The Left Cells Of Some Weighted Coxeter Groups

We prove left-connectedness of left cells in the two-sided cells Ω of the affine Weyl group E$with a(Ω)=6and get the distinguished involution graphs of Ω in E8with a(Ω)=5,6in this paper.The fixed point set of the affine Weyl group (A2n+1,S)(resp.,(A2n,S)) under a certain group automorphism η with η(S)=S can be considered as the affine Weyl group (Cn, S). Then the left and two-sided cells of the weighted Coxeter group (Cn,l), where l is the length function of A2n+1(resp.,A2n), can be given an explicit description by studying the fixed point set of the affine Weyl group (A2n+1, S)(resp.,(A2n, S)) under η. We give an explicit description for all the left cells of the weighted Coxeter group (Cn, L1), where the weight function L1of the affine Weyl group Cn is (3,2,…,2,3) corresponding to the partitions k12n+2-k for all1≤k≤2n+2and k(2n+2-k) for all n+1≤k≤2n, and also for all the cells of the weighted Coxeter group (C3, L'1), where the weight function L'1of the affine Weyl group C3is (3,2,2,3); all the left cells of the weighted Coxeter group (Cn,L2), where the weight function L2of the affine Weyl group Cn is (3,2,…,2,1), corresponding to the partitions k12n+1-k for all1≤k≤2n+1and k(2n+1-k) for all n+1≤k≤2n-1。...