Generalized iterated wreath products of cyclic groups and rooted trees correspondence

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.