27 pages, 4 figures. Major addition to the previous version. Section 4 is new. Organization of the paper modified. Stylistic changes. Small errors and typos corrected
We study Coxeter diagrams of some unitary reflection groups. Using solely the combinatorics of diagrams, we give a new proof of the classification of root lattices defined over $\cE = \ZZ[e^{2 πi/3}]$: there are only four such lattices, namely, the $\cE$-lattices whose real forms are $A_2$, $D_4$, $E_6$ and $E_8$. Next, we address the issue of characterizing the diagrams for unitary reflection groups, a question that was raised by Broué, Malle and Rouquier. To this end, we describe an algorithm which, given a unitary reflection group $G$, picks out a set of complex reflections. The algorithm is based on an analogy with Weyl groups. If $G$ is a Weyl group, the algorithm immediately yields a set of simple roots. Experimentally we observe that if $G$ is primitive and $G$ has a set of roots whose $\ZZ$--span is a discrete subset of the ambient vector space, then the algorithm selects a minimal generating set for $G$. The group $G$ has a presentation on these generators such that if we forget that the generators have finite order then we get a (Coxeter-like) presentation of the corresponding braid group. For some groups, such as $G_{33}$ and $G_{34}$, new diagrams are obtained. For $G_{34}$, our new diagram extends to an "affine diagram" with $\ZZ/7\ZZ$ symmetry.
Beijing International Center for Mathematical Research Peking University, Beijing, China（北京大学北京国际数学研究中心）；School of Mathematics and Statistics University of Melbourne, VIC 3010, Australia（墨尔本大学数学与统计学院）；Department of Foundational Mathematics School of Mathematics and Physics Xi’an Jiaotong-Liverpool University, Suzhou, China（西交利物浦大学数学与物理学院基础数学系）
A Coxeter group W is called reflection independent if its reflections are uniquely determined by W only, independently on the choice of the generating set. We give a new sufficient condition for the reflection independence, and examine this condition for Coxeter groups in certain classes, possibly of infinite ranks. We also determine the finite irreducible components of another Coxeter group, that is a subgroup of W generated by the reflections centralizing a given generator of W. Determining such a subgroup makes our criterion efficient.
H.S.M. Coxeter showed that a group $Γ$ is a finite reflection group of an Euclidean space if and only if $Γ$ is a finite Coxeter group. In this paper, we define {\it reflections} of geodesic spaces in general, and we prove that $Γ$ is a cocompact discrete reflection group of some geodesic space if and only if $Γ$ is a Coxeter group.
We refine Brink's theorem, that the non-reflection part of a reflection centralizer in a Coxeter group W is a free group. We give an explicit set of generators for centralizer, which is finitely generated when W is. And we give a method for computing the Coxeter diagram for its reflection subgroup. In many cases, our method allows one to compute centralizers in one's head.
Given an irreducible well-generated complex reflection group W with Coxeter number h, we call a Coxeter element any regular element (in the sense of Springer) of order h in W; this is a slight extension of the most common notion of Coxeter element. We show that the class of these Coxeter elements forms a single orbit in W under the action of reflection automorphisms. For Coxeter and Shephard groups, this implies that an element c is a Coxeter element if and only if there exists a simple system S of reflections such that c is the product of the generators in S. We moreover deduce multiple furth…
Based on the third author's thesis in this article we complete the local recognition of commuting reflection graphs of spherical Coxeter groups arising from irreducible crystallographic root systems.
We give a criterion for a finitely generated odd-angled Coxeter group to have a proper finite index subgroup generated by reflections. The answer is given in terms of the least prime divisors of the exponents of the Coxeter relations.
The imaginary cone of a Kac-Moody Lie algebra is the convex hull of zero and the positive imaginary roots. This paper studies the imaginary cone for a class of root systems of general Coxeter groups W. It is shown that the imaginary cone of a reflection subgroup of W is contained in that of W, and that for irreducible infinite W of finite rank, the closed imaginary cone is the only non-zero, closed, pointed W-stable cone contained in the pointed cone spanned by the simple roots. For W of finite rank, various natural notions of faces of the imaginary cone are shown to coincide, the face lattice…
Given a reflection $r$ in a Coxeter group $W$ (possibly of infinite rank), we consider the subgroup of $W$ generated by the reflections in $W$ having (-1)-eigenvectors orthogonal to the (-1)-eigenvector of $r$. In this paper, we determine completely when this subgroup is finitely generated, by using preceding results on centralizers of reflections. This result provides a new and naturally constructed example of a family of non-finitely generated subgroups of finitely generated groups.
We extend a result of Lewis and Reiner from finite Coxeter groups to all Coxeter groups by showing that two reflection factorizations of a Coxeter element lie in the same Hurwitz orbit if and only if they share the same multiset of conjugacy classes.
Let $W$ be a Coxeter group and $r\in W$ a reflection. If the group of order 2 generated by $r$ is the intersection of all the maximal finite subgroups of $W$ that contain it, then any isomorphism from $W$ to a Coxeter group $W'$ must take $r$ to a reflection in $W'$. The aim of this paper is to show how to determine, by inspection of the Coxeter graph, the intersection of the maximal finite sugroups containing $r$. In particular we show that the condition above is satisfied whenever $W$ is infinite and irreducible, and has the property that all rank two parabolic subgroups are finite. So in th…
We show that the Hurwitz action is "as transitive as possible" on reflection factorizations of Coxeter elements in the well-generated complex reflection groups $G(d, 1, n)$ (the group of $d$-colored permutations) and $G(d, d, n)$.
Let $(W,S)$ be a Coxeter system and let $s \in S$. We call $s$ a right-angled generator of $(W,S)$ if $st = ts$ or $st$ has infinite order for each $t \in S$. We call $s$ an intrinsic reflection of $W$ if $s \in R^W$ for all Coxeter generating sets $R$ of $W$. We give necessary and sufficient conditions for a right-angled generator $s \in S$ of $(W,S)$ to be an intrinsic reflection of $W$.
In any Coxeter group, the conjugates of elements in the standard minimal generating set are called reflections and the minimal number of reflections needed to factor a particular element is called its reflection length. In this article we prove that the reflection length function on an affine Coxeter group has a uniform upper bound. More precisely we prove that the reflection length function on an affine Coxeter group that naturally acts faithfully and cocompactly on $\R^n$ is bounded above by $2n$ and we also show that this bound is optimal. Conjecturally, spherical and affine Coxeter groups …
In any Coxeter group, the conjugates of elements in its Coxeter generating set are called reflections and the reflection length of an element is its length with respect to this expanded generating set. In this article we give a simple formula that computes the reflection length of any element in any affine Coxeter group and we provide a simple uniform proof.
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.
We provide local recognition results for the reflection graphs on spherical Coxeter groups. In particular, we study the case $F_4$ which is locally recognizable under additional constraints only. It is then demonstrated in the cases $A_n$ and $F_4$ how these graph theoretical recognition results can be used to characterize the corresponding Coxeter groups in terms of their reflection centralizers. Finally, we outline the connection to the local recognition of Chevalley groups which is particularly important in the classification of the finite simple 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 and the question whether there is an anlogue 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 and Reiner as well a…
The reflection length of an element of a Coxeter group is the minimal number of conjugates of the standard generators whose product is equal to that element. In this paper we prove the conjecture of McCammond and Petersen that reflection length is unbounded in any non-affine Coxeter group. Among the tools used, the construction of word-hyperbolic quotients of all minimal non-affine Coxeter groups might be of independent interest.
We use geometry of Davis complex of a Coxeter group to prove the following result: if G is an infinite indecomposable Coxeter group and $H\subset G$ is a finite index reflection subgroup then the rank of H is not less than the rank of G. This generalizes results of math/0305093. We also describe some properties of the nerves of the group and the subgroup in the case of equal ranks.
We consider a cocompact discrete reflection group $W$ of a CAT(0) space $X$. Then $W$ becomes a Coxeter group. In this paper, we study an analogy between the Davis-Moussong complex $Σ(W,S)$ and the CAT(0) space $X$, and show several analogous results about the limit set of a parabolic subgroup of the Coxeter group $W$.
In this paper, we give a class of reflection rigid Coxeter systems. Let $(W,S)$ be a Coxeter system. Suppose that (1) for each $s,t\in S$ such that $m(s,t)$ is odd, $\{s,t\}$ is a maximal spherical subset of $S$, (2) there does not exist a three-points subset $\{s,t,u\}\subset S$ such that $m(s,t)$ and $m(t,u)$ are odd, and (3) for each $s,t\in S$ such that $m(s,t)$ is odd, the number of maximal spherical subsets of $S$ intersecting with $\{s,t\}$ is at most two, where $m(s,t)$ is the order of $st$ in the Coxeter group $W$. Then we show that the Coxeter system $(W,S)$ is reflection rigid. This…
We define and study generalized nil-Coxeter algebras associated to Coxeter groups. Motivated by a question of Coxeter (1957), we construct the first examples of such finite-dimensional algebras that are not the 'usual' nil-Coxeter algebras: a novel 2-parameter type $A$ family that we call $NC_A(n,d)$. We explore several combinatorial properties of $NC_A(n,d)$, including its Coxeter word basis, length function, and Hilbert-Poincare series, and show that the corresponding generalized Coxeter group is not a flat deformation of $NC_A(n,d)$. These algebras yield symmetric semigroup module categorie…
In which it is proven that, for each parabolic quasi-Coxeter element in a finite real reflection group, the orbits of the Hurwitz action on its reflection factorizations are distinguished by the two obvious invariants
We improve a bound of Borcherds on the virtual cohomological dimension of the non-reflection part of the normalizer of a parabolic subgroup of a Coxeter group. Our bound is in terms of the types of the components of the corresponding Coxeter subdiagram rather than the number of nodes. A consequence is an extension of Brink's result that the non-reflection part of a reflection centralizer is free. Namely, the non-reflection part of the normalizer of parabolic subgroup of type D5 or Aodd is either free or has a free subgroup of index 2.
For each positive integer $k$ we present an example of Coxeter system $(G_k,S_k)$ such that $G_k$ is a word-hyperbolic Coxeter group, for any two generating reflections $s,t\in S_k$ the product $st$ has finite order, and the Coxeter graph of $S_k$ has negative inertia index equal to $k$. In particular, $G_k$ cannot be embedded into pseudo-orthogonal group $O(n,k-1)$ for any $n$ as a reflection group with generating set being reflections w.r.t. $(G_k,S_k)$
This is the first of a series of papers in which we initiate and develop the theory of reflection monoids, motivated by the theory of reflection groups. The main results identify a number of important inverse semigroups as reflection monoids, introduce new examples, and determine their orders.
To any finite graph $X$ (viewed as a topological space) we assosiate some explicit compact metric space ${\cal X}^r(X)$ which we call {\it the reflection tree of graphs $X$}. This space is of topological dimension $\le1$ and its connected components are locally connected. We show that if $X$ is appropriately triangulated (as a simplicial graph $Γ$ for which $X$ is the geometric realization) then the visual boundary $\partial_\infty(W,S)$ of the right angled Coxeter system $(W,S)$ with the nerve isomorphic to $Γ$ is homeomorphic to ${\cal X}^r(X)$. For each $X$, this yields in particular many w…
This paper examines a systematic method to construct a pair of (inter-related) root systems for arbitrary Coxeter groups from a class of non-standard geometric representations. This method can be employed to construct generalizations of root systems for a large family of groups generated only by involutions. We then give a characterization of Coxeter groups, among these groups, in terms of such paired root systems. Furthermore, we use this method to construct and study the paired root systems for reflection subgroups of Coxeter groups.
In this sixth part we study rank $3$ reflection groups not well generated: $G(2r,r,2)$, $G_{12}$, $G_{13}$ and $G_{22}$. We start from a reflection representation of a rank $3$ Coxeter group and we show that we can obtain in this manner affine Coxeter groups whose linear part is one of the preceding groups.
We show that for a parabolic quasi-Coxeter element in an affine Coxeter group the Hurwitz action on its set of reduced factorizations into a product of reflections is transitive. We call an element of the Coxeter group parabolic quasi-Coxeter element if it has a reduced factorization into a product of reflections that generate a parabolic subgroup.
The isomorphism problem for Coxeter groups has been reduced to its 'reflection preserving version' by B. Howlett and the second author. Thus, in order to solve it, it suffices to determine for a given Coxeter system (W,R) all Coxeter generating sets S of W which are contained in R^W, the set of reflections of (W,R). In this paper, we provide a further reduction: it suffices to determine all Coxeter generating sets S in R^W which are sharp-angled with respect to R.
We define the notion of a climbing element in a finite real reflection group relative to a total order on the reflection set and we characterise these elements in the case where the total order arises from a bipartite Coxeter element.
Let $W_0$ be a reflection subgroup of a finite complex reflection group $W$, and let $B_0$ and $B$ be their respective braid groups. In order to construct a Hecke algebra $\widetilde{H}_0$ for the normalizer $N_W(W_0)$, one first considers a natural subquotient $\widetilde{B}_0$ of $B$ which is an extension of $N_W(W_0)/W_0$ by $B_0$. We prove that this extension is split when $W$ is a Coxeter group, and deduce a standard basis for the Hecke algebra $\widetilde{H}_0$. We also give classes of both split and non-split examples in the non-Coxeter case.
Several finite complex reflection groups have a braid group which is isomorphic to a torus knot group. The reflection group is obtained from the torus knot group by declaring meridians to have order $k$ for some $k\geq 2$, and meridians are mapped to reflections. We study all possible quotients of torus knot groups obtained by requiring meridians to have finite order. Using the theory of $J$-groups of Achar and Aubert, we show that these groups behave like (in general infinite) complex reflection groups of rank two. The large family of "toric reflection groups" which we obtain includes, among …
We provide a necessary and sufficient condition on an element of a finite Coxeter group to ensure the transitivity of the Hurwitz action on its set of reduced decompositions into products of reflections. We show that this action is transitive if and only if the element is a parabolic quasi-Coxeter element, that is, if and only if it has a reduced decomposition into a product of reflections that generate a parabolic subgroup.
We refine Brieskorn's study of the cohomology of the complement of the reflection arrangement of a finite Coxeter group $W$. As a result we complete the verification of a conjecture by Felder and Veselov that gives an explicit basis of the space of $W$-invariants in this cohomology ring.
The principal objects studied in this note are Coxeter groups $W$ that are neither finite nor affine. A well known result of de la Harpe asserts that such groups have exponential growth. We consider quotients of $W$ by its parabolic subgroups and by a certain class of reflection subgroups. We show that these quotients have exponential growth as well. To achieve this, we use a theorem of Dyer to construct a reflection subgroup of $W$ that is isomorphic to the universal Coxeter group on three generators. The results are all proved under the restriction that the Coxeter diagram of $W$ is simply l…
For every dimension d, there is an infinite family of convex co-compact reflection groups of isometries of hyperbolic d-space --- the superideal (simplicial and cubical) reflection groups --- with the property that a random group at any density less than a half (or in the few relators model) contains quasiconvex subgroups commensurable with some member of the family, with overwhelming probability.
We review the properties of the finite Coxeter groups which are most useful for applications to cohomological invariants, namely their classes of involutions and their "cubes" (abelian subgroups generated by reflections).
We consider both standard and twisted action of a (real) Coxeter group G on the complement M_G to the complexified reflection hyperplanes by combining the reflections with complex conjugation. We introduce a natural geometric class of special involutions in G and give explicit formulae which describe both actions on the total cohomology H(M_G,C) in terms of these involutions. As a corollary we prove that the corresponding twisted representation is regular only for the symmetric group S_n, the Weyl groups of type D_{2m+1}, E_6 and dihedral groups I_2 (2k+1) and that the standard action has no a…
In a finite Coxeter group $W$ and with two given conjugacy classes of parabolic subgroups $[X]$ and $[Y]$, we count those parabolic subgroups of $W$ in $[Y]$ that are full support, while simultaneously being simple extensions (i.e., extensions by a single reflection) of some standard parabolic subgroup of $W$ in $[X]$. The enumeration is given by a product formula that depends only on the two parabolic types. Our derivation is case-free and combines a geometric interpretation of the "full support" property with a double counting argument involving Crapo's beta invariant. As a corollary, this a…
In this note, we provide a short and self-contained proof that the braid group on n strands acts transitively on the set of reduced factorizations of a Coxeter element in a Coxeter group of finite rank n into products of reflections. We moreover use the same argument to also show that all factorizations of an element in a parabolic subgroup of W lie as well in this parabolic subgroup.
We investigate a novel diagrammatic approach to examining strict actions of a Coxeter group or a braid group on a category. This diagrammatic language, which was developed in a series of papers by Elias, Khovanov and Williamson, provides new tools and methods to attack many problems of current interest in representation theory. In our research we considered a particular problem which arises in this context. To a Coxeter group $W$ one can associate a real hyperplane arrangement, and can consider the complement of these hyperplanes in the complexification $Y_W$. The celebrated $K(π,1)$ conjectur…
A Coxeter system is called two-dimensional if its associated Davis complex is two-dimensional (equivalently, every spherical subgroup has rank less than or equal to 2). We prove that given a two-dimensional system (W,S) and any other system (W,S') which yields the same reflections, the diagrams corresponding to these systems are isomorphic, up to the operation of diagram twisting defined by N. Brady, J. McCammond, B. Muhlherr, and W. Neumann. As a step in the proof of this result, certain two-dimensional groups are shown to be reflection rigid in the sense defined by the above authors, and a r…
We study the Hurwitz action of the classical braid group on factorisations of a Coxeter element c in a well-generated complex reflection group W. It is well-known that the Hurwitz action is transitive on the set of reduced decompositions of c in reflections. Our main result is a similar property for the primitive factorisations of c, i.e. factorisations with only one factor which is not a reflection. The motivation is the search for a geometric proof of Chapoton's formula for the number of chains of given length in the non-crossing partitions lattice NCP_W. Our proof uses the properties of the…
The notion of limit roots of a Coxeter group W was recently introduced (see arXiv:1112.5415 and arXiv:1303.6710): they are the accumulation points of directions of roots of a root system for W. In the case where the root system lives in a Lorentzian space W admits a faithful representation as a discrete reflection group of isometries on a hyperbolic space; the accumulation set of any of its orbits is then classically called the limit set of W. In this article we show that the limit roots of a Coxeter group W acting on a Lorentzian space is equal to the limit set of W seen as a discrete reflect…
