Narrow Results

Using geometric methods we describe a large class of subgroups of Coxeter groups which are closed in the profinite topology and discuss some related open problems.

Much is known about random right-angled Coxeter groups (i.e., right-angled Coxeter groups whose defining graphs are random graphs under the Erdös–Rényi model). In this paper, we extend this model to study random general Coxeter groups and give some results about random Coxeter groups, including some information about the homology of the nerve of a random Coxeter group and results about when random Coxeter groups are $\delta$-hyperbolic and when they have the FC-type property.

Based on the Kauffman bracket at A = e^iπ/4, we define an invariant for a special type of n-punctured ball tangles. The invariant Fⁿ takes values in the set PM_{2 × 2ⁿ}(ℤ) of 2 × 2ⁿ matrices over ℤ modulo the scalar multiplication of ±1. We provide the formula to compute the invariant of the k₁ + ⋯ + k_n-punctured ball tangle composed of given n, k₁, …, k_n-punctured ball tangles. Also, we define the horizontal and the vertical connect sums of punctured ball tangles and provide the formulae for their invariants from those of given punctured ball tangles. In addition, we introduce the elementary operations on the class ST of 1-punctured ball tangles, called spherical tangles. The elementary operations on ST induce the operations on PM_{2 × 2}(ℤ), also called the elementary operations. We show that the group generated by the elementary operations on PM_{2 × 2}(ℤ) is isomorphic to a Coxeter group.

We construct an Alexander-type invariant for oriented doodles from a deformation of the Tits representation of the twin group and from the Chebyshev polynomials of the second kind. Like the Alexander polynomial, our invariant vanishes on unlinked doodles with more than one component. We also include values of our invariant on several doodles.

The irreducible representations of full support in the rational Cherednik category ${{𝒪}}_c(W)$ attached to a Coxeter group $W$ are in bijection with the irreducible representations of an associated Iwahori–Hecke algebra. Recent work has shown that the irreducible representations in ${{𝒪}}_c(W)$ of arbitrary given support are similarly governed by certain generalized Hecke algebras. In this paper, we compute the parameters for these generalized Hecke algebras in the remaining previously unknown cases, corresponding to the parabolic subgroup $B_n\times S_k$ in $B_{n+k}$ for $k\ge 2$ and $n\ge 0$.

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 a finite number φ (G) can be for Coxeter groups G. In this paper, we give full answers in the 3- and 4-generator cases.

The Poincaré series of a Coxeter group W measures its growth relative to the generating set R. A well-known formula gives an inductive method of calculating the Poincaré series of W when W is finite. The aim of this paper is to obtain generalizations of this formula.

We introduce the mutation game on a directed multigraph, which is dual to Mozes’ numbers game. This new game allows us to create geometric and combinatorial structure that allows generalization of root systems to more general graphs. We interpret Coxeter–Dynkin diagrams in this multigraph context and exhibit new geometric forms for the associated root systems.

The discrete group generated by reflections of the sphere, or the Euclidean space, or hyperbolic space are said to be Coxeter groups of, respectively, spherical, or Euclidean, or hyperbolic type. The hyperbolic Coxeter groups are said to be (quasi-)Lannér if the tiles covering the space are of finite volume and all (resp. some of them) are compact. For any Coxeter group stratified by the length of its elements, the Poincaré series is the generating function of the cardinalities of sets of elements of equal length. Around 1966, Solomon established that, for ANY Coxeter group, its Poincaré series is a rational function with zeros somewhere on the unit circle centered at the origin, and gave an implicit (recurrence) formula. For the spherical and Euclidean Coxeter groups, the explicit expression of the Poincaré series is well-known. The explicit answer was known for any 3-generated Coxeter group, and (with mistakes) for the Lannér groups. Here we give a lucid description of the numerator of the Poincaré series of any Coxeter group, the explicit expression of the Poincaré series for each Lannér and quasi-Lannér group, and review the scene. We give an interpretation of some coefficients of the denominator of the growth function. The non-real poles behave as in Eneström's theorem (lie in a narrow annulus) though the coefficients of the denominators do not satisfy theorem's requirements.

This expository paper exhibits and discusses the communalities and differences between wavelets and fractal functions, both examples of multiscale processes.

In this paper, Gröbner—Shirshov bases and normal forms of elements for the Coxeter groups E₆ and E₇ are found. These results support the conjecture in [4] about the general form of Gröbner—Shirshov bases for all Coxeter groups.

We discuss the enumeration theory for flags in Eulerian partially ordered sets, emphasizing the two main geometric and algebraic examples, face posets of convex polytopes and regular CW-spheres, and Bruhat intervals in Coxeter groups. We review the two algebraic approaches to flag enumeration – one essentially as a quotient of the algebra of noncommutative symmetric functions and the other as a subalgebra of the algebra of quasisymmetric functions – and their relation via duality of Hopf algebras. One result is a direct expression for the Kazhdan-Lusztig polynomial of a Bruhat interval in terms of a new invariant, the complete cd-index. Finally, we summarize the theory of combinatorial Hopf algebras, which gives a unifying framework for the quasisymmetric generating functions developed here.

Emergence of the experimental evidence of E₈ in the analysis of one dimensional Ising-model invokes further studies of the Coxeter-Weyl groups generated by reflections regarding their applications to polytopes. The Coxeter group W(H₄) which describes the Platonic Polytopes 600-cell and 120-cell in 4D singles out in the mass relations of the bound states of the Ising model for it is a maximal subgroup of the Coxeter-Weyl group W(E₈). There exists a one-to-one correspondence between the finite subgroups of quaternions and the Coxeter-Weyl groups of rank 4 which facilitates the study of the rank-4 Coxeter-Weyl groups. In this paper we study the systematic classifications of the 3D-polyhedra and 4D-polytopes through their symmetries described by the rank-3 and rank-4 Coxeter-Weyl groups represented by finite groups of quaternions. We also develop a technique on the constructions of the duals of the polyhedra and the polytopes and give a number of examples. Applications of the rank-2 Coxeter groups have been briefly mentioned.