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.     

The reduced expressions in a Coxeter system with a strictly complete Coxeter graph

Let (W,S)$(W,S)$ be a Coxeter system with a strictly complete Coxeter graph. The present paper concerns the set Red(z)$\mathrm{Red}(z)$ of all reduced expressions for any z∈W$z\in W$. By associating each bc-expression to a certain symbol, we describe the set Red(z)$\mathrm{Red}(z)$ and compute its cardinal |Red(z)|$|\mathrm{Red}(z)|$ in terms of symbols. An explicit formula for |Red(z)|$|\mathrm{Red}(z)|$ is deduced, where the Fibonacci numbers play a crucial role.     

The Spectra of Coxeter Graphs

We determine the spectra of the finite Coxeter graphs defined by a terminal node of the Coxeter diagram, and the spectra of their thick equivalents.     

Coxeter transformations: Construction of deltoids

The Coxeter transformations associated with deltoids (i.e., with graphs in which all simple cycles have length 3) are considered. The structure of the set of all connected deltoids whose spectra do not contain ?1 is described.     

Coxeter Transformation and Quasiquivers

Mathematical Notes -     

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.     

Reflection Independence in Even Coxeter Groups

If (W,S) is a Coxeter system, then an element of W is a reflection if it is conjugate to some element of S. To each Coxeter system there is an associated Coxeter diagram. A Coxeter system is called reflection preserving if every automorphism of W preserves reflections in this Coxeter system. As a direct application of our main theorem, we classify all reflection preserving even Coxeter systems. More generally, if (W,S) is an even Coxeter system, we give a combinatorial condition on the diagram for (W,S) that determines whether or not two even systems for W have the same set of reflections. If (W,S) is even and (W,S) is not even, then these systems do not have the same set of reflections. A Coxeter group is said to be reflection independent if any two Coxeter systems (W,S) and (W,S) have the same set of reflections. We classify all reflection independent even Coxeter groups.Mathematics Subject Classifications (2000). 20F05, 20F55, 20F65, 51F15.     

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.     

Four Coefficients of Characteristic Polynomials of Coxeter Transformations

Mathematical Notes -     

The Length of the Element of Maximal Length in the Coxeter System of Type A_n,B_n/C_n and D_n

Let ω be the element of maximal length in a finite irreducible Coxetersystem(W,S).In the present paper,we get the length of ω when(W,S)is of typeA_n B_n/C_n or D_n.     

Dense subsets of the boundary of a Coxeter system

In this paper, we investigate dense subsets of the boundary of a Coxeter system. We show that for a Coxeter system , if is quasi-dense in and the order for some , then there exists a point in the boundary of the Coxeter system such that the orbit is dense in . Here . We also show that if the set is quasi-dense in , then is dense in .

     

Addendum to ``Dense subsets of the boundary of a Coxeter system'

In this paper, we investigate boundaries of parabolic subgroups of Coxeter groups. Let be a Coxeter system and let be a subset of such that the parabolic subgroup is infinite. Then we show that if a certain set is quasi-dense in , then is dense in the boundary of the Coxeter system , where is the boundary of .

     

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.     

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

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 .     

A note on parabolic subgroups of a Coxeter group

The aim of this note is to prove that the parabolic closure of any subset of a Coxeter group is a parabolic subgroup. To obtain that, several technical lemmas on the root system of a parabolic subgroup are established.