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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Weak friezes and frieze pattern determinants
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by Thorsten Holm and Peter Jørgensen
Proc. Amer. Math. Soc. 152 (2024), 1479-1491
Published electronically: February 2, 2024


Frieze patterns have been introduced by Coxeter [Acta Arith. 18 (1971), pp. 297–310] in the 1970’s and have recently attracted renewed interest due to their close connection with Fomin-Zelevinsky’s cluster algebras. Frieze patterns can be interpreted as assignments of values to the diagonals of a triangulated polygon satisfying certain conditions for crossing diagonals (Ptolemy relations). Weak friezes, as introduced by Çanakçı and Jørgensen [Adv. in Appl. Math. 131 (2021), Paper No. 102253], are generalizing this concept by allowing to glue dissected polygons so that the Ptolemy relations only have to be satisfied for crossings involving one of the gluing diagonals.

To any frieze pattern one can associate a symmetric matrix using a triangular fundamental domain of the frieze pattern in the upper and lower half of the matrix and putting zeroes on the diagonal. Broline, Crowe and Isaacs [Geometriae Dedicata 3 (1974), pp. 171–176] have found a formula for the determinants of these matrices and their work has later been generalized in various directions by other authors. These frieze pattern determinants are the main focus of our paper. As our main result we show that this determinant behaves well with respect to gluing weak friezes: the determinant is the product of the determinants for the pieces glued, up to a scalar factor coming from the gluing diagonal. Then we give several applications of this result, showing that formulas from the literature, obtained by Broline-Crowe-Isaacs, Baur-Marsh [J. Combin. Theory Ser. A 119 (2012), pp. 1110–1122], Bessenrodt-Holm-Jørgensen [J. Combin. Theory Ser. A 123 (2014), pp. 30–42] and Maldonado [Frieze matrices and infinite frieze patterns with coefficients, Preprint, arXiv:2207.04120, 2022] can all be obtained as consequences of our result.

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Bibliographic Information
  • Thorsten Holm
  • Affiliation: Leibniz Universität Hannover, Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Fakultät für Mathematik und Physik, Welfengarten 1, D-30167 Hannover, Germany
  • MR Author ID: 357945
  • ORCID: 0000-0003-1840-0304
  • Email:
  • Peter Jørgensen
  • Affiliation: Department of Mathematics, Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark
  • MR Author ID: 601258
  • Email:
  • Received by editor(s): January 11, 2023
  • Received by editor(s) in revised form: August 21, 2023
  • Published electronically: February 2, 2024
  • Communicated by: Jerzy Weyman
  • © Copyright 2024 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 152 (2024), 1479-1491
  • MSC (2020): Primary 05B45, 05E99, 13F60, 15A15, 51M20, 52B45
  • DOI:
  • MathSciNet review: 4709220