Weak friezes and frieze pattern determinants

TL;DR

This paper studies the determinants of matrices associated with frieze patterns, especially weak friezes, showing how these determinants behave under gluing operations and deriving various known formulas as consequences.

Contribution

It establishes a new property of frieze pattern determinants under gluing of weak friezes, unifying and extending previous results in the literature.

Findings

Determinant of frieze matrices multiplies under gluing of weak friezes.

The main result generalizes previous formulas for determinants by Broline-Crowe-Isaacs and others.

Several known determinant formulas are derived as corollaries of the main theorem.

Abstract

Frieze patterns have been introduced by Coxeter 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 Canakci and Jorgensen, 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 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, Bessenrodt-Holm-Jorgensen and Maldonado can all be obtained as consequences of our result.