Representation theory of the group of automorphisms of a finite rooted tree

TL;DR

This paper develops a comprehensive representation theory for automorphism groups of finite rooted trees, introducing novel combinatorial structures called tree compositions to classify irreducible representations.

Contribution

It constructs and parametrizes all irreducible representations of these automorphism groups using new combinatorial tools, without assuming spherical homogeneity.

Findings

Introduced tree compositions as a new combinatorial framework

Parametrized all irreducible representations of the automorphism group

Provided a coordinate-free approach applicable to non-homogeneous trees

Abstract

We construct the ordinary irreducible representations of the group of automorphisms of a finite rooted tree and we get a natural parametrization of them. To achieve this goals, we introduce and study the combinatorics of tree compositions, a natural generalization of set compositions but with new features and more complexity. These combinatorial structures lead to a family of permutation representations which have the same parametrization of the irreducible representations. Our trees are not necessarily spherically homogeneous and our approach is coordinate free.