Conway-Coxeter friezes and beyond: Polynomially weighted walks around dissected polygons and generalized frieze patterns

TL;DR

This paper extends the theory of frieze patterns by introducing polynomially weighted walks around dissected polygons, deriving explicit determinant formulas, symmetry conditions, and diagonal forms that generalize previous results in cluster algebra contexts.

Contribution

It generalizes frieze pattern theory to arbitrary polygon dissections with polynomial weights, providing explicit determinant formulas and normal forms.

Findings

Determinant of the generalized frieze table is a multisymmetric polynomial.

The frieze matrix can be transformed into a diagonal form over Laurent polynomials.

Non-zero local determinants are explicitly given monomials depending on polygon geometry.

Abstract

Conway and Coxeter introduced frieze patterns in 1973 and classified them via triangulated polygons. The determinant of the matrix associated to a frieze table was computed explicitly by Broline, Crowe and Isaacs in 1974, a result generalized 2012 by Baur and Marsh in the context of cluster algebras of type A. Higher angulations of polygons and associated generalized frieze patterns were studied in a joint paper with Holm and J\o rgensen. Here we take these results further; we allow arbitrary dissections and introduce polynomially weighted walks around such dissected polygons. The corresponding generalized frieze table satisfies a complementary symmetry condition; its determinant is a multisymmetric multivariate polynomial that is given explicitly. But even more, the frieze matrix may be transformed over a ring of Laurent polynomials to a nice diagonal form generalizing the Smith normal form result given in an earlier joint paper with Holm and J\o rgensen. Considering the generalized polynomial frieze in this context it is also shown that the non-zero local determinants are monomials that are given explicitly, depending on the geometry of the dissected polygon.