The Cell Theory Of Some Weighted Coxeter Groups

In this paper, we study the cell theory of Coxeter groups and the corresponding Hecke algebras. In the first chapter, we introduce some basic concepts and known results in cell theory. Then in the subsequent chapters, we deal with some special kinds of Coxeter groups and determine the left and two-sided cells for them.In Chapter 2, we deal with the weighted universal Coxeter groups. We determine all the left cells and two-sided cells in those Coxeter groups. Moreover, we verify Lusztig’s 15 conjectures for weighted universal Coxeter groups. These conjectures are of great importance in cell representation theory.Lusztig’s a-function plays a crucial role in cell theory. Chapter 3 shows the boundedness of a-function for Coxeter groups with complete Coxeter graph.In order to generalize the results on universal Coxeter groups; we consider Coxeter groups with universal elements in Chapter 4. With some restrictions on the weight function, we describe the cells in W containing universal elements. We also obtain some properties of the elements of finite Coxeter groups with second largest weight.