Mathematische Zeitschrift

, Volume 281, Issue 3–4, pp 1061–1087 | Cite as

Introducing supersymmetric frieze patterns and linear difference operators

  • Sophie Morier-Genoud
  • Valentin OvsienkoEmail author
  • Serge Tabachnikov


We introduce a supersymmetric analog of the classical Coxeter frieze patterns. Our approach is based on the relation with linear difference operators. We define supersymmetric analogs of linear difference operators called Hill’s operators. The space of these “superfriezes” is an algebraic supervariety, isomorphic to the space of supersymmetric second order difference equations, called Hill’s equations.


Supercommutative algebra Frieze pattern Difference equation Cluster algebra 



The first three authors would like to thank the Centro Internazionale per la Ricerca Matematica, the Mathematics Department of the University of Trento and the foundation Bruno Kessler for excellent conditions they offered us. We are pleased to thank Frederic Chapoton and Dimitry Leites for interesting discussions, special thanks to Dimitry for a careful reading of the first version of this paper. S. M-G. and V. O. are grateful to the Institute for Computational and Experimental Research in Mathematics for its hospitality. S. M-G. and V. O. were partially supported by the PICS05974 “PENTAFRIZ” of CNRS. S. T. was supported by NSF grant DMS-1105442. A. U.’s research was supported by Russian Science Foundation (Project N 14-11-00335).


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Sophie Morier-Genoud
    • 1
  • Valentin Ovsienko
    • 2
    Email author
  • Serge Tabachnikov
    • 3
    • 4
  1. 1.UPMC Univ Paris 06, UMR 7586, Institut de Mathématiques de Jussieu- Paris Rive Gauche, Case 247Sorbonne UniversitésParisFrance
  2. 2.Laboratoire de Mathématiques U.F.R. Sciences Exactes et NaturellesCNRSREIMS cedex 2France
  3. 3.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA
  4. 4.ICERMBrown UniversityProvidenceUSA

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