American Institute of Mathematical Sciences

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  2012, 19: 1-17. doi: 10.3934/era.2012.19.1

Higher pentagram maps, weighted directed networks, and cluster dynamics

Michael Gekhtman 1, , Michael Shapiro 2, , Serge Tabachnikov 3, and  Alek Vainshtein 4,

1. 

Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States
2. 

Department of Mathematics, Michigan State University, East Lansing, MI 48823, United States
3. 

Department of Mathematics, Penn State University, University Park, PA 16802
4. 

Department of Mathematics and Department of Computer Science, University of Haifa, Haifa, Mount Carmel 31905, Israel

Received  October 2011 Revised  December 2011 Published  January 2012

The pentagram map was extensively studied in a series of papers by V. Ovsienko, R. Schwartz and S. Tabachnikov. It was recently interpreted by M. Glick as a sequence of cluster transformations associated with a special quiver. Using compatible Poisson structures in cluster algebras and Poisson geometry of directed networks on surfaces, we generalize Glick's construction to include the pentagram map into a family of geometrically meaningful discrete integrable maps.
Keywords: Pentagram map, discrete integrable system., cluster dynamics.
Mathematics Subject Classification: Primary: 13F60; Secondary: 53D1.
Citation: Michael Gekhtman, Michael Shapiro, Serge Tabachnikov, Alek Vainshtein. Higher pentagram maps, weighted directed networks, and cluster dynamics. Electronic Research Announcements, 2012, 19: 1-17. doi: 10.3934/era.2012.19.1
References:
[1]

A. Bobenko and T. Hoffmann, Hexagonal circle patterns and integrable systems: Patterns with constant angles,, Duke Math. J., 116 (2003), 525.  doi: 10.1215/S0012-7094-03-11635-X.  Google Scholar

[2]

A. Bobenko and Yu. Suris, "Discrete Differential Geometry. Integrable Structure,", Amer. Math. Soc., (2008).   Google Scholar

[3]

H. Busemann and P. Kelly, "Projective geometry and projective metrics,", Academic Press, (1953).   Google Scholar

[4]

B. Dubrovin, I. Krichever and S. Novikov, Integrable systems. I,, in, 4 (2001), 177.   Google Scholar

[5]

L. Faddeev and L. Takhtajan, "Hamiltonian Methods in the Theory of Solitons,", Springer-Verlag, (1987).   Google Scholar

[6]

S. Fomin and A. Zelevinsky, Cluster algebras. IV. Coefficients,, Compos. Math., 143 (2007), 112.  doi: 10.1112/S0010437X06002521.  Google Scholar

[7]

M. Gekhtman, M. Shapiro and A. Vainshtein, Poisson geometry of directed networks in a disk,, Selecta Math., 15 (2009), 61.  doi: 10.1007/s00029-009-0523-z.  Google Scholar

[8]

M. Gekhtman, M. Shapiro and A. Vainshtein, "Cluster Algebras and Poisson Geometry,", Amer. Math. Soc., (2010).   Google Scholar

[9]

M. Gekhtman, M. Shapiro and A. Vainshtein, Poisson geometry of directed networks in an annulus,, J. Europ. Math. Soc., ().   Google Scholar

[10]

M. Gekhtman, M. Shapiro and A. Vainshtein, Generalized Bäcklund-Darboux transformations for Coxeter-Toda flows from a cluster algebra perspective,, Acta Math., 206 (2011), 245.  doi: 10.1007/s11511-011-0063-1.  Google Scholar

[11]

M. Glick, The pentagram map and $Y$-patterns,, Adv. Math., 227 (2011), 1019.  doi: 10.1016/j.aim.2011.02.018.  Google Scholar

[12]

, M. Glick,, private communication., ().   Google Scholar

[13]

A. Goncharov and R. Kenyon, Dimers and cluster integrable systems,, preprint, ().   Google Scholar

[14]

B. Khesin and F. Soloviev, Integrability of a space pentagram map,, in preparation., ().   Google Scholar

[15]

B. Konopelchenko and W. Schief, Menelaus' theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy,, J. Phys. A, 35 (2002), 6125.  doi: 10.1088/0305-4470/35/29/313.  Google Scholar

[16]

G. Mari Beffa, On generalizations of the pentagram map: Discretizations of AGD flows,, preprint, ().   Google Scholar

[17]

S. Morier-Genoud, V. Ovsienko and S. Tabachnikov, 2-frieze patterns and the cluster structure of the space of polygons,, Ann. Inst. Fourier, ().   Google Scholar

[18]

F. Nijhoff and H. Capel, The discrete Korteweg-de Vries equation,, Acta Appl. Math., 39 (1995), 133.  doi: 10.1007/BF00994631.  Google Scholar

[19]

M. Olshanetsky, A. Perelomov, A. Reyman and M. Semenov-Tian-Shansky, Integrable systems. II,, in, 16 (1994), 83.   Google Scholar

[20]

V. Ovsienko, R. Schwartz and S. Tabachnikov, Quasiperiodic motion for the Pentagram map,, Electron. Res. Announc. Math. Sci., 16 (2009), 1.   Google Scholar

[21]

V. Ovsienko, R. Schwartz and S. Tabachnikov, The Pentagram map: A discrete integrable system,, Commun. Math. Phys., 299 (2010), 409.  doi: 10.1007/s00220-010-1075-y.  Google Scholar

[22]

V. Ovsienko, R. Schwartz and S. Tabachnikov, Liouville-Arnold integrability of the pentagram map on closed polygons,, preprint, ().   Google Scholar

[23]

A. Postnikov, Total positivity, Grassmannians, and networks,, preprint, ().   Google Scholar

[24]

O. Schramm, Circle patterns with the combinatorics of the square grid,, Duke Math. J., 86 (1997), 347.  doi: 10.1215/S0012-7094-97-08611-7.  Google Scholar

[25]

R. Schwartz, The pentagram map,, Experiment. Math., 1 (1992), 71.   Google Scholar

[26]

R. Schwartz, The pentagram map is recurrent,, Experiment. Math., 10 (2001), 519.   Google Scholar

[27]

R. Schwartz, Discrete monodromy, pentagrams, and the method of condensation,, J. Fixed Point Theory Appl., 3 (2008), 379.  doi: 10.1007/s11784-008-0079-0.  Google Scholar

[28]

R. Schwartz and S. Tabachnikov, Elementary surprises in projective geometry,, Math. Intelligencer, 32 (2010), 31.  doi: 10.1007/s00283-010-9137-8.  Google Scholar

[29]

R. Schwartz and S. Tabachnikov, The pentagram integrals on inscribed polygons,, Electron. J. Comb., 18 (2011).   Google Scholar

[30]

F. Soloviev, Integrability of the pentagram map,, preprint, ().   Google Scholar

[31]

Yu. Suris, On some integrable systems related to the Toda lattice,, J. Phys. A, 30 (1997), 2235.  doi: 10.1088/0305-4470/30/6/041.  Google Scholar

show all references

References:
[1]

A. Bobenko and T. Hoffmann, Hexagonal circle patterns and integrable systems: Patterns with constant angles,, Duke Math. J., 116 (2003), 525.  doi: 10.1215/S0012-7094-03-11635-X.  Google Scholar

[2]

A. Bobenko and Yu. Suris, "Discrete Differential Geometry. Integrable Structure,", Amer. Math. Soc., (2008).   Google Scholar

[3]

H. Busemann and P. Kelly, "Projective geometry and projective metrics,", Academic Press, (1953).   Google Scholar

[4]

B. Dubrovin, I. Krichever and S. Novikov, Integrable systems. I,, in, 4 (2001), 177.   Google Scholar

[5]

L. Faddeev and L. Takhtajan, "Hamiltonian Methods in the Theory of Solitons,", Springer-Verlag, (1987).   Google Scholar

[6]

S. Fomin and A. Zelevinsky, Cluster algebras. IV. Coefficients,, Compos. Math., 143 (2007), 112.  doi: 10.1112/S0010437X06002521.  Google Scholar

[7]

M. Gekhtman, M. Shapiro and A. Vainshtein, Poisson geometry of directed networks in a disk,, Selecta Math., 15 (2009), 61.  doi: 10.1007/s00029-009-0523-z.  Google Scholar

[8]

M. Gekhtman, M. Shapiro and A. Vainshtein, "Cluster Algebras and Poisson Geometry,", Amer. Math. Soc., (2010).   Google Scholar

[9]

M. Gekhtman, M. Shapiro and A. Vainshtein, Poisson geometry of directed networks in an annulus,, J. Europ. Math. Soc., ().   Google Scholar

[10]

M. Gekhtman, M. Shapiro and A. Vainshtein, Generalized Bäcklund-Darboux transformations for Coxeter-Toda flows from a cluster algebra perspective,, Acta Math., 206 (2011), 245.  doi: 10.1007/s11511-011-0063-1.  Google Scholar

[11]

M. Glick, The pentagram map and $Y$-patterns,, Adv. Math., 227 (2011), 1019.  doi: 10.1016/j.aim.2011.02.018.  Google Scholar

[12]

, M. Glick,, private communication., ().   Google Scholar

[13]

A. Goncharov and R. Kenyon, Dimers and cluster integrable systems,, preprint, ().   Google Scholar

[14]

B. Khesin and F. Soloviev, Integrability of a space pentagram map,, in preparation., ().   Google Scholar

[15]

B. Konopelchenko and W. Schief, Menelaus' theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy,, J. Phys. A, 35 (2002), 6125.  doi: 10.1088/0305-4470/35/29/313.  Google Scholar

[16]

G. Mari Beffa, On generalizations of the pentagram map: Discretizations of AGD flows,, preprint, ().   Google Scholar

[17]

S. Morier-Genoud, V. Ovsienko and S. Tabachnikov, 2-frieze patterns and the cluster structure of the space of polygons,, Ann. Inst. Fourier, ().   Google Scholar

[18]

F. Nijhoff and H. Capel, The discrete Korteweg-de Vries equation,, Acta Appl. Math., 39 (1995), 133.  doi: 10.1007/BF00994631.  Google Scholar

[19]

M. Olshanetsky, A. Perelomov, A. Reyman and M. Semenov-Tian-Shansky, Integrable systems. II,, in, 16 (1994), 83.   Google Scholar

[20]

V. Ovsienko, R. Schwartz and S. Tabachnikov, Quasiperiodic motion for the Pentagram map,, Electron. Res. Announc. Math. Sci., 16 (2009), 1.   Google Scholar

[21]

V. Ovsienko, R. Schwartz and S. Tabachnikov, The Pentagram map: A discrete integrable system,, Commun. Math. Phys., 299 (2010), 409.  doi: 10.1007/s00220-010-1075-y.  Google Scholar

[22]

V. Ovsienko, R. Schwartz and S. Tabachnikov, Liouville-Arnold integrability of the pentagram map on closed polygons,, preprint, ().   Google Scholar

[23]

A. Postnikov, Total positivity, Grassmannians, and networks,, preprint, ().   Google Scholar

[24]

O. Schramm, Circle patterns with the combinatorics of the square grid,, Duke Math. J., 86 (1997), 347.  doi: 10.1215/S0012-7094-97-08611-7.  Google Scholar

[25]

R. Schwartz, The pentagram map,, Experiment. Math., 1 (1992), 71.   Google Scholar

[26]

R. Schwartz, The pentagram map is recurrent,, Experiment. Math., 10 (2001), 519.   Google Scholar

[27]

R. Schwartz, Discrete monodromy, pentagrams, and the method of condensation,, J. Fixed Point Theory Appl., 3 (2008), 379.  doi: 10.1007/s11784-008-0079-0.  Google Scholar

[28]

R. Schwartz and S. Tabachnikov, Elementary surprises in projective geometry,, Math. Intelligencer, 32 (2010), 31.  doi: 10.1007/s00283-010-9137-8.  Google Scholar

[29]

R. Schwartz and S. Tabachnikov, The pentagram integrals on inscribed polygons,, Electron. J. Comb., 18 (2011).   Google Scholar

[30]

F. Soloviev, Integrability of the pentagram map,, preprint, ().   Google Scholar

[31]

Yu. Suris, On some integrable systems related to the Toda lattice,, J. Phys. A, 30 (1997), 2235.  doi: 10.1088/0305-4470/30/6/041.  Google Scholar

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