Acta Mathematica

Generalized Bäcklund–Darboux transformations for Coxeter–Toda flows from a cluster algebra perspective

Michael Gekhtman, Michael Shapiro, and Alek Vainshtein

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Abstract

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 Sn and the corresponding networks $N_{u,v}^\circ$ 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 $N_{u,v}^\circ$ is a rational function that coincides up to a non-zero factor with the Weyl function for the boundary measurement for Nu,v. The corresponding Poisson bracket on the space of weights of $N_{u,v}^\circ$ induces a Poisson bracket on the certain space $ {\mathcal{R}_n} $ 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 $ \mathcal{A} $ on $ {\mathcal{R}_n} $ 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 $ \mathcal{A} $.

Article information

Source
Acta Math., Volume 206, Number 2 (2011), 245-310.

Dates
Received: 26 June 2009
Revised: 12 July 2010
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485892547

Digital Object Identifier
doi:10.1007/s11511-011-0063-1

Mathematical Reviews number (MathSciNet)
MR2810853

Zentralblatt MATH identifier
1228.53095

Subjects
Primary: 37K10: Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies (KdV, KP, Toda, etc.)
Secondary: 53D17: Poisson manifolds; Poisson groupoids and algebroids 13A99: None of the above, but in this section

Rights
2011 © Institut Mittag-Leffler

Citation

Gekhtman, Michael; Shapiro, Michael; Vainshtein, Alek. Generalized Bäcklund–Darboux transformations for Coxeter–Toda flows from a cluster algebra perspective. Acta Math. 206 (2011), no. 2, 245--310. doi:10.1007/s11511-011-0063-1. https://projecteuclid.org/euclid.acta/1485892547


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References

  • A khiezer, N. I., The Classical Moment Problem and Some Related Questions in Analysis. Hafner, New York, 1965.
  • B erenstein, A., F omin, S. & Z elevinsky, A., Parametrizations of canonical bases and totally positive matrices. Adv. Math., 122 (1996), 49–149.
  • — Cluster algebras. III. Upper bounds and double Bruhat cells. Duke Math. J., 126 (2005), 1–52.
  • B erenstein, A. & K azhdan, D., Quantum Hankel algebras, clusters, and canonical bases. In preparation.
  • B rockett, R.W. & F aybusovich, L.E., Toda flows, inverse spectral transform and realization theory. Systems Control Lett., 16 (1991), 79–88.
  • 6 C antero, M. J., M oral, L. & V elázquez, L., Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle. Linear Algebra Appl., 362 (2003), 29–56.
  • 7 D eift, P., Li, L. C., N anda, T. & T omei, C., The Toda flow on a generic orbit is integrable. Comm. Pure Appl. Math., 39 (1986), 183–232.
  • 8 D eift, P., Li, L. C. & T omei, C., Matrix factorizations and integrable systems. Comm. Pure Appl. Math., 42 (1989), 443–521.
  • 9 D i F rancesco, P. & K edem, R., Q-system cluster algebras, paths and total positivity. SIGMA Symmetry Integrability Geom. Methods Appl., 6 (2010), Paper 014, 36.
  • Q-systems, heaps, paths and cluster positivity. Comm. Math. Phys., 293 (2010), 727–802.
  • F allat, S. M., Bidiagonal factorizations of totally nonnegative matrices. Amer. Math. Monthly, 108 (2001), 697–712.
  • F aybusovich, L. & G ekhtman, M., Elementary Toda orbits and integrable lattices. J. Math. Phys., 41 (2000), 2905–2921.
  • — Poisson brackets on rational functions and multi-Hamiltonian structure for integrable lattices. Phys. Lett. A, 272 (2000), 236–244.
  • — Inverse moment problem for elementary co-adjoint orbits. Inverse Problems, 17 (2001), 1295–1306.
  • F omin, S. & Z elevinsky, A., Double Bruhat cells and total positivity. J. Amer. Math. Soc., 12 (1999), 335–380.
  • — Total positivity: tests and parametrizations. Math. Intelligencer, 22 (2000), 23–33.
  • — Cluster algebras. I. Foundations. J. Amer. Math. Soc., 15 (2002), 497–529.
  • — Cluster algebras. II. Finite type classification. Invent. Math., 154 (2003), 63–121.
  • F uhrmann, P. A., A Polynomial Approach to Linear Algebra. Universitext. Springer, New York, 1996.
  • G ekhtman, M., S hapiro, M. & V ainshtein, A., Cluster algebras and Poisson geometry. Mosc. Math. J., 3 (2003), 899–934, 1199.
  • — Poisson geometry of directed networks in a disk. Selecta Math., 15 (2009), 61–103.
  • — Poisson geometry of directed networks in an annulus. Preprint, 2009. arXiv:0901.0020 [math.QA].
  • H offmann, T., K ellendonk, J., K utz, N. & R eshetikhin, N., Factorization dynamics and Coxeter–Toda lattices. Comm. Math. Phys., 212 (2000), 297–321.
  • K arlin, S. & M cG regor, J., Coincidence probabilities. Pacific J. Math., 9 (1959), 1141–1164.
  • K edem, R., Q-systems as cluster algebras. J. Phys. A, 41 (2008), 194011, 14 pp.
  • K ogan, M. & Z elevinsky, A., On symplectic leaves and integrable systems in standard complex semisimple Poisson-Lie groups. Int. Math. Res. Not., 32 (2002), 1685–1702.
  • L indström, B., On the vector representations of induced matroids. Bull. London Math. Soc., 5 (1973), 85–90.
  • M oser, J., Finitely many mass points on the line under the influence of an exponential potential–an integrable system, in Dynamical Systems, Theory and Applications (Seattle, WA, 1974), Lecture Notes in Physics, 38, pp. 467–497. Springer, Berlin–Heidelberg, 1975.
  • P ostnikov, A., Total positivity, Grassmannians, and networks. Preprint, 2006. arXiv:math/0609764 [math.CO].
  • R eshetikhin, N., Integrability of characteristic Hamiltonian systems on simple Lie groups with standard Poisson Lie structure. Comm. Math. Phys., 242 (2003), 1–29.
  • R eyman, A. & Semenov-Tian-Shansky, M., Group-theoretical methods in the theory of finite-dimensional integrable systems, in Encyclopaedia of Mathematical Sciences, 16, pp. 116–225. Springer, Berlin–Heidelberg, 1994.
  • S imon, B., Orthogonal Polynomials on the Unit Circle. Part 1. American Mathematical Society Colloquium Publications, 54. Amer. Math. Soc., Providence, RI, 2005.
  • S tieltjes, T. J., Recherches sur les fractions continues, in Euvres complètes/Collected papers. Vol. II, pp. 402–566. Noordhoff, Groningen, 1918.
  • W atkins, D. S., Isospectral flows. SIAM Rev., 26 (1984), 379–391.
  • Y akimov, M., Symplectic leaves of complex reductive Poisson–Lie groups. Duke Math. J., 112 (2002), 453–509.
  • Y ang, S. W. & Z elevinsky, A., Cluster algebras of finite type via Coxeter elements and principal minors. Transform. Groups, 13 (2008), 855–895.
  • Z elevinsky, A., Connected components of real double Bruhat cells. Int. Math. Res. Not., 21 (2000), 1131–1154.