Hecke Groups, Dessins d'Enfants and the Archimedean Solids

TL;DR

This paper explores the relationship between dessins d'enfants, Hecke groups, and Archimedean solids, providing computational methods to identify conjugacy classes of Hecke subgroups linked to highly symmetric dessins in mathematical physics.

Contribution

It introduces a novel approach to compute representatives of conjugacy classes of Hecke subgroups associated with dessins corresponding to Archimedean solids.

Findings

Computed conjugacy class representatives for Hecke subgroups

Linked dessins d'enfants to Archimedean solids in physics

Provided tools for further mathematical physics research

Abstract

Grothendieck's dessins d'enfants arise with ever-increasing frequency in many areas of 21st century mathematical physics. In this paper, we review the connections between dessins and the theory of Hecke groups. Focussing on the restricted class of highly symmetric dessins corresponding to the so-called Archimedean solids, we apply this theory in order to provide a means of computing representatives of the associated conjugacy classes of Hecke subgroups in each case. The aim of this paper is to demonstrate that dessins arising in mathematical physics can point to new and hitherto unexpected directions for further research. In addition, given the particular ubiquity of many of the dessins corresponding to the Archimedean solids, the hope is that the computational results of this paper will prove useful in the further study of these objects in mathematical physics contexts.