Root cycles in Coxeter groups

Let W be a finite irreducible Coxeter group. The aim of this paper is to describe the maximal and minimal length elements in conjugacy classes of involutions in W.

We prove that a non-spherical irreducible Coxeter group is (directly) indecomposable and that an indefinite irreducible Coxeter group is strongly indecomposable in the sense that all its finite index subgroups are (directly) indecomposable. Let W be a Coxeter group. Write W = WX1× ⋯ × WXb× WZ3, where WX1, … , WXbare non-spherical irreducible Coxeter groups and WZ3is a finite one. By a classical result, known as the Krull–Remak–Schmidt theorem, the group WZ3has a decomposition WZ3= H1× ⋯ × Hqas a direct product of indecomposable groups, which is unique up to a central automorphism and a permutation of the factors. Now, W = WX1× ⋯ × WXb× H1× ⋯ × Hqis a decomposition of W as a direct product of indecomposable subgroups. We prove that such a decomposition is unique up to a central automorphism and a permutation of the factors. Write W = WX1× ⋯ × WXa× WZ2× WZ3, where WX1, … , WXaare indefinite irreducible Coxeter groups, WZ2is an affine Coxeter group whose irreducible components are all infinite, and WZ3is a finite Coxeter group. The group WZ2contains a finite index subgroup R isomorphic to ℤd, where d = |Z2| - b + a and b - a is the number of irreducible components of WZ2. Choose d copies R1, … , Rdof ℤ such that R = R1× ⋯ × Rd. Then G = WX1× ⋯ × WXa× R1× ⋯ × Rdis a virtual decomposition of W as a direct product of strongly indecomposable subgroups. We prove that such a virtual decomposition is unique up to commensurability and a permutation of the factors.

Supposing G is a group and k a natural number, d_k(G) is defined to be the minimal number of elements of G of order k which generate G (setting d_k(G)=0 if G has no such generating sets). This paper investigates d_k(G) when G is a finite Coxeter group either of type B_n or D_n, or of exceptional type. Together with work of Garzoni and Yu, this determines d_k(G) for all finite irreducible Coxeter groups G when k is between 2 and rank(G) (or rank(G)+1 when G is of type A_n).

The main algorithmic problems of group theory posed by M. Dehn are the problem of words, the problem of the conjugation of words for finitely presented groups, and the group’s isomorphism problem. Among the works related to the study of the M. Dehn’s problems, the most outstanding ones are the work of P. S. Novikov who proved the undecidability of the problem of words and the conjugacy problem for finitely presented groups as well as the undecidability of the problem of isomorphism of groups. In this regard, the main algorithmic problems and their various generalizations are studied in certain classes of groups. Coxeter groups were introduced by H. S. M. Coxeter: every reflection group is a Coxeter group if its generating elements are reflections with respect to hyperplanes limiting its fundamental polyhedron. H. S. M. Coxeter listed all the reflection groups in three-dimensional Euclidean space and proved that they are all Coxeter groups and every finite Coxeter group is isomorphic to some reflection group in the three-dimensional Euclidean space which elements have a common fixed point. In an algebraic aspect Coxeter groups are studied starting with works by J. Tits who solved the problem of words in certain Coxeter groups. The article describes the known results obtained in solving algorithmic problems in Coxeter groups; the main purpose of the paper is to analyze of the results of solving algorithmic problems in Coxeter groups that were obtained by members of the Tula algebraic school ’Algorithmic problems of theory of the groups and semigroups ’ under the supervision of V. N. Bezverkhnii. It reviews assertions and theorems proved by the authors of the article for the various classes of Coxeter groups: Coxeter groups of large and extra-large types, Coxeter groups with a tree-structure, and Coxeter groups with n-angled structure. The basic approaches and methods of evidence among which the method of diagrams worked out by van Kampen, reopened by R. Lindon and refined by V. N. Bezverkhnii concerning the introduction of R-cancellations, special R-cancellations, special ring cancellations as well as method of graphs, method of types worked out by V. N. Bezverkhnii, method of special set of words designed by V. N. Bezverkhnii on the basis of the generalization of Nielsen method for free construction of groups. Classes of group considered in the article include all Coxeter groups which may be represented as generalized tree structures of Coxeter groups formed from Coxeter groups with tree structure with replacing some vertices of the corresponding tree-graph by Coxeter groups of large or extra-large types as well as Coxeter groups with n-angled structure.

Let $(W,S)$ be a Coxeter system, and write $S=\{s_i:i\in I\}$ , where I is a finite index set. Fix a nonempty convex subset $\mathscr {L}$ of W. If W is of type A, then $\mathscr {L}$ is the set of linear extensions of a poset, and there are important Bender–Knuth involutions $\mathrm {BK}_i\colon \mathscr {L}\to \mathscr {L}$ indexed by elements of I. For arbitrary W and for each $i\in I$ , we introduce an operator $\tau _i\colon W\to W$ (depending on $\mathscr {L}$ ) that we call a noninvertible Bender–Knuth toggle; this operator restricts to an involution on $\mathscr {L}$ that coincides with $\mathrm {BK}_i$ in type A. Given a Coxeter element $c=s_{i_n}\cdots s_{i_1}$ , we consider the operator $\mathrm {Pro}_c=\tau _{i_n}\cdots \tau _{i_1}$ . We say W is futuristic if for every nonempty finite convex set $\mathscr {L}$ , every Coxeter element c and every $u\in W$ , there exists an integer $K\geq 0$ such that $\mathrm {Pro}_c^K(u)\in \mathscr {L}$ . We prove that finite Coxeter groups, right-angled Coxeter groups, rank-3 Coxeter groups, affine Coxeter groups of types $\widetilde A$ and $\widetilde C$ , and Coxeter groups whose Coxeter graphs are complete are all futuristic. When W is finite, we actually prove that if $s_{i_N}\cdots s_{i_1}$ is a reduced expression for the long element of W, then $\tau _{i_N}\cdots \tau _{i_1}(W)=\mathscr {L}$ ; this allows us to determine the smallest integer $\mathrm {M}(c)$ such that $\mathrm {Pro}_c^{{\mathrm {M}}(c)}(W)=\mathscr {L}$ for all $\mathscr {L}$ . We also exhibit infinitely many non-futuristic Coxeter groups, including all irreducible affine Coxeter groups that are not of type $\widetilde A$ , $\widetilde C$ , or $\widetilde G_2$ .

Abstract.LetWhen

A combinatorial derivation of the Poincaré polynomials of the finite irreducible Coxeter groups

We derive presentations of the interval groups related to all quasi-Coxeter elements in the Coxeter group of type D n . Type D n is the only infinite family of finite Coxeter groups that admits proper quasi-Coxeter elements. The presentations we obtain are over a set of generators in bijection with what we call a Carter generating set, and the relations are those defined by the related Carter diagram together with a twisted cycle or a cycle commutator relator, depending on whether the quasi-Coxeter element is a Coxeter element or not. The proof is based on the description of two combinatorial techniques related to the intervals of quasi-Coxeter elements.In a subsequent work [4], we complete our analysis to cover all the exceptional cases of finite Coxeter groups, and establish that almost all the interval groups related to proper quasi-Coxeter elements are not isomorphic to the related Artin groups, hence establishing a new family of interval groups with nice presentations [4, 5]. Alongside the proof of the main results, we establish important properties related to the dual approach to Coxeter and Artin groups.

The isomorphism problem for finitely generated Coxeter groups is the problem of deciding if two finite Coxeter matrices define isomorphic Coxeter groups. Coxeter [4] solved this problem for finite irreducible Coxeter groups. Recently there has been considerable interest and activity on the isomorphism problem for arbitrary finitely generated Coxeter groups. In this paper we describe a family of isomorphism invariants of a finitely generated Coxeter group W . Each of these invariants is the isomorphism type of a quotient group W/N of W by a characteristic subgroup N . The virtue of these invariants is that W/N is also a Coxeter group. For some of these invariants, the isomorphism problem of W/N is solved and so we obtain isomorphism invariants that can be effectively used to distinguish isomorphism types of finitely generated Coxeter groups. We emphasize that even if the isomorphism problem for finitely generated Coxeter groups is eventually solved, several of the algorithms described in our paper will still be useful because they are computational fast and would most likely be incorporated into an efficient computer program that determines if two finite rank Coxeter systems have isomorphic groups. In §2, we establish notation. In §3, we describe two elementary quotienting operations on a Coxeter system that yields another Coxeter system. In §4, we describe the even part isomorphism invariant of a finitely generated Coxeter group. In §5, we review some matching theorems. In §6, we describe the even isomorphism invariant of a finitely generated Coxeter group. In §7, we define basic characteristic subgroups of a finitely generated Coxeter

Let W W be an infinite irreducible Coxeter group with ( s 1 , … , s n ) (s_1, \ldots , s_n) the simple generators. We give a short proof that the word s 1 s 2 ⋯ s n s 1 s 2 ⋯ s_1 s_2 \cdots s_n s_1 s_2 \cdots s n ⋯ s 1 s 2 ⋯ s n s_n \cdots s_1 s_2 \cdots s_n is reduced for any number of repetitions of s 1 s 2 ⋯ s n s_1 s_2 \cdots s_n . This result was proved for simply laced, crystallographic groups by Kleiner and Pelley using methods from the theory of quiver representations. Our proof uses only basic facts about Coxeter groups and the geometry of root systems. We also prove that, in finite Coxeter groups, there is a reduced word for w 0 w_0 which is obtained from the semi-infinite word s 1 s 2 ⋯ s n s 1 s 2 ⋯ s n ⋯ s_1 s_2 \cdots s_n s_1 s_2 \cdots s_n \cdots by interchanging commuting elements and taking a prefix.

Let W be a finite Coxeter group and X a subset of W . The length polynomial L W , X ( t ) is defined by L W , X ( t ) = ∑ x ∈ X t ℓ ( x ) , where ℓ is the length function on W . If X = { x ∈ W : x 2 = 1 } then we call L W , X ( t ) the involution length polynomial of W . In this article we derive expressions for the length polynomial where X is any conjugacy class of involutions, and the involution length polynomial, in any finite Coxeter group W . In particular, these results correct errors in [11] for the involution length polynomials of Coxeter groups of type B n and D n . Moreover, we give a counterexample to a unimodality conjecture stated in [11] .

Let $W$ be a Coxeter group. We provide a precise description of the conjugacy classes in $W$, in the spirit of Matsumoto's theorem. This extends to all Coxeter groups an important result on finite Coxeter groups by M. Geck and G. Pfeiffer from 1993. In particular, we describe the cyclically reduced elements of $W$, thereby proving a conjecture of A. Cohen from 1994.

Eriksson’s numbers game and finite Coxeter groups

We investigate the so-called dual Matsumoto property or Hurwitz action in finite, affine and arbitrary Coxeter groups. In particular, we want to investigate how to reduce reflection factorizations and how two reflection factorizations of the same element are related to each other. We are motivated by the dual approach to Coxeter groups proposed by Bessis (2003) and the question whether there is an analogue of the well-known Matsumoto property for reflection factorizations. Our aim is a substantial understanding of the Hurwitz action. We therefore reprove uniformly results of Lewis–Reiner as well as Baumeister–Gobet–Roberts and the first author on the Hurwitz action in finite and affine Coxeter groups. Further we show that in an arbitrary Coxeter group all reduced reflection factorizations of the same element appear in the same Hurwitz orbit after a suitable extension by simple reflections. As parabolic quasi-Coxeter elements play an outstanding role in the study of the Hurwitz action, we aim to characterize these elements. We provide a characterization of maximal parabolic quasi-Coxeter elements in arbitrary Coxeter groups as well as a characterization of all parabolic quasi-Coxeter elements in affine Coxeter groups.

Vanishing ranges for the mod p cohomology of alternating subgroups of Coxeter groups