Conjugacy classes of cyclically fully commutative elements in Coxeter groups of type A

TL;DR

This thesis explores the conjugacy of cyclically fully commutative elements in type A Coxeter groups using cylindrical heaps and ring equivalence, providing a combinatorial characterization of conjugacy classes.

Contribution

It introduces cylindrical heaps and ring equivalence as new tools to characterize conjugacy of cyclically fully commutative elements in type A Coxeter groups.

Findings

Conjugate cyclically fully commutative elements correspond to ring equivalent cylindrical heaps.

The main result provides a combinatorial criterion for conjugacy in these groups.

The approach links heap combinatorics with group conjugacy properties.

Abstract

In this thesis, we study the combinatorics of cyclically fully commutative elements in Coxeter groups of type $A$ as it relates to conjugacy. In particular, we introduce the notion of cylindrical heaps and ring equivalence in order to state our main result, which says that two cyclically fully commutative elements of a Coxeter group of type $A$ are conjugate if and only if their corresponding cylindrical heaps are ring equivalent.