Higher pentagram maps, weighted directed networks, and cluster dynamics

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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

Abstract
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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), 525-566. doi: 10.1215/S0012-7094-03-11635-X.
[2]	A. Bobenko and Yu. Suris, "Discrete Differential Geometry. Integrable Structure," Amer. Math. Soc., Providence, RI, 2008.
[3]	H. Busemann and P. Kelly, "Projective geometry and projective metrics," Academic Press, New York, 1953.
[4]	B. Dubrovin, I. Krichever and S. Novikov, Integrable systems. I, in "Dynamical Systems," IV, Encyclopaedia Math. Sci., 4, Springer, Berlin, (2001), 177-332.
[5]	L. Faddeev and L. Takhtajan, "Hamiltonian Methods in the Theory of Solitons," Springer-Verlag, Berlin, 1987.
[6]	S. Fomin and A. Zelevinsky, Cluster algebras. IV. Coefficients, Compos. Math., 143 (2007), 112-164. doi: 10.1112/S0010437X06002521.
[7]	M. Gekhtman, M. Shapiro and A. Vainshtein, Poisson geometry of directed networks in a disk, Selecta Math., 15 (2009), 61-103. doi: 10.1007/s00029-009-0523-z.
[8]	M. Gekhtman, M. Shapiro and A. Vainshtein, "Cluster Algebras and Poisson Geometry," Amer. Math. Soc., Providence, RI, 2010.
[9]	M. Gekhtman, M. Shapiro and A. Vainshtein, Poisson geometry of directed networks in an annulus, J. Europ. Math. Soc., preprint, arXiv:0901.0020.
[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), 245-310. doi: 10.1007/s11511-011-0063-1.
[11]	M. Glick, The pentagram map and $Y$-patterns, Adv. Math., 227 (2011), 1019-1045. doi: 10.1016/j.aim.2011.02.018.
[12]	, M. Glick, private communication.
[13]	A. Goncharov and R. Kenyon, Dimers and cluster integrable systems, preprint, arXiv:1107.5588.
[14]	B. Khesin and F. Soloviev, Integrability of a space pentagram map, in preparation.
[15]	B. Konopelchenko and W. Schief, Menelaus' theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy, J. Phys. A, 35 (2002), 6125-6144. doi: 10.1088/0305-4470/35/29/313.
[16]	G. Mari Beffa, On generalizations of the pentagram map: Discretizations of AGD flows, preprint, arXiv:1103.5047.
[17]	S. Morier-Genoud, V. Ovsienko and S. Tabachnikov, 2-frieze patterns and the cluster structure of the space of polygons, Ann. Inst. Fourier, preprint, arXiv:1008.3359.
[18]	F. Nijhoff and H. Capel, The discrete Korteweg-de Vries equation, Acta Appl. Math., 39 (1995), 133-158. doi: 10.1007/BF00994631.
[19]	M. Olshanetsky, A. Perelomov, A. Reyman and M. Semenov-Tian-Shansky, Integrable systems. II, in "Dynamical systems," VII, Encycl. Math. Sci., 16, Springer, Berlin, (1994), 83-259.
[20]	V. Ovsienko, R. Schwartz and S. Tabachnikov, Quasiperiodic motion for the Pentagram map, Electron. Res. Announc. Math. Sci., 16 (2009), 1-8.
[21]	V. Ovsienko, R. Schwartz and S. Tabachnikov, The Pentagram map: A discrete integrable system, Commun. Math. Phys., 299 (2010), 409-446. doi: 10.1007/s00220-010-1075-y.
[22]	V. Ovsienko, R. Schwartz and S. Tabachnikov, Liouville-Arnold integrability of the pentagram map on closed polygons, preprint, arXiv:1107.3633.
[23]	A. Postnikov, Total positivity, Grassmannians, and networks, preprint, arXiv:math.CO/0609764.
[24]	O. Schramm, Circle patterns with the combinatorics of the square grid, Duke Math. J., 86 (1997), 347-389. doi: 10.1215/S0012-7094-97-08611-7.
[25]	R. Schwartz, The pentagram map, Experiment. Math., 1 (1992), 71-81.
[26]	R. Schwartz, The pentagram map is recurrent, Experiment. Math., 10 (2001), 519-528.
[27]	R. Schwartz, Discrete monodromy, pentagrams, and the method of condensation, J. Fixed Point Theory Appl., 3 (2008), 379-409. doi: 10.1007/s11784-008-0079-0.
[28]	R. Schwartz and S. Tabachnikov, Elementary surprises in projective geometry, Math. Intelligencer, 32 (2010), 31-34. doi: 10.1007/s00283-010-9137-8.
[29]	R. Schwartz and S. Tabachnikov, The pentagram integrals on inscribed polygons, Electron. J. Comb., 18 (2011), 171.
[30]	F. Soloviev, Integrability of the pentagram map, preprint, arXiv:1106.3950.
[31]	Yu. Suris, On some integrable systems related to the Toda lattice, J. Phys. A, 30 (1997), 2235-2249. doi: 10.1088/0305-4470/30/6/041.

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), 525-566. doi: 10.1215/S0012-7094-03-11635-X.
[2]	A. Bobenko and Yu. Suris, "Discrete Differential Geometry. Integrable Structure," Amer. Math. Soc., Providence, RI, 2008.
[3]	H. Busemann and P. Kelly, "Projective geometry and projective metrics," Academic Press, New York, 1953.
[4]	B. Dubrovin, I. Krichever and S. Novikov, Integrable systems. I, in "Dynamical Systems," IV, Encyclopaedia Math. Sci., 4, Springer, Berlin, (2001), 177-332.
[5]	L. Faddeev and L. Takhtajan, "Hamiltonian Methods in the Theory of Solitons," Springer-Verlag, Berlin, 1987.
[6]	S. Fomin and A. Zelevinsky, Cluster algebras. IV. Coefficients, Compos. Math., 143 (2007), 112-164. doi: 10.1112/S0010437X06002521.
[7]	M. Gekhtman, M. Shapiro and A. Vainshtein, Poisson geometry of directed networks in a disk, Selecta Math., 15 (2009), 61-103. doi: 10.1007/s00029-009-0523-z.
[8]	M. Gekhtman, M. Shapiro and A. Vainshtein, "Cluster Algebras and Poisson Geometry," Amer. Math. Soc., Providence, RI, 2010.
[9]	M. Gekhtman, M. Shapiro and A. Vainshtein, Poisson geometry of directed networks in an annulus, J. Europ. Math. Soc., preprint, arXiv:0901.0020.
[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), 245-310. doi: 10.1007/s11511-011-0063-1.
[11]	M. Glick, The pentagram map and $Y$-patterns, Adv. Math., 227 (2011), 1019-1045. doi: 10.1016/j.aim.2011.02.018.
[12]	, M. Glick, private communication.
[13]	A. Goncharov and R. Kenyon, Dimers and cluster integrable systems, preprint, arXiv:1107.5588.
[14]	B. Khesin and F. Soloviev, Integrability of a space pentagram map, in preparation.
[15]	B. Konopelchenko and W. Schief, Menelaus' theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy, J. Phys. A, 35 (2002), 6125-6144. doi: 10.1088/0305-4470/35/29/313.
[16]	G. Mari Beffa, On generalizations of the pentagram map: Discretizations of AGD flows, preprint, arXiv:1103.5047.
[17]	S. Morier-Genoud, V. Ovsienko and S. Tabachnikov, 2-frieze patterns and the cluster structure of the space of polygons, Ann. Inst. Fourier, preprint, arXiv:1008.3359.
[18]	F. Nijhoff and H. Capel, The discrete Korteweg-de Vries equation, Acta Appl. Math., 39 (1995), 133-158. doi: 10.1007/BF00994631.
[19]	M. Olshanetsky, A. Perelomov, A. Reyman and M. Semenov-Tian-Shansky, Integrable systems. II, in "Dynamical systems," VII, Encycl. Math. Sci., 16, Springer, Berlin, (1994), 83-259.
[20]	V. Ovsienko, R. Schwartz and S. Tabachnikov, Quasiperiodic motion for the Pentagram map, Electron. Res. Announc. Math. Sci., 16 (2009), 1-8.
[21]	V. Ovsienko, R. Schwartz and S. Tabachnikov, The Pentagram map: A discrete integrable system, Commun. Math. Phys., 299 (2010), 409-446. doi: 10.1007/s00220-010-1075-y.
[22]	V. Ovsienko, R. Schwartz and S. Tabachnikov, Liouville-Arnold integrability of the pentagram map on closed polygons, preprint, arXiv:1107.3633.
[23]	A. Postnikov, Total positivity, Grassmannians, and networks, preprint, arXiv:math.CO/0609764.
[24]	O. Schramm, Circle patterns with the combinatorics of the square grid, Duke Math. J., 86 (1997), 347-389. doi: 10.1215/S0012-7094-97-08611-7.
[25]	R. Schwartz, The pentagram map, Experiment. Math., 1 (1992), 71-81.
[26]	R. Schwartz, The pentagram map is recurrent, Experiment. Math., 10 (2001), 519-528.
[27]	R. Schwartz, Discrete monodromy, pentagrams, and the method of condensation, J. Fixed Point Theory Appl., 3 (2008), 379-409. doi: 10.1007/s11784-008-0079-0.
[28]	R. Schwartz and S. Tabachnikov, Elementary surprises in projective geometry, Math. Intelligencer, 32 (2010), 31-34. doi: 10.1007/s00283-010-9137-8.
[29]	R. Schwartz and S. Tabachnikov, The pentagram integrals on inscribed polygons, Electron. J. Comb., 18 (2011), 171.
[30]	F. Soloviev, Integrability of the pentagram map, preprint, arXiv:1106.3950.
[31]	Yu. Suris, On some integrable systems related to the Toda lattice, J. Phys. A, 30 (1997), 2235-2249. doi: 10.1088/0305-4470/30/6/041.

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