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

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

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

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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

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