponding to the network in a disk studied in Section 10.2. The space of face weights

of this network is equipped with a particular Poisson bracket from the family stud-

ied in Chapter 9. We proceed by using results of Chapter 4 to build a cluster algebra

of rank 2n − 2 compatible with this Poisson bracket. Theorem 10.27 claims that

this cluster algebra does not depend on the choice of the pair of Coxeter elements,

and that the ring of regular functions on Rn is isomorphic to the localization of this

cluster algebra with respect to the stable variables. In Section 10.5 we use these re-

sults to characterize generalized B¨ acklund–Darboux transformations as sequences of

cluster transformations in the above cluster algebra conjugated by by a certain map

defined by the solution of the inverse problem in Section 10.3, see Theorem 10.36.