A new algorithm for finding an l.c.r. set in certain two-sided cells

Abstract

Let .W; S/ be an irreducible Weyl or affine Weyl group. In 1994, we con- structed 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 . Here, we introduce a new simpler algorithm for finding an l.c.r. set of W inwhen the subset F./ ofis known. We introduce some technical tricks by some examples for applying the algorithm and for finding the set F./. The resulting set E./ 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 cellof W (in the sense of (Kazhdan and Lusztig 1979)), we introduced an algorithm for finding an l.c.r. set of W inin (Shi 1994a). The algorithm has been efficiently applied in many cases; see for example (Chen 2000, Chen and Shi 1998; Rui 1995; Shi 1994a; 1994b; 1998a; 1998b; Shi and Zhang