A note on fully commutative elements in complex reflection groups

TL;DR

This paper explores the relationship between fully commutative elements in complex reflection groups and Coxeter groups, providing new characterizations and counting formulas for these elements in $G(m,1,n)$.

Contribution

It establishes a connection between fully commutative elements in $B_n$ and $G(m,1,n)$, enabling pattern avoidance characterization and enumeration in the complex setting.

Findings

Characterization of fully commutative elements in $G(m,1,n)$

Pattern avoidance description for these elements

A new counting formula for fully commutative elements

Abstract

Fully commutative elements in types $B$ and $D$ are completely characterized and counted by Stembridge. Recently, Feinberg-Kim-Lee-Oh have extended the study of fully commutative elements from Coxeter groups to the complex setting, giving an enumeration of such elements in $G(m,1,n)$. In this note, we prove a connection between fully commutative elements in $B_n$ and in $G(m,1,n)$, which allows us to characterize fully commutative elements in $G(m,1,n )$ by pattern avoidance. Further, we present a counting formula for such elements in $G(m,1,n)$.