  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.
