Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 089, 36 pages      arXiv:1803.06001      https://doi.org/10.3842/SIGMA.2019.089
Contribution to the Special Issue on Algebra, Topology, and Dynamics in Interaction in honor of Dmitry Fuchs

Symplectic Frieze Patterns

Sophie Morier-Genoud
Sorbonne Université, Université Paris Diderot, CNRS, Institut de Mathé-matiquesde Jussieu-Paris Rive Gauche, IMJ-PRG, F-75005, Paris, France

Received June 18, 2019, in final form November 07, 2019; Published online November 14, 2019

Abstract
We introduce a new class of friezes which is related to symplectic geometry. On the algebraic and combinatrics sides, this variant of friezes is related to the cluster algebras involving the Dynkin diagrams of type ${\rm C}_{2}$ and ${\rm A}_{m}$. On the geometric side, they are related to the moduli space of Lagrangian configurations of points in the 4-dimensional symplectic space introduced in [Conley C.H., Ovsienko V., Math. Ann. 375 (2019), 1105-1145]. Symplectic friezes share similar combinatorial properties to those of Coxeter friezes and SL-friezes.

Key words: frieze; cluster algebra; moduli space; difference equation; Lagrangian configuration.

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References

  1. Assem I., Dupont G., Schiffler R., Smith D., Friezes, strings and cluster variables, Glasg. Math. J. 54 (2012), 27-60, arXiv:1009.3341.
  2. Assem I., Reutenauer C., Smith D., Friezes, Adv. Math. 225 (2010), 3134-3165, arXiv:0906.2026.
  3. Baur K., Marsh R.J., Frieze patterns for punctured discs, J. Algebraic Combin. 30 (2009), 349-379, arXiv:1008.5329.
  4. Bergeron F., Reutenauer C., ${\rm SL}_k$-tilings of the plane, Illinois J. Math. 54 (2010), 263-300, arXiv:1002.1089.
  5. Bessenrodt C., Holm T., Jørgensen P., All ${\rm SL}_2$-tilings come from infinite triangulations, Adv. Math. 315 (2017), 194-245, arXiv:1603.09103.
  6. Conley C.H., Ovsienko V., Lagrangian configurations and symplectic cross-ratios, Math. Ann. 375 (2019), 1105-1145, arXiv:1812.04271.
  7. Conway J.H., Coxeter H.S.M., Triangulated polygons and frieze patterns, Math. Gaz. 57 (1973), 175-183.
  8. Conway J.H., Guy R.K., The book of numbers, Copernicus, New York, 1996.
  9. Coxeter H.S.M., Frieze patterns, Acta Arith. 18 (1971), 297-310.
  10. Cuntz M., Holm T., Frieze patterns over integers and other subsets of the complex numbers, J. Comb. Algebra 3 (2019), 153-188, arXiv:1711.03724.
  11. Dupont G., An approach to non-simply laced cluster algebras, J. Algebra 320 (2008), 1626-1661, arXiv:math.RT/0512043.
  12. Fomin S., Williams L., Zelevinsky A., Introduction to cluster algebras, Chapters 1-3, arXiv:1608.05735.
  13. Fomin S., Zelevinsky A., Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), 497-529, arXiv:math.RT/0104151.
  14. Fomin S., Zelevinsky A., Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), 63-121, arXiv:math.RA/0208229.
  15. Fomin S., Zelevinsky A., Cluster algebras. IV. Coefficients, Compos. Math. 143 (2007), 112-164,arXiv:math.RA/0602259.
  16. Fontaine B., Plamondon P.G., Counting friezes in type $D_n$, J. Algebraic Combin. 44 (2016), 433-445, arXiv:1409.3698.
  17. Fuchs D., Tabachnikov S., Self-dual polygons and self-dual curves, Funct. Anal. Other Math. 2 (2009), 203-220, arXiv:0707.1048.
  18. Keller B., Quiver mutation in java, available at https://webusers.imj-prg.fr/~bernhard.keller/.
  19. Keller B., Cluster algebras, quiver representations and triangulated categories, in Triangulated Categories, London Math. Soc. Lecture Note Ser., Vol. 375, Cambridge University Press, Cambridge, 2010, 76-160, arXiv:0807.1960.
  20. Keller B., The periodicity conjecture for pairs of Dynkin diagrams, Ann. of Math. 177 (2013), 111-170, arXiv:1001.1531.
  21. Krichever I.M., Commuting difference operators and the combinatorial Gale transform, Funct. Anal. Appl. 49 (2015), 175-188, arXiv:1403.4629.
  22. Morier-Genoud S., Arithmetics of 2-friezes, J. Algebraic Combin. 36 (2012), 515-539, arXiv:1109.0917.
  23. Morier-Genoud S., Coxeter's frieze patterns at the crossroads of algebra, geometry and combinatorics, Bull. Lond. Math. Soc. 47 (2015), 895-938, arXiv:1503.05049.
  24. Morier-Genoud S., Ovsienko V., Schwartz R.E., Tabachnikov S., Linear difference equations, frieze patterns, and the combinatorial Gale transform, Forum Math. Sigma 2 (2014), e22, 45 pages, arXiv:1309.3880.
  25. Morier-Genoud S., Ovsienko V., Tabachnikov S., 2-frieze patterns and the cluster structure of the space of polygons, Ann. Inst. Fourier (Grenoble) 62 (2012), 937-987, arXiv:1008.3359.
  26. Ovsienko V., Partitions of unity in ${\rm SL}(2,{\mathbb Z})$, negative continued fractions, and dissections of polygons, Res. Math. Sci. 5 (2018), 21, 25 pages, arXiv:1710.02996.
  27. Tschabold M., Arithmetic infinite friezes from punctured discs, arXiv:1503.04352.

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