Left Cells in Certain Coxeter Groups

A survey is given on the achievements of KL-cell theory of certain Coxeter groups. Some techniques applied in that theory are introduced. Also, we propose several open problems for further study. In order to construct the representations of a Coxeter group W and the associative Hecke algebra, D. Kazhdan and G. Lusztig defined certain equivalence classes of W called left, right and two-sided cells. Thus the description of cells of W and the structural study of these cells become interesting and also important in the representation theory of groups and algebras. In the present paper, we shall make a survey on the achievements of studying left cells of W . According to the definition, the description of left cells might involve complicated computation of Kazhdan-Lusztig polynomials and is hard even by a computer when the order of W is getting larger. Thus we shall introduce some methods to simplify our work. They reduce the computation in significant rate so that sometimes we can reach our goal only by hand even when W is in some infinite case. We shall see that the study of cells of W involves some combinatorial techniques and has to invoke some other mathematical theory. We also propose some related open problems for further study. The content of this paper is organized as below. In section 1, we make a historical review on the cells of a Coxeter group, introduce the definition of cells by Kazhdan and Lusztig and some related concepts. Then we state some results of Lusztig concerning cells of Coxeter groups with properties 2.3, (a), (b), mainly of affine Weyl groups. A survey is given in section 3 on the achievements for the description of left cells of Weyl groups, Supported by the National Science Foundation of China and by the Science Foundation of the University Doctorial Program of CNEC Typeset by AMS-TEX 1