Fully Commutative Elements and Kazhdan–lusztig Cells in the Finite and Affine Coxeter Groups

The main goal of the paper is to show that the fully commutative elements in the affine Coxeter group e Cn form a union of two-sided cells. Then we completely answer the question of when the fully commutative elements of W form or do not form a union of two-sided cells in the case where W is either a finite or an affine Coxeter group. Let W be a Coxeter group with S the distinguished generator set. The fully commutative elements of W were defined by Stembridge: w ∈ W is fully commutative if any two reduced expressions of w can be transformed from each other by only applying the relations st = ts with s, t ∈ S and o(st) = 2, or equivalently, w has no reduced expression of the form w = x(sts...)y, where sts... is a string of length o(st) > 2 (o(st) being the order of st) for some s 6= t in S. The fully commutative elements were studied extensively by a number of people (see [3, 6, 8, 16]). Now let W be either a finite or an affine Coxeter group and let Wc be the set of all the fully commutative elements in W . We consider the relation between Wc and the two-sided cells of W (in the sense of Kazhdan and Lusztig, see [9]). It is well known that when W is either the finite Coxeter group An (n > 1), Bl (l > 2), I2(m) (m > 2), or the affine Coxeter group Ãn (n > 1), Wc is a union of two-sided cells of W (see [12, §1.7, Theorems 16.2.8 and 17.4], [13, Theorem 3.1] and [8, Theorem 3.1.1]). On the other hand, since Wc is not a union of two-sided cells of W when W = D4 (see [2]), it should also be the case when