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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 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 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 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.
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