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
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show all references

References:
[1]

Duke Math. J., 116 (2003), 525-566. doi: 10.1215/S0012-7094-03-11635-X.  Google Scholar

[2]

Amer. Math. Soc., Providence, RI, 2008.  Google Scholar

[3]

Academic Press, New York, 1953.  Google Scholar

[4]

in "Dynamical Systems," IV, Encyclopaedia Math. Sci., 4, Springer, Berlin, (2001), 177-332.  Google Scholar

[5]

Springer-Verlag, Berlin, 1987.  Google Scholar

[6]

Compos. Math., 143 (2007), 112-164. doi: 10.1112/S0010437X06002521.  Google Scholar

[7]

Selecta Math., 15 (2009), 61-103. doi: 10.1007/s00029-009-0523-z.  Google Scholar

[8]

Amer. Math. Soc., Providence, RI, 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]

Acta Math., 206 (2011), 245-310. doi: 10.1007/s11511-011-0063-1.  Google Scholar

[11]

Adv. Math., 227 (2011), 1019-1045. 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]

J. Phys. A, 35 (2002), 6125-6144. 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]

Acta Appl. Math., 39 (1995), 133-158. doi: 10.1007/BF00994631.  Google Scholar

[19]

in "Dynamical systems," VII, Encycl. Math. Sci., 16, Springer, Berlin, (1994), 83-259.  Google Scholar

[20]

Electron. Res. Announc. Math. Sci., 16 (2009), 1-8.  Google Scholar

[21]

Commun. Math. Phys., 299 (2010), 409-446. 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]

Duke Math. J., 86 (1997), 347-389. doi: 10.1215/S0012-7094-97-08611-7.  Google Scholar

[25]

Experiment. Math., 1 (1992), 71-81.  Google Scholar

[26]

Experiment. Math., 10 (2001), 519-528.  Google Scholar

[27]

J. Fixed Point Theory Appl., 3 (2008), 379-409. doi: 10.1007/s11784-008-0079-0.  Google Scholar

[28]

Math. Intelligencer, 32 (2010), 31-34. doi: 10.1007/s00283-010-9137-8.  Google Scholar

[29]

Electron. J. Comb., 18 (2011), 171. Google Scholar

[30]

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

[31]

J. Phys. A, 30 (1997), 2235-2249. doi: 10.1088/0305-4470/30/6/041.  Google Scholar

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