Dessin d'enfants, groups and representations

In this series of talks during the fall semester 2020 we discuss representation of dessin d'enfants via permutations, edge rotation group and its invariance under the Galois group action on dessins. We learn how to apply representation theory of finite groups to dessins d'enfants: we retrieve information about fields of definitions from the character tables, enumerate dessins using Frobenius formula, find equations for canonical curves with many automorphisms using Eichler trace formula. We define Fried functions and a dessin on the base of a Fried family (megamap) - combinatorially and via braid group action. Our goal is to introduce students to the group theoretical approach to the theory of dessins d'enfants and the basics of the Grothendieck-Teichmuller theory.