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Abstract: We present the third in the series of papers describing Poisson properties ofplanar directed networks in the disk or in the annulus. In this paper weconcentrate on special networks N {u,v} in the disk that correspond to thechoice of a pair u,v of Coxeter elements in the symmetric group and thecorresponding networks N {u,v}^\circ in the annulus. Boundary measurements forN {u,v} represent elements of the Coxeter double Bruhat cell G^{u,v} in GL n.The Cartan subgroup acts on G^{u,v} by conjugation. The standard Poissonstructure on the space of weights of N {u,v} induces a Poisson structure onG^{u,v}, and hence on its quotient by the Cartan subgroup, which makes thelatter into the phase space for an appropriate Coxeter-Toda lattice. Theboundary measurement for N {u,v}^\circ is a rational function that coincides upto a nonzero factor with the Weyl function for the boundary measurement forN {u,v}. The corresponding Poisson bracket on the space of weights ofN {u,v}^\circ induces a Poisson bracket on the certain space of rationalfunctions, which appeared previously in the context of Toda flows.Following the ideas developed in our previous papers, we introduce a clusteralgebra A on this space, compatible with the obtained Poisson bracket.Generalized B\-acklund-Darboux transformations map solutions of oneCoxeter-Toda lattice to solutions of another preserving the corresponding Weylfunction. Using network representation, we construct generalizedB\-acklund-Darboux transformations as appropriate sequences of clustertransformations in A.



Autor: Michael Gekhtman, Michael Shapiro, Alek Vainshtein

Fuente: https://arxiv.org/



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