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

Fabio Scarabotti

TL;DR

This work extends the representation theory of automorphism groups of finite rooted trees by introducing tree compositions and trees of partitions to parametrize conjugacy classes and irreducible representations. It develops two parallel frameworks—Clifford theory and Mackey theory—to construct and classify all ordinary irreducible representations of $\mathrm{Aut}(\mathcal{T})$, showing a natural one-to-one correspondence with $\mathsf{Par}(\mathcal{T})$. The results generalize classical cases for symmetric groups and wreath products beyond spherically homogeneous trees and provide a coordinate-free, structural approach to representations via permutation actions on tree compositions. The approach yields a complete, multiplicity-free parametrization and connects algebraic structure with combinatorial objects, enabling a unified treatment of conjugacy and irreducibles through tree-of-partitions data.

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.

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

TL;DR

This work extends the representation theory of automorphism groups of finite rooted trees by introducing tree compositions and trees of partitions to parametrize conjugacy classes and irreducible representations. It develops two parallel frameworks—Clifford theory and Mackey theory—to construct and classify all ordinary irreducible representations of , showing a natural one-to-one correspondence with . The results generalize classical cases for symmetric groups and wreath products beyond spherically homogeneous trees and provide a coordinate-free, structural approach to representations via permutation actions on tree compositions. The approach yields a complete, multiplicity-free parametrization and connects algebraic structure with combinatorial objects, enabling a unified treatment of conjugacy and irreducibles through tree-of-partitions data.

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.
Paper Structure (9 sections, 37 theorems, 129 equations, 1 algorithm)

This paper contains 9 sections, 37 theorems, 129 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Finite rooted trees: automorphisms and compositions
  3. The quotient of a tree composition
  4. Trees of partitions
  5. The conjugacy classes of $\hbox{\rm Aut}(\mathcal{T})$
  6. The irreducible representations via Clifford theory
  7. Common refinements of tree compositions
  8. The total order and the transpose
  9. The irreducible representations via Mackey theory

Key Result

Lemma 2.3

If $\varphi\colon\mathcal{T}\rightarrow\mathcal{P}$ is a tree composition, $u,v\in\mathcal{T}$ are distinct internal vertices and $\varphi(u)=\varphi(v)\eqcolon x$ then there exists $\tau\in\hbox{\rm Aut}(\mathcal{T})$ such that

Theorems & Definitions (107)

  • Example 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Example 2.4
  • Remark 2.5
  • Definition 2.6
  • Example 2.7
  • Remark 2.8
  • Example 2.9
  • ...and 97 more