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.     

Order Varieties and Monotone Retractions of Finite Posets

In this paper we introduce a new version of the concept of order varieties. Namely, in addition to closure under retracts and products we require that the class of posets should be closed under taking idempotent subalgebras. As an application we prove that the variety generated by an order-primal algebra on a finite connected poset P is congruence modular if and only if every idempotent subalgebra of P is connected. We give a polynomial time algorithm to decide whether or not a variety generated by an order-primal algebra admits a near unanimity function and so we answer a problem of Larose and Zádori.     

Weak Cayley Table Groups II: Alternating Groups and Finite Coxeter Groups

A weak Cayley table isomorphism is a bijection φ: G → H of groups such that φ(xy) ～ φ(x)φ(y) for all x, y ∈ G. Here ～denotes conjugacy. When G = H the set of all weak Cayley table isomorphisms φ: G → G forms a group 𝒲(G) that contains the automorphism group Aut(G) and the inverse map I: G → G, x → x ^?1. Let 𝒲₀(G) = ?Aut(G), I? ≤ 𝒲(G) and say that G has trivial weak Cayley table group if 𝒲(G) = 𝒲₀(G). We show that all finite irreducible Coxeter groups (except possibly E ₈) have trivial weak Cayley table group, as well as most alternating groups. We also consider some sporadic simple groups.     

On the Derived Length of Coxeter Groups

We characterize certain properties of the derived series of Coxeter groups by properties of the corresponding Coxeter graphs. In particular, we give necessary and sufficient conditions for a Coxeter group to be quasiperfect.     

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.     

Homomorphisms of Edge-Colored Graphs and Coxeter Groups

Let be two edge-colored graphs (without multiple edges or loops). A homomorphism is a mapping : for which, for every pair of adjacent vertices u and v of G ₁, (u) and (v) are adjacent in G ₂ and the color of the edge (u)(v) is the same as that of the edge uv.We prove a number of results asserting the existence of a graphG , edge-colored from a set C, into which every member from a given class of graphs, also edge-colored from C, maps homomorphically.We apply one of these results to prove that every three-dimensional hyperbolic reflection group, having rotations of orders from the setM ={m₁, m₂,..., m_k}, has a torsion-free subgroup of index not exceeding some bound, which depends only on the setM .     

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.     

The combinatorics of twisted involutions in Coxeter groups

The open intervals in the Bruhat order on twisted involutions in a Coxeter group are shown to be PL spheres. This implies results conjectured by F. Incitti and sharpens the known fact that these posets are Gorenstein over .

We also introduce a Boolean cell complex which is an analogue for twisted involutions of the Coxeter complex. Several classical Coxeter complex properties are shared by our complex. When the group is finite, it is a shellable sphere, shelling orders being given by the linear extensions of the weak order on twisted involutions. Furthermore, the -polynomial of the complex coincides with the polynomial counting twisted involutions by descents. In particular, this gives a type-independent proof that the latter is symmetric.

     

The Greedy Algorithm and Coxeter Matroids

The notion of matroid has been generalized to Coxeter matroid by Gelfand and Serganova. To each pair (W, P) consisting of a finite irreducible Coxeter group W and parabolic subgroup P is associated a collection of objects called Coxeter matroids. The (ordinary) matroids are a special case, the case W = A (isomorphic to the symmetric group Sym_{_n+1}) and P a maximal parabolic subgroup. The main result of this paper is that for Coxeter matroids, just as for ordinary matroids, the greedy algorithm provides a solution to a naturally associated combinatorial optimization problem. Indeed, in many important cases, Coxeter matroids are characterized by this property. This result generalizes the classical Rado-Edmonds and Gale theorems.A corollary of our theorem is that, for Coxeter matroids L, the greedy algorithm solves the L-assignment problem. Let W be a finite group acting as linear transformations on a Euclidean space , and let

On Growth Types of Quotients of Coxeter Groups by Parabolic Subgroups

The principal objects studied in this note are infinite, non-affine Coxeter groups W. A well-known result of de la Harpe asserts that such groups have exponential growth. We study the growth type of quotients of W by parabolic subgroups and by a certain class of reflection subgroups. Our main result is that these quotients have exponential growth as well.     

The Enumeration of Fully Commutative Elements of Coxeter Groups

A Coxeter group element w is fully commutative if any reduced expression for w can be obtained from any other via the interchange of commuting generators. For example, in the symmetric group of degree n, the number of fully commutative elements is the nth Catalan number. The Coxeter groups with finitely many fully commutative elements can be arranged into seven infinite families A_n, B_n, D_n, E_n,F_n, H_n and I₂(m). For each family, we provide explicit generating functions for the number of fully commutative elements and the number of fully commutative involutions; in each case, the generating function is algebraic.     

On the Weak Order of Orthogonal Groups

A complete lattice structure is defined on the underlying set of the orthogonal group of a real Euclidean space, by a construction analogous to that of the weak order of Coxeter systems in terms of root systems. This produces a complete rootoid in the sense of Dyer, with the orthogonal group as underlying group. It is shown that this complete lattice has a saturation property which is used along with other properties of the lattice to characterize the maximal totally ordered subsets of the lattice as collections of initial sections with respect to a total ordering on the positive roots.     

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.     

Spherical and geodesic growth rates of right-angled Coxeter and Artin groups are Perron numbers

We prove that for any infinite right-angled Coxeter or Artin group, its spherical and geodesic growth rates (with respect to the standard generating set) either take values in the set of Perron numbers, or equal 1. Also, we compute the average number of geodesics representing an element of given word-length in such groups.     

On the Derived Series of Coxeter Groups

Let G be a group and let φ(G) be the least integer k such that G^(k) = G^(k+1). If no such k exists, then φ(G) = ∞ and we write G ∈ 𝒰. We are interested in the questions which Coxeter groups are in 𝒰 and how large can finite φ(G) be for Coxeter groups. The second author answered these questions for 3-generator and 4-generator Coxeter groups. This article begins the study for the 5-generator case.     

关于群的阶与共轭类数

讨论群的共轭数与群阶的关系,获得两个新的数量不等式,同时改进了一些相关的已知结果。     

On the Direct Indecomposability of Infinite Irreducible Coxeter Groups and the Isomorphism Problem of Coxeter Groups

In this article, we prove that any irreducible Coxeter group of infinite order, which is possibly of infinite rank, is directly indecomposable as an abstract group. The key ingredient of the proof is that we can determine, for an irreducible Coxeter group W, the centralizers in W of the normal subgroups of W that are generated by involu-tions. As a consequence, the problem of deciding whether two general Coxeter groups are isomorphic is reduced to the case of irreducible ones. We also describe the automorphism group of a general Coxeter group in terms of those of its irreducible components.     

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.     

Root Systems for Asymmetric Geometric Representations of Coxeter Groups

Results are obtained concerning root systems for asymmetric geometric representations of Coxeter groups. These representations were independently introduced by Vinberg and Eriksson, and generalize the standard geometric representation of a Coxeter group in such a way as to include certain restrictions of all Kac–Moody Weyl groups. In particular, a characterization of when a nontrivial multiple of a root may also be a root is given in the general context. Characterizations of when the number of such multiples of a root is finite and when the number of positive roots sent to negative roots by a group element is finite are also given. These characterizations are stated in terms of combinatorial conditions on a graph closely related to the Coxeter graph for the group. Other finiteness results for the symmetric case which are connected to the Tits cone and to a natural partial order on positive roots are extended to this asymmetric setting.