Left cells in the affine Weyl group Wa(\widetilde D₄)
Jian Yi Shi · Institutional Repositories DataBase (IRDB) · 1994
The cells of affine Weyl groups have been studied for more than one decade.They have been described explicitly in cases of type A n (n>\) [13], [9] and of rank <3 [1],[4], [10].But there are only some partial results for an arbitrary irreducible affine Weyl group [2], [7], [8], [16], [17]. In [18], we constructed an algorithm to find a representative set of left cells of a certain crystallographic group W in a given two-sided cell.This provides us a practicable way to describe the cell of more groups.In the present paper, we shall apply it to the case when W is the affine Weyl group W a (D 4 ) (or denoted by W a for brevity) of type J5 4 .We shall give an explicit description for all the left cells of W a by finding a representative set of left cells of W a .Before this paper, Du Jie gave an explicit description for all the two-sided cells of W a , but he was unable to find the left cells of this group [5].Chen Chengdong recently desciibed all the left cells of W a in terms of certain special reduced expressions of elements [3].Comparing with their results, our description on the cells of W a is neater and easier expressable in nature.Moreover, by doing the above work, we develop some technical skill in performing the mentioned algorithm In particular, we could avoid any computation of non-trivial Kazhdan-Lusztig polynomials throughout this work.The content of the present paper is organized as below.Section 1 is the preliminaries.Some basic concepts and results concerning our algorithm are stated there.In section 2, we introduce the alcove forms of elements of W a and also state some properties of elements of W a in terms of alcove forms, which are quite useful in the subsequent sections Then in sections 3-5, we apply our algorithm to find a representative set 2 of left cells of W a .Finally, in section 6, we describe all the left cells of W a by making use of the set Σ.