Symmetric groups and checker triangulated surfaces

TL;DR

This paper explores the connection between colored triangulations of surfaces, Belyi data, and pairs of permutations, highlighting their relation to infinite symmetric groups and their representations.

Contribution

It establishes new links between combinatorial surface structures, permutation pairs, and the representation theory of infinite symmetric groups.

Findings

Triangulations with three-colored edges relate to Belyi data.

Permutation pairs are classified up to conjugation.

Connections to infinite symmetric group representations are demonstrated.

Abstract

We consider triangulations of surfaces with edges painted three colors so that edges of each triangle have different colors. Such structures arise as Belyi data (or Grothendieck dessins d'enfant), on the other hand they enumerate pairs of permutations determined up to a common conjugation. The topic of these notes is links of such combinatorial structures with infinite symmetric groups and their representations.