Fully commutative elements of the complex reflection groups

TL;DR

This paper extends the concept of fully commutative elements from Coxeter groups to complex reflection groups, providing a combinatorial decomposition and enumeration of these elements.

Contribution

It introduces a new framework for understanding fully commutative elements in complex reflection groups and analyzes their combinatorial structures.

Findings

Decomposition of fully commutative elements into natural subsets

Enumeration and explicit description of these elements in complex reflection groups

Structural insights into the combinatorial properties of these elements

Abstract

We extend the usual notion of fully commutative elements from the Coxeter groups to the complex reflection groups. Then we decompose the sets of fully commutative elements into natural subsets according to their combinatorial properties, and investigate the structure of these decompositions. As a consequence, we enumerate and describe the form of these elements in the complex reflection groups.