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Abstract: In our previous paper [J.Y. Shi, Formula for the reflection length of elements in the group , J. Algebra 316 (1) (2007) 284–296], we get a closed formula for the reflection length functions on the reflection group with and . In the present paper, we introduce a partial ordering, called reflection ordering, on the elements of . By applying this formula, we give an explicit description for the reflection ordering on the groups and , . [Copyright &y& Elsevier]

Abstract: We introduce two methods which reduce the checking of the joint relations on element pairs (i.e., checking if , see 1.2) inside a two-sided cell of the symmetric group to some special cases up to star operations. One involves the involutions in , and the other involves the set for (see 3.1 and 5.1). We give a detailed investigation for the properties of the set in terms of standard tableaux. Also, we study the sets and and give an affirmative answer to a question of N.H. Xi (see Proposition 2.9). [Copyright &y& Elsevier]

Abstract: We deduce a closed formula for the reflection length functions on the reflection group with and . The formula is shown separately in three cases: , and (see the proof of Theorems 2.1, 3.1 and 4.4). [Copyright &y& Elsevier]

We find representatives of all the equivalence classes of simple root systems (or r.e.s. for brevity) for the complex reflection groups G12, G24, G25 and G26. Then we give representatives of ail the congruence classes of (essential) presentations (or r.c.p. (r.c.e.p.) for brevity) for these groups by generators and relations. The method used in the paper is applicable to any finite (complex) reflection groups. [ABSTRACT FROM AUTHOR]

By the correspondence between Coxeter elements of a Coxeter system (W, S, Γ) and the acyclic orientations of the Coxeter graph Γ, we study some properties of elements in the set C0(W). We show that when W is of finite, affine, or hyperbolic type, any w ∈ C0(W) satisfies w∼lower limit LR wJ with ℓ(wJ) = &z.sfnc;J&z.sfnc; = m(w) for some J ⊂ S. Now assume that W is of finite or affine type. We give an explicit description for all the distinguished involutions d of W with d∼lower limit L w for some w ∈ E(W), which verifies a conjecture proposed in [Jian-yi Shi, Adv. Sci. China, Math.3 (1990), 79–98, Conjecture 8.10] in our case. We show that any left cell of W containing some element of C0(W) is left-connected, which verifies a conjecture of Lusztig [Ryoshi Hotta (Ed.), in “Problems from the Conference on Algebraic Groups and Representations held at Katata, August 29–September 3, 1983”] in our case. [ABSTRACT FROM AUTHOR]