# Factorization problems in complex reflection groups

## TL;DR

This paper provides a combinatorial enumeration of factorizations of Coxeter elements in complex reflection groups, extending known results and characterizing related noncrossing partitions.

## Contribution

It introduces a new combinatorial approach to factorization enumeration in complex reflection groups, generalizing previous symmetric group results.

## Key findings

- Enumeration of Coxeter element factorizations with fixed space dimensions
- Extension of Jackson's results to complex reflection groups
- New characterization of the poset of W-noncrossing partitions

## Abstract

We enumerate factorizations of a Coxeter element in a well generated complex reflection group into arbitrary factors, keeping track of the fixed space dimension of each factor. In the infinite families of generalized permutations, our approach is fully combinatorial. It gives results analogous to those of Jackson in the symmetric group and can be refined to encode a notion of cycle type. As one application of our results, we give a previously overlooked characterization of the poset of $W$-noncrossing partitions.