A New Algorithm for Finding an L.c.r. Set in Certain Two-sided Cells

Let (W,S) be an irreducible Weyl or affine Weyl group. In [18], we constructed an algorithm for finding a representative set of left cells (or an l.c.r. set for short) of W in a two-sided cell Ω. In the present paper, we introduce a new simpler algorithm for finding an l.c.r. set of W in Ω when the subset F (Ω) of Ω is known (see Algorithm 3.11). Some technical tricks are introduced by some examples for applying the algorithm and for finding the set F (Ω). The resulting set E(Ω) of Algorithm 3.4 is useful in verifying a conjecture of Lusztig that any left cell in an affine Weyl group is left-connected. Let W be an irreducible Weyl or affine Weyl group with S its Coxeter generator set. For a two-sided cell Ω of W (in the sense of Kazhdan–Lusztig, see [7]), we introduced an algorithm for finding an l.c.r. set of W in Ω in [18]. The algorithm has been efficiently applied in many cases (see [3], [4], [13], [18], [19], [22], [23], [26], [27], [28], [29], etc) since then. The algorithm consists of three processes (A), (B), (C) on a distinguished set F (see 3.1), where process (C) is the most difficult part among the three in which one need to find, for any given x ∈ F , all elements y satisfying y—–x, y < x, R(y) * R(x) and a(y) = a(x) (see 1.1. and 1.3. for the notation). It becomes extremely difficult as the length of x is getting larger. For any two-sided cell Ω of W , let F (Ω) be the set of all w ∈ Ω such that a(sw), a(wt) < a(w) for any s ∈ L(w) and t ∈ R(w). We shall introduce a new