Title:

The decomposition into cells of the affine Weyl groups of type A

In [1], Kazhdan and Lusztig introduce the concept of a Wgraph for a Coxeter group W. In particular, they define left, right and twosided cells. These Wgraphs play an important role in the representation theory. However, the algorithm given by Kazhdan and Lusztig to compute these cells is enormously complicated. These cells have been worked out only in a very few cases. In the present thesis, we shall find all the left, right and twosided cells in the affine Weyl group A n of type A n  1 > 2. Our main results show that each left (resp. right) cell of A n determines a partition, say λ of n and, is characterized by a λtabloid and also by its generalized right (resp. left ) Tinvariant. There exists a onetoone correspondence between the set of twosided cells of A n and the set A n of partitions of n. The number of left (resp. right) cells corresponding to a given partition λ ϵ A n is equal to n l / m п j = 1 u j l , where {u 1 > ... > um} is the dual partition of λ. Each two sided cell in A n is also an RLequivalence class of A n and is a connected set. Each left (resp. right) cell in A n is a maximal left (resp. right) connected component in the twosided cell of A n containing it. Let P be any proper standard parabolic subgroup of A n isomorphic to the symmetric group S n  then the intersection of P with each twosided cell of A n is nonempty and is just a twosided cell of P. The intersection of P with each left (rasp, right) cell of A n is either empty or a left (resp. right) cell of A n. Most of these results were conjectured by Lusstig [2], [3].
