[Paper Review] Generalized iterated wreath products of cyclic groups and rooted trees correspondence

Mee Seong Im, Angela Y. Wu|arXiv (Cornell University)|Sep 2, 2014

Genetic Neurodegenerative Diseases3 citations

TL;DR

This paper establishes a correspondence between irreducible representations of generalized iterated wreath products of cyclic groups and labeled rooted trees, providing a Bratteli diagram, a recursive formula for representation counts, and a new upper bound for fast Fourier transform (FFT) computations. The key contribution is a systematic indexing of irreducible representations via tree structures, enabling efficient FFT algorithms with the tightest known upper bound in the literature.

ABSTRACT

Consider the generalized iterated wreath product $\mathbb{Z}_{r_1}\wr \mathbb{Z}_{r_2}\wr \ldots \wr \mathbb{Z}_{r_k}$ where $r_i \in \mathbb{N}$. We prove that the irreducible representations for this class of groups are indexed by a certain type of rooted trees. This provides a Bratteli diagram for the generalized iterated wreath product, a simple recursion formula for the number of irreducible representations, and a strategy to calculate the dimension of each irreducible representation. We calculate explicitly fast Fourier transforms (FFT) for this class of groups, giving literature's fastest FFT upper bound estimate.

Motivation & Objective

• To establish a combinatorial indexing of irreducible representations for generalized iterated wreath products of cyclic groups.
• To develop a recursive method for counting irreducible representations using rooted tree structures.
• To derive a tight upper bound for fast Fourier transform (FFT) computation on these groups.
• To generalize prior results on wreath product representation theory to arbitrary cyclic group orders.
• To lay the foundation for future study of symmetric group wreath products and adapted FFT bases.

Proposed method

• The paper uses induction and group-theoretic functors—specifically induction and restriction—on subgroups of the iterated wreath product to build irreducible representations.
• It defines a traversal of irreducible representations via orbits of words under cyclic group actions, using $\mathbb{Z}_{r_k}$-orbits on $[h]^{r_k}$ to parameterize induced representations.
• The construction relies on the induction formula: $\mathcal{R}_{W(\vec{r}|_k)} = \{\mathop{\mathrm{Ind}}_{W(\vec{r}|_{k-1})\rtimes\mathbb{Z}_{d_x}}^{W(\vec{r}|_k)}(\rho_{x_1}\otimes\cdots\otimes\rho_{x_{r_k}}\otimes\tau)\}$, where $x$ ranges over orbit representatives and $\tau$ over irreps of $\mathbb{Z}_{d_x}$.
• The FFT computation is analyzed by decomposing the group algebra into blocks and applying recursive Fourier transforms on subgroups, with complexity bounded by $r \cdot T(K^r) + \sum_{\eta \in E} m(r\dim(\eta)^\alpha + \dim(\eta)^2 O(r\log(r/m))) + r m \dim(\eta)^\alpha$.
• The method exploits the structure of $G = K^r \rtimes \mathbb{Z}_r$ with $K = \mathbb{Z}_m$, and uses matrix extension techniques to compute induced representation transforms efficiently.
• A bijection between irreducible representations and orbits of labelings on $\vec{r}|_k$-ary rooted trees is established, forming a Bratteli diagram for the group series.

Experimental results

Research questions

• RQ1 How can the irreducible representations of generalized iterated wreath products $\mathbb{Z}_{r_1} \wr \cdots \wr \mathbb{Z}_{r_k}$ be systematically indexed and counted?
• RQ2 What is the structure of the Bratteli diagram for this class of groups, and how does it reflect the representation theory?
• RQ3 Can a fast Fourier transform algorithm be constructed for these groups with improved computational complexity bounds?
• RQ4 How do the induced representations from subgroups relate to the labeling of rooted trees in the construction?
• RQ5 What is the tightest known upper bound for FFT computation on iterated wreath products of cyclic groups?

Key findings

• The irreducible representations of $W(\vec{r}|_k)$ are indexed by orbits of labelings on $\vec{r}|_k$-ary rooted trees of height $k$, establishing a complete combinatorial correspondence.
• A recursive formula for the number of irreducible representations is derived from the orbit structure of $[h]^{r_k}$ under $\mathbb{Z}_{r_k}$-action.
• The paper provides the fastest known FFT upper bound estimate for this class of groups, improving upon prior literature.
• The dimension of each irreducible representation can be computed via the induced representation construction involving $\mathbb{Z}_{d_x}$ and $\rho_{x_i}$, with $d_x$ dividing $r_k$.
• The total FFT computation cost is bounded by $r \cdot T(K^r) + \sum_{\eta \in E} m(r\dim(\eta)^\alpha + \dim(\eta)^2 O(r\log(r/m))) + r m \dim(\eta)^\alpha$, where $E$ is a set of $G$-orbit representatives.
• The method establishes a bijection between irreducible representations and orbits of labelings on rooted trees, generalizing earlier results for symmetric groups and cyclic wreath products.