Let W be an irreducible finite or affine Coxeter group and let Wc be the set of fully commutative elements in W. We prove that the set Wc is closed under the Kazhdan-Lusztig's preorder≥LR if and only if Wc is a union of two-sided cells of W.

The concept of Yang-Baxter basis is useful to interpret Young's constructions for the symmetric group. We extend this concept first to any Weyl group and then to any Coxeter group.

We find all the inequivalent simple root systems for the complex reflection groups G12, G24, G25 and G26. Then we give all the non-congruent essential presentations of these groups by generators and relations. The methed used in the paper is applicable to any finite (complex) reflection groups.

Let W be a Weyl or affine Weyl group and let Wc be the set of fully commutative elements in W. We associate each w ∈ Wc to a digraph G(w). By using G(w), we give a graph-theoretic description for Lusztig's a-function on Wc and describe explicitly all the distinguished involutions of W. The results verify two conjectures in our case: one was proposed by myself in [15, Conjecture 8.10] and the other was by Lusztig in [2].

We give an explicit description in terms of rooted graphs for representatives of all the congruence classes of presentations (or r.c.p. for brevity) for the imprimitive complex reflection group G(m, p, n). Also, we show that (S, PS) forms a presentation of G(m, p, n), where S is any generating reflection set of G(m, p, n) of minimally possible cardinality and PS is the set of all the basic relations on S.