Finite and infinite frieze patterns from p-angulations and a generalization of Weyl groupoids

TL;DR

This paper generalizes Conway-Coxeter frieze patterns to p-angulations, introduces a combinatorial algorithm for their entries, and extends the theory to infinite patterns and Weyl groupoids, revealing deep combinatorial and algebraic structures.

Contribution

It presents a new combinatorial algorithm involving Chebyshev polynomials for frieze pattern entries and establishes a bijection between infinite frieze patterns and p-angulations, extending classical results.

Findings

A combinatorial algorithm for frieze pattern entries using Chebyshev polynomials.

Characterization of frieze patterns of types Λ₄ and Λ₆.

A bijection between infinite frieze patterns and p-angulations of an infinite strip.

Abstract

A classic result of Conway and Coxeter on frieze patterns has been generalized to a bijection between $p$-angulations of regular polygons and frieze patterns of type $\Lambda_p$. One of the features of Conway-Coxeter theory is a combinatorial procedure to obtain from the triangulation all entries of the corresponding frieze pattern. We first present a combinatorial algorithm, involving Chebyshev polynomials, for obtaining from a dissection all entries of the corresponding frieze pattern. As an application we obtain a characterisation of frieze patterns of types $\Lambda_4$ and $\Lambda_6$ in terms of all entries (not only the quiddity cycle). We then study infinite frieze patterns of type $\Lambda_p$, which appeared in a preprint by Banaian and Chen, generalizing the infinite frieze patterns of positive integers studied by Baur, Parsons and Tschabold. As our main result we obtain a combinatorial model for infinite frieze patterns of type $\Lambda_p$, these are in bijection with certain $p$-angulations of an infinite strip. This extends results by Baur, Parsons and Tschabold from $p=3$ to arbitrary $p\ge 3$, and also provides new insight in the classic case. Infinite frieze patterns of positive integers appear in the context of Weyl groupoids. In the final section we extend this to infinite frieze patterns of type $\Lambda_p$ for any $p\ge 3$ by introducing a generalization of Cartan graphs and Weyl groupoids. We show that, up to equivalence, there is a 1-1 correspondence between connected simply connected Cartan graphs of type $\Lambda_p$ of rank two with infinitely many vertices permitting a root system and infinite frieze patterns of type $\Lambda_p$.