Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T04:12:36.715Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear difference equations, frieze patterns, and the combinatorial Gale transform

Part of: Difference and functional equations Arithmetic rings and other special rings Polytopes and polyhedra Special varieties Difference equations

Published online by Cambridge University Press:  22 August 2014

SOPHIE MORIER-GENOUD ,
VALENTIN OVSIENKO ,
RICHARD EVAN SCHWARTZ  and
SERGE TABACHNIKOV
SOPHIE MORIER-GENOUD
Affiliation:
Sorbonne Universités, UPMC Univ Paris 06, UMR 7586, Institut de Mathématiques de Jussieu-Paris Rive Gauche, Case 247, 4 place Jussieu, F-75005, Paris, France
VALENTIN OVSIENKO
Affiliation:
CNRS, Laboratoire de Mathématiques, U.F.R. Sciences Exactes et Naturelles, Moulin de la Housse - BP 1039, 51687 REIMS Cedex 2, France
RICHARD EVAN SCHWARTZ
Affiliation:
Department of Mathematics, Brown University, Providence, RI 02912, USA
SERGE TABACHNIKOV
Affiliation:
Pennsylvania State University, Department of Mathematics, University Park, PA 16802, USA ICERM, Brown University, Box 1995, Providence, RI 02912, USA; tabachni@math.psu.edu

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the space of linear difference equations with periodic coefficients and (anti)periodic solutions. We show that this space is isomorphic to the space of tame frieze patterns and closely related to the moduli space of configurations of points in the projective space. We define the notion of a combinatorial Gale transform, which is a duality between periodic difference equations of different orders. We describe periodic rational maps generalizing the classical Gauss map.

MSC classification

Secondary: 39A70: Difference operators 14M15: Grassmannians, Schubert varieties, flag manifolds 13F60: Cluster algebras 52B99: None of the above, but in this section
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 2 , 01 July 2014 , e22
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2014

References

André, D., ‘Terme général d’un série quelconque déterminée à la façon des séries récurrentes’, Ann. Sci. Éc. Norm. Super. 7 (2) (1878), 375–408.CrossRefGoogle Scholar
Assem, I., Reutenauer, C. and Smith, D., ‘Friezes’, Adv. Math. 225 (2010), 3134–3165.CrossRefGoogle Scholar
Berenstein, A., Fomin, S. and Zelevinsky, A., ‘Parametrizations of canonical bases and totally positive matrices’, Adv. Math. 122 (1996), 49–149.Google Scholar
Bergeron, F. and Reutenauer, C., ‘SL k -tilings of the plane’, Illinois J. Math. 54 (2010), 263–300.CrossRefGoogle Scholar
Caldero, P. and Chapoton, F., ‘Cluster algebras as Hall algebras of quiver representations’, Comment. Math. Helv. 81 (2006), 595–616.CrossRefGoogle Scholar
Coble, A. B., ‘Associated sets of points’, Trans. Amer. Math. Soc. 24 (1922), 1–20.Google Scholar
Coble, A. B., Algebraic Geometry and Theta Functions, vol. X. (AMS, Providence, RI, 1961).Google Scholar
Conway, J. H. and Coxeter, H. S. M., ‘Triangulated polygons and frieze patterns’, Math. Gaz. 57 (1973), 87–94 and 175–183.CrossRefGoogle Scholar
Coxeter, H. S. M., ‘Frieze patterns’, Acta Arith. 18 (1971), 297–310.CrossRefGoogle Scholar
Davis, Ph., Circulant Matrices, (John Wiley & Sons, New York–Chichester–Brisbane, 1979).Google Scholar
Di Francesco, P. and Kedem, R., ‘ T-systems with boundaries from network solutions’, Electron. J. Combin. 20 (1) (2013), 62.Google Scholar
Dolgachev, I. and Ortland, D., ‘Points sets in projective spaces and theta functions’, Astérisque 165 (1988).Google Scholar
Dubrovin, B., Fomenko, A. and Novikov, S., ‘Modern geometry–methods and applications. Part I’, inThe Geometry of Surfaces, Transformation Groups, and Fields, 2nd edn, (Springer-Verlag, New York, 1992).Google Scholar
Eisenbud, D. and Popescu, S., ‘The projective geometry of the Gale transform’, J. Algebra 220 (2000), 127–173.Google Scholar
Fuchs, D. and Tabachnikov, S., ‘Self-dual polygons and self-dual curves’, Funct. Anal. Other Math. 2 (2009), 203–220.CrossRefGoogle Scholar
Fulton, W., Young tableaux, with applications to representation theory and geometry, London Mathematical Society Student Texts 35 (Cambridge University Press, Cambridge, 1997).Google Scholar
Gale, D., ‘Neighboring vertices on a convex polyhedron’, inLinear Inequalities and Related Systems, (eds Kuhn, H. W. and Tucker, A. W.), Annals of Math Studies 38 (Princeton University Press, Princeton, NJ, 1956), 255–263.Google Scholar
Gauss, C. F., ‘Pentagramma Mirificum’, Werke, Bd. III, 481–490; Bd VIII 106–111.Google Scholar
Gekhtman, M., Shapiro, M., Tabachnikov, S. and Vainshtein, A., ‘Higher pentagram maps, weighted directed networks, and cluster dynamics’, Electron. Res. Announc. Math. Sci. 19 (2012), 1–17.Google Scholar
Gekhtman, M., Shapiro, M., Tabachnikov, S. and Vainshtein, A., ‘Integrable cluster dynamics of directed networks and pentagram maps’, arXiv:14061883.Google Scholar
Gelfand, I. M. and MacPherson, R. D., ‘Geometry in Grassmannians and a generalization of the dilogarithm’, Adv. Math. 44 (1982), 279–312.CrossRefGoogle Scholar
Jordan, C., Calculus of Finite Differences, 3rd edn, (Chelsea Publishing Co, New York, 1965).Google Scholar
Kapranov, M., ‘Chow quotients of Grassmannians, I’, inI. M. Gel’fand Seminar, Advances in Soviet Math. 16 Part 2 (Amer. Math. Soc., Providence, RI, 1993), 29–110.Google Scholar
Keller, B., ‘The periodicity conjecture for pairs of Dynkin diagrams’, Ann. of Math. (2) 177 (2013), 111–170.CrossRefGoogle Scholar
Khesin, B. and Soloviev, F., ‘Integrability of a space pentagram map’, Math. Ann. 357 (2013), 1005–1047.CrossRefGoogle Scholar
Khesin, B. and Soloviev, F., ‘The geometry of dented pentagram maps’, J. Eur. Math. Soc. (to appear).Google Scholar
Krichever, I., ‘Commuting difference operators and the combinatorial Gale transform’,arXiv:14034629.Google Scholar
Krichever, I. and Novikov, S., ‘A two-dimensionalized Toda chain, commuting difference operators, and holomorphic vector bundles’, Russian Math. Surveys 58 (2003), 473–510.Google Scholar
Mari-Beffa, G., ‘On generalizations of the pentagram map: discretizations of AGD flows’, J. Nonlinear Sci. 23 (2013), 303–334.Google Scholar
Mari-Beffa, G., ‘On integrable generalizations of the pentagram map’, IMRN (to appear).Google Scholar
Morier-Genoud, S., ‘Arithmetics of 2-friezes’, J. Algebraic Combin. 36 (2012), 515–539.Google Scholar
Morier-Genoud, S., Ovsienko, V. and Tabachnikov, S., ‘2-Frieze patterns and the cluster structure of the space of polygons’, Ann. Inst. Fourier 62 (2012), 937–987.CrossRefGoogle Scholar
Ovsienko, V. and Tabachnikov, S., Projective Differential Geometry Old and New: From the Schwarzian Derivative to the Cohomology of Diffeomorphism Groups, (Cambridge University Press, Cambridge, 2005).Google Scholar
Ovsienko, V., Schwartz, R. and Tabachnikov, S., ‘The pentagram map: a discrete integrable system’, Comm. Math. Phys. 299 (2) (2010), 409–446.Google Scholar
Ovsienko, V., Schwartz, R.  and Tabachnikov, S. , ‘Liouville–Arnold integrability of the pentagram map on closed polygons’, Duke Math. J. 162 (2013), 2149–2196.Google Scholar
Penner, R., Decorated Teichmüller Theory, (European Mathematical Society, 2012).CrossRefGoogle Scholar
Penner, R., ‘Lambda lengths’. http://wwwctqmaudk/research/MCS/lambdalengthspdf.Google Scholar
Propp, J., $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\! $ ‘The combinatorics of frieze patterns and Markoff numbers’, arXiv:math/0511633.Google Scholar
Schechtman, V., ‘Pentagramma Mirificum and elliptic functions’, arXiv:11063633.Google Scholar
Schwartz, R., ‘The pentagram map’, Experiment. Math. 1 (1992), 71–81.Google Scholar
Schwartz, R., ‘Discrete monodromy, pentagrams, and the method of condensation’, J. Fixed Point Theory Appl. 3 (2008), 379–409.Google Scholar
Schwartz, R., ‘Pentagram spirals’, Experiment. Math. 22 (2013), 384–405.Google Scholar
Soloviev, F., ‘Integrability of the Pentagram map’, Duke Math. J. 162 (2013), 2815–2853.CrossRefGoogle Scholar
Springer, T. A., Linear algebraic groups, Progress in Mathematics , vol. 9. (Birkhäuser Boston, Inc, Boston, MA, 1998).Google Scholar
Volkov, A. Y., ‘On the periodicity conjecture for Y-systems’, Comm. Math. Phys. 276 (2007), 509–517.CrossRefGoogle Scholar
Zabrodin, A., ‘Discrete Hirotaś equation in quantum integrable models’, Internat. J. Modern Phys. B 11 (1997), 3125–3158.Google Scholar