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Let (W , S , L) be a weighted Coxeter group such that the order m s t of the product s t is not 3 for any s , t ∈ S and that N (W) : = max { L (w I) | I ⊆ S ; | W I | < ∞ } < ∞ , where w I is the longest element in the parabolic subgroup W I of W generated by I. We prove that (W , S , L) is bounded with N (W) an upper bound in the sense of Lusztig in Sec. 13.2 of [Hecke Algebras with Unequal Parameters, arXiv:math/0208154 v2 [math.RT] 10 Jun 2014], verifying a conjecture of Lusztig in our case (see Conjecture 13.4 in loc. cite). [ABSTRACT FROM AUTHOR]

The aim of the present paper is to describe all the left cells L of the affine Weyl group with a(L)≤11. We find a representative set of those left cells which occur as the vertices of the corresponding left cell graphs. The main technical tools are the algorithm designed by the first-named author and the various right primitive pairs. Some interesting empirical phenomenon is observed concerning isomorphic left cell graphs in different two-sided cells of . [ABSTRACT FROM AUTHOR]

The affine Coxeter groupcan be realized as the fixed point set of the affine Weyl group,m ∈ {2n, 2n + 1}, under a certain group automorphism αm, nwith. Letbe the length function of. We study the properties of the left cells of the weighted Coxeter groupand give an explicit description for all the left cellsLofwitha(L) ≤6. [ABSTRACT FROM AUTHOR]

The aim of the present article is to describe all the left cells of a-value 4 in the affine Weyl groups [image omitted]. We find a representative set of those left cells which occur as the vertex set of the corresponding left cell graphs by applying Shi's algorithm. Then we find all the distinguished involutions in those left cells. We show that those left cells are left-connected, verifying a conjecture of Lusztig in our case. [ABSTRACT FROM AUTHOR]

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]

We define a concept of “regularity” for finite unitary reflection groups, and show that an irreducible finite unitary reflection group of rank greater than 1 is regular if and only if it is a Coxeter group. Hence we get a characterization of Coxeter groups among all the irreducible finite reflection groups of rank greater than one. [ABSTRACT FROM AUTHOR]

The article looks at some special kind of sign types called ⊕-sign types. It mentions that ⊕-sign types are in one-to-one correspondence with the raising subsets of the related positive root system as a partially ordered set. It notes the significant role played by sign type in the study of the cells of any irreducible affine Weyl group.

Let (W,S, Γ) be a Coxeter system: a Coxeter group W with S its distinguished generator set and Γ its Coxeter graph. In the present paper, we always assume that the cardinality l=|S| of S is finite. A Coxeter element of W is by definition a product of all generators s $$\in $$ S in any fixed order. We use the notation C(W) to denote the set of all the Coxeter elements in W. These elements play an important role in the theory of Coxeter groups, e.g., the determination of polynomial invariants, the Poincaré polynomial, the Coxeter number and the group order of W (see [1–5] for example). They are also important in representation theory (see [6]). In the present paper, we show that the set C(W) is in one-to-one correspondence with the set C(Γ) of all acyclic orientations ofΓ . Then we use some graph-theoretic tricks to compute the cardinality c(W) of the set C(W) for any Coxeter group W. We deduce a recurrence formula for this number. Furthermore, we obtain some direct formulae of c(W) for a large family of Coxeter groups, which include all the finite, affine and hyperbolic Coxeter groups. The content of the paper is organized as below. In Section 1, we discuss some properties of Coxeter elements for simplifying the computation of the value c(W). In particular, we establish a bijection between the sets C(W) and C(Γ) . Then among the other results, we give a recurrence formula of c(W) in Section 2. Subsequently we deduce some closed formulae of c(W) for certain families of Coxeter groups in Section 3. [ABSTRACT FROM AUTHOR]