Noncommutative frieze patterns with coefficients

Cuntz, Michael; Holm, Thorsten; Jørgensen, Peter

doi:10.2140/ant.2026.20.1877

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Noncommutative frieze patterns with coefficients

Michael Cuntz, Thorsten Holm and Peter Jørgensen

Vol. 20 (2026), No. 9, 1877–1899

DOI: 10.2140/ant.2026.20.1877

Abstract

Based on Berenstein and Retakh’s notion of noncommutative polygons, we introduce and study noncommutative frieze patterns. We generalize several notions and fundamental properties from the classic (commutative) frieze patterns to noncommutative frieze patterns, e.g., propagation formulae and $μ$ -matrices, quiddity cycles and reduction formulae, and we show that local noncommutative exchange relations and local triangle relations imply all noncommutative exchange relations and triangle relations. Throughout, we allow coefficients, so we obtain generalizations of earlier results of ours from the commutative to the noncommutative setting.

Keywords

frieze pattern, tame frieze pattern, quiddity cycle, quasideterminant, noncommutative polygon

Mathematical Subject Classification

Primary: 05E99, 13F60, 51M20

Milestones

Received: 10 September 2024

Revised: 30 June 2025

Accepted: 5 August 2025

Published: 19 October 2026

Authors

Michael Cuntz
Institut für Algebra, Zahlentheorie und Diskrete Mathematik Leibniz Universität Hannover Hannover Germany
https://www.iazd.uni-hannover.de/de/cuntz
Thorsten Holm
Institut für Algebra, Zahlentheorie und Diskrete Mathematik Leibniz Universität Hannover Hannover Germany
https://www.iazd.uni-hannover.de/de/holm
Peter Jørgensen
Department of Mathematics Aarhus University Aarhus Denmark
https://sites.google.com/view/peterjorgensen

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Vol. 20, No. 9, 2026