On Reflection Subgroups of Finite Coxeter Groups

Let W be a finite Coxeter group. We classify the reflection subgroups of W up to conjugacy and give necessary and sufficient conditions for the map that assigns to a reflection subgroup R of W the conjugacy class of its Coxeter elements to be injective, up to conjugacy.     

Bounded Rank Subgroups of Coxeter Groups, Artin Groups and One-Relator Groups with Torsion

We obtain a number of results regarding the freeness of subgroupsof Coxeter groups, Artin groups and one-relator groups withtorsion. In the case of Coxeter groups, we also obtain resultson quasiconvexity and subgroup separability. 2000 MathematicsSubject Classification 20F65, 20F55, 20F36, 20F06.     

Strong Reality Properties of Normalizers of Parabolic Subgroups in Finite Coxeter Groups

Let W be a finite Coxeter group, P a parabolic subgroup of W, and N _W (P) the normalizer of P in W. We prove that every element in N _W (P) is strongly real in N _W (P), and that every irreducible complex character of N _W (P) has Frobenius-Schur indicator 1.     

Rigidity of Coxeter Groups and Artin Groups

A Coxeter group is rigid if it cannot be defined by two nonisomorphic diagrams. There have been a number of recent results showing that various classes of Coxeter groups are rigid, and a particularly interesting example of a nonrigid Coxeter group has been given by Bernhard Mühlherr. We show that this example belongs to a general operation of diagram twisting. We show that the Coxeter groups defined by twisted diagrams are isomorphic, and, moreover, that the Artin groups they define are also isomorphic, thus answering a question posed by Charney. Finally, we show a number of Coxeter groups are reflection rigid once twisting is taken into account.     

On Groups Generated by Elements of Pirme Order

In the first part of the paper we give a characterization of groups generated by elements of fixed prime order p. In the second part we study the group G _n ^(p) of n × n matrices with the pth power of the determinant equal to 1 over a field F containing a primitive pth root of 1. It is known that the group G _n ⁽²⁾ of n × n matrices of determinant ± 1 over a field F and the group SL _n (F) are generated by their involutions and that each element in these groups is a product of four involutions. We consider some subgroups G of G _n ^(p) and study the following problems: Is G generated by its elements of order p? If so, is every element of G a product of k elements of order p for some fixed integer k? We show that G _n ^(p) and SL _n (F) are generated by their elements of order p and that the bound k exists and is equal to 4. We show that every universal p-Coxeter group has faithful two-dimensional representations over many fields F (including ? and ?). For a universal p-Coxeter group of rank ≥ 2 for p ≥ 3 or of rank ≥ 3 for p = 2 there is no bound k.     

Sylvester Waves in the Coxeter Groups

A new recursive procedure of the calculation of partition numbers function W(s, d ^m ) is suggested. We find its zeroes and prove a lemma on the function parity properties. The explicit formulas of W(s, d ^m ) and their periods (G) for the irreducible Coxeter groups and a list for the first twelve symmetric group _m are presented. A least common multiple (m) of the series of the natural numbers 1,2,...,m plays a role in the period ( _m ) of W(s, d ^m) in _m .     

Coxeter Groups, 2-Completion, Perimeter Reduction and Subgroup Separability

We show that all groups in a very large class of Coxeter groups are locally quasiconvex and have a uniform membership problem solvable in quadratic time. If a group in the class satisfies a further hypothesis it is subgroup separable and relevant homomorphisms are also calculable in quadratic time. The algorithm also decides if a finitely generated subgroup has finite index.     

On Negative Orbits of Finite Coxeter Groups

For a Coxeter group W, X a subset of W and a positive root, we define the negative orbit of under X to be {w · | w X} ^–, where ^– is the set of negative roots. Here we investigate the sizes of such sets as varies in the case when W is a finite Coxeter group and X is a conjugacy class of W.     

On Subgroup Separability in Hyperbolic Coxeter Groups

We prove that certain hyperbolic Coxeter groups are separable on their geometrically finite subgroups.     

A Characterization of the Simply-Laced FC-Finite Coxeter Groups

We call an element of a Coxeter group fully covering (or a fully covering element) if its length is equal to the number of the elements it covers in the Bruhat ordering. It is easy to see that the notion of fully covering is a generalization of the notion of a 321-avoiding permutation and that a fully covering element is a fully commutative element. Also, we call a Coxeter group bi-full if its fully commutative elements coincide with its fully covering elements. We show that the bi-full Coxeter groups are the ones of type A_n, D_n, E_n with no restriction on n. In other words, Coxeter groups of type E₉, E₁₀,.... are also bi-full. According to a result of Fan, a Coxeter group is a simply-laced FC-finite Coxeter group if and only if it is a bi-full Coxeter group.AMS Subject Classification: 06A07, 20F55.     

On coherence of graph products of groups and Coxeter groups

Graph products of groups and Coxeter groups are defined via vertex-edge-labeled graphs. We show that if the graph has a special shape, then the corresponding group is coherent, i.e. every finitely generated subgroup is finitely presented.     

On a Graph Related to Subgroups Generated by a Conjugacy Class in Finite Groups

Let G be a group,and let (→Γ)C(G) be a digraph whose vertices are the non-trivial conjugacy classes of G and there is an arc from a vertex C to a vertex C'if an...     

On Degrees of Growth of Finitely Generated Groups

We prove that for an arbitrary function ρ of subexponential growth there exists a group G of intermediate growth whose growth function satisfies the inequality v _G,S (n) ? ρ(n) for all n. For every prime p, one can take G to be a p-group; one can also take a torsion-free group G. We also discuss some generalizations of this assertion.     

On the Fully Commutative Elements of Coxeter Groups

Let W be a Coxeter group. We define an element w ε W to be fully commutative if any reduced expression for w can be obtained from any other by means of braid relations that only involve commuting generators. We give several combinatorial characterizations of this property, classify the Coxeter groups with finitely many fully commutative elements, and classify the parabolic quotients whose members are all fully commutative. As applications of the latter, we classify all parabolic quotients with the property that (1) the Bruhat ordering is a lattice, (2) the Bruhat ordering is a distributive lattice, (3) the weak ordering is a distributive lattice, and (4) the weak ordering and Bruhat ordering coincide. Partially supported by NSF Grants DMS-9057192 and DMS-9401575.     

Coxeter群的半直积分解(英文)

本文证明了满足一定条件的Coxeter群都可以分解成两个Coxeter群的半直积.     

Maximal Parabolic Subgroups in Classical Groups are of Modality Zero

The principal aim of this paper is to show that every maximal parabolic subgroup P of a classical reductive algebraic group G operates with a finite number of orbits on its unipotent radical. This is a consequence of the fact that each parabolic subgroup of a group of type A _n whose unipotent radical is of nilpotent class at most two has this finiteness property.     

Parametrization of Triangle Groups as Subgroups of PSL(2, q)

In this paper we are interested in triangle groups (j, k, l) where j = 2 and k = 3. The groups (j, k, l) can be considered as factor groups of the modular group PSL(2, Z) which has the presentation x, y : x² = y³ = 1. Since PSL(2,q) is a factor group of G^k,l,m if -1 is a quadratic residue in the finite field F_q, it is therefore worthwhile to look at (j, k, l) groups as subgroups of PSL(2, q) or PGL(2, q). Specifically, we shall find a condition in form of a polynomial for the existence of groups (2, 3, k) as subgroups of PSL(2, q) or PGL(2, q).Mathematics Subject Classification: Primary 20F05 Secondary 20G40.