# Generalized Iterated Wreath Products of Cyclic Groups and Rooted Trees Correspondence

## 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 the literature’s fastest FFT upper bound estimate.

## Keywords

Iterated wreath products Cyclic groups Rooted trees Irreducible representations Fast Fourier transform Bratteli diagrams## AMS Subject Classification

Primary 20C99 20E08; Secondary 65T50 05E25## Notes

### Acknowledgments

The authors acknowledge Mathematics Research Communities for providing an exceptional working environment at Snowbird, Utah. They would like to thank Michael Orrison for helpful discussions, and the referees for extremely useful remarks on this manuscript. This manuscript was written during M.S.I.’s visit to the University of Chicago in 2014. She thanks their hospitality.

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