Generalized iterated wreath products of symmetric groups and generalized rooted trees correspondence

TL;DR

This paper characterizes the irreducible representations of generalized iterated wreath products of symmetric groups, establishing a bijection with labeled rooted trees and providing recursive formulas and bounds relevant for Fourier transform computations.

Contribution

It introduces a complete description of the representation theory for these wreath products and links them to labeled rooted trees, offering new recursive formulas and bounds.

Findings

Bijection between irreducible representations and labeled rooted trees.

Recursive formulas for counting labels and degrees of representations.

Upper bounds for fast Fourier transform complexity.

Abstract

Consider the generalized iterated wreath product $S_{r_1}\wr \ldots \wr S_{r_k}$ of symmetric groups. We give a complete description of the traversal for the generalized iterated wreath product. We also prove an existence of a bijection between the equivalence classes of ordinary irreducible representations of the generalized iterated wreath product and orbits of labels on certain rooted trees. We find a recursion for the number of these labels and the degrees of irreducible representations of the generalized iterated wreath product. Finally, we give rough upper bound estimates for fast Fourier transforms.