Dessins d'enfants, Brauer graph algebras and Galois invariants

TL;DR

This paper explores the connection between dessins d'enfants and Brauer graph algebras, demonstrating Galois invariance of certain algebraic properties and establishing derived equivalences among Galois conjugates.

Contribution

It introduces a method to associate Brauer graph algebras to dessins d'enfants and proves Galois invariants in their algebraic structures.

Findings

Galois conjugate dessins yield derived equivalent Brauer graph algebras

The stable Auslander-Reiten quiver is Galois invariant

The dimension of the Brauer graph algebra remains invariant under Galois action

Abstract

In this paper, we associate a finite dimensional algebra, called a Brauer graph algebra, to every clean dessin d'enfant by constructing a quiver based on the monodromy of the dessin. We show that Galois conjugate dessins d'enfants give rise to derived equivalent Brauer graph algebras and that the stable Auslander-Reiten quiver and the dimension of the Brauer graph algebra are invariant under the induced action of the absolute Galois group.