Entiers monomialement irr{\'e}ductibles

TL;DR

This paper investigates the combinatorics of congruence subgroups of the modular group, focusing on minimal monomial solutions of matrix equations modulo N and their irreducibility properties.

Contribution

It introduces the concept of minimal monomial solutions in the context of congruence subgroups and analyzes the integers N for which these solutions are irreducible.

Findings

Characterization of integers N with irreducible minimal monomial solutions

Connection between matrix equations and Coxeter friezes

Insights into the structure of congruence subgroups

Abstract

In this article we study the combinatorics of congruence subgroups of the modular group. We consider the notion of minimal monomial solutions. These are the solutions of a matrix equation (also appearing in the study of Coxeter friezes), modulo an integer $N$, whose components are identical and minimal for this property. Our objective is to study the integers $N$ for which all these solutions have a certain irreducibility property.