On centralizers of parabolic subgroups in Coxeter groups
Koji Nuida
备注：
33 pages; improved version, notations changed (ver. 2); inappropriate Eq. (4.7) was removed (ver. 3); Theorem 7.1 was improved (ver. 4); the content of Section 7 separated as an individual paper and removed from this paper (ver. 5)
Let $W$ be an arbitrary Coxeter group, possibly of infinite rank. We describe a decomposition of the centralizer $Z_W(W_I)$ of an arbitrary parabolic subgroup $W_I$ into the center of $W_I$, a Coxeter group and a subgroup defined by a 2-cell complex. Only information about finite parabolic subgroups is required in an explicit computation. Moreover, by using our description of $Z_W(W_I)$, we reveal a further strong property of the action of the third factor on the second factor, in particular on the finite irreducible components of the second factor.
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.
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.
It has been known that the centralizer $Z_W(W_I)$ of a parabolic subgroup $W_I$ of a Coxeter group $W$ is a split extension of a naturally defined reflection subgroup by a subgroup defined by a 2-cell complex $\mathcal{Y}$. In this paper, we study the structure of $Z_W(W_I)$ further and show that, if $I$ has no irreducible components of type $A_n$ with $2 \leq n < \infty$, then every element of finite irreducible components of the inner factor is fixed by a natural action of the fundamental group of $\mathcal{Y}$. This property has an application to the isomorphism problem in Coxeter groups.
The normalizer $N_W(W_J)$ of a standard parabolic subgroup $W_J$ of a finite Coxeter group $W$ splits over the parabolic subgroup with complement $N_J$ consisting of certain minimal length coset representatives of $W_J$ in $W$. In this note we show that (with the exception of a small number of cases arising from a situation in Coxeter groups of type $D_n$) the centralizer $C_W(w)$ of an element $w \in W$ is in a similar way a semidirect product of the centralizer of $w$ in a suitable small parabolic subgroup $W_J$ with complement isomorphic to the normalizer complement $N_J$.
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.
We prove that even Coxeter groups, whose Coxeter diagrams contain no (4,4,2) triangles, are conjugacy separable. In particular, this applies to all right-angled Coxeter groups or word hyperbolic even Coxeter groups. For an arbitrary Coxeter group W, we also study the relationship between Coxeter generating sets that give rise to the same collection of parabolic subgroups. As an application we show that if an automorphism of W preserves the conjugacy class of every sufficiently short element then it is inner. We then derive consequences for the outer automorphism groups of Coxeter groups.
We characterize the double centralizer of all parabolic subgroups of the braid groups. We apply this result to provide a new and potentially more efficient solution to the subgroup conjugacy problem for parabolic subgroups. In the course of the proof we also characterize the centralizer for all parabolic subgroups.
The conjugacy classes of so-called special involutions parameterize the constituents of the action of a finite Coxeter group on the cohomology of the complement of its complexified hyperplane arrangement. In this note we give a short intrinsic characterisation of special involutions in terms of so-called bulky parabolic subgroups.
In this paper, we investigate boundaries of parabolic subgroups of Coxeter groups. Let $(W,S)$ be a Coxeter system and let $T$ be a subset of $S$ such that the parabolic subgroup $W_T$ is infinite. Then we show that if a certain set is quasi-dense in $W$, then $W \partialΣ(W_T,T)$ is dense in the boundary $\partialΣ(W,S)$ of the Coxeter system $(W,S)$, where $\partialΣ(W_T,T)$ is the boundary of $(W_T,T)$.
In this paper, given a split extension of an arbitrary Coxeter group by automorphisms of the Coxeter graph, we determine the involutions in that extension whose centralizer has finite index. Our result has applications to many problems such as the isomorphism problem of general Coxeter groups. In the argument, some properties of certain special elements and of the fixed-point subgroups by graph automorphisms in Coxeter groups, which are of independent interest, are also given.
We discuss one construction of nonstandard subgroups in the category of Coxeter groups. Two formulae for the growth series of such a subgroups are given. As an application we construct a flag simple convex polytope, whose f-polynomial has non-real roots.
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.
We determine all finite subgroups of simple algebraic groups that have irreducible centralizers - that is, centralizers whose connected component does not lie in a parabolic subgroup.
Despite the significance of the notion of parabolic closures in Coxeter groups of finite ranks, the parabolic closure is not guaranteed to exist as a parabolic subgroup in a general case. In this paper, first we give a concrete example to clarify that the parabolic closure of even an irreducible reflection subgroup of countable rank does not necessarily exist as a parabolic subgroup. Then we propose a generalized notion of "locally parabolic closure" by introducing a notion of "locally parabolic subgroups", which involves parabolic ones as a special case, and prove that the locally parabolic c…
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…
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.
In this paper we prove, without the finite rank assumption, that any irreducible Coxeter group of infinite order is directly indecomposable as an abstract group. The key ingredient of the proof is that we can determine, for an irreducible Coxeter group, the centralizers of the normal subgroups that are generated by involutions. As a consequence, we show that the problem of deciding whether two general Coxeter groups are isomorphic, as abstract groups, is reduced to the case of irreducible Coxeter groups, without assuming the finiteness of the number of the irreducible components or their ranks…
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.
In this paper, we show that the center of every Coxeter group is finite and isomorphic to $(\Z_2)^n$ for some $n\ge 0$. Moreover, for a Coxeter system $(W,S)$, we prove that $Z(W)=Z(W_{S\setminus\tilde{S}})$ and $Z(W_{\tilde{S}})=1$, where $Z(W)$ is the center of the Coxeter group $W$ and $\tilde{S}$ is the subset of $S$ such that the parabolic subgroup $W_{\tilde{S}}$ is the {\it essential parabolic subgroup} of $(W,S)$ (i.e.\ $W_{\tilde{S}}$ is the minimum parabolic subgroup of finite index in $(W,S)$). The finiteness of the center of a Coxeter group implies that a splitting theorem holds fo…
We introduce the notion of two-dimensional Coxeter system and show that parabolic subgroups of GL_n(F_2) can be described by an appropriate two-dimensional Coxeter system.
Let $(A_S,S)$ be an Artin-Tits and $X$ a subset of $S$ ; denote by $A_X$ the subgroup of $A_S$ generated by $X$. When $A_S$ is of spherical type, we prove that the normalizer and the commensurator of $A_X$ in $A_S$ are equal and are the product of $A_X$ by the quasi-centralizer of $A_X$ in $A_S$. Looking to the associated monoids $A_S^+$ and $A_X^+$, we describe the quasi-centralizer of $A_X^+$ in $A_S^+$ thanks to results in Coxeter groups. These two results generalize earlier results of Paris. Finaly, we compare, in the spherical case, the normalizer of a parabolic subgroup in the Artin-Tits…
The aim of this note is to show that the cycle decomposition of elements of the symmetric group admits a quite natural formulation in the framework of dual Coxeter theory, yielding a generalization of it to the family of so-called parabolic quasi-Coxeter elements of Coxeter groups (in the symmetric group every element is a parabolic quasi-Coxeter element). We show that such an element admits an analogue of the cycle decomposition. Elements which are not in this family still admit a generalized cycle decomposition, but it is not unique in general.
Let $(W,S)$ be a Coxeter system, and let $X$ be a subset of $S$. The subgroup of $W$ generated by $X$ is denoted by $W_X$ and is called a parabolic subgroup. We give the precise definition of the commensurator of a subgroup in a group. In particular, the commensurator of $W_X$ in $W$ is the subgroup of $w$ in $W$ such that $wW_Xw^{-1}\cap W_X$ has finite index in both $W_X$ and $wW_Xw^{-1}$. The subgroup $W_X$ can be decomposed in the form $W_X = W_{X^0} \cdot W_{X^\infty} \simeq W_{X^0} \times W_{X^\infty}$ where $W_{X^0}$ is finite and all the irreducible components of $W_{X^\infty}$" > are …
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 $W$.
One can observe that Coxeter groups and right-angled Artin groups share the same solution to the word problem. On the other hand, in his study of reflection subgroups of Coxeter groups Dyer introduces a family of groups, which we call Dyer groups, which contains both, Coxeter groups and right-angled Artin groups. We show that all Dyer groups have this solution to the word problem, we show that a group which admits such a solution belongs to a little more general family of groups that we call quasi-Dyer groups, and we show that this inclusion is strict. Then we show several results on parabolic…
We show that right-angled Coxeter groups are relatively hyperbolic in the sense defined by Farb, relative to a natural collection of rank-2 parabolic subgroups.
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.
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.
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
Let G be a complete Kac-Moody group over a finite field. It is known that G possesses a BN-pair structure, all of whose parabolic subgroups are open in G. We show that, conversely, every open subgroup of G is contained with finite index in some parabolic subgroup; moreover there are only finitely many such parabolic subgroups. The proof uses some new results on parabolic closures in Coxeter groups. In particular, we give conditions ensuring that the parabolic closure of the product of two elements in a Coxeter group contains the respective parabolic closures of those elements.
In recent papers we have refined a conjecture of Lehrer and Solomon expressing the characters of a finite Coxeter group $W$ afforded by the homogeneous components of its Orlik-Solomon algebra as sums of characters induced from linear characters of centralizers of elements of $W$. Our refined conjecture also relates the Orlik-Solomon characters above to the terms of a decomposition of the regular character of $W$ related to the descent algebra of $W$. A consequence of our conjecture is that both the regular character of $W$ and the character of the Orlik-Solomon algebra have parallel, graded de…
We introduce and investigate the ribbon groupoid associated with a Garside group. Under a technical hypothesis, we prove that this category is a Garside groupoid. We decompose this groupoid into a semi-direct product of two of its parabolic subgroupoids and provide a groupoid presentation. In order to established the latter result, we describe quasi-centralizers in Garside groups. All results hold in the particular case of Artin-Tits groups of spherical type.
In this paper we prove that $2^{20+1}.U_6(2)$, known as the centralizer of an involution in the group $2E_6(2)$ is a quotient of a Coxeter group. We obtain a presentation of $2^{20+1}.U_6(2)$ as a $Q_{222}$-group, which now resolve a long pending question.
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…
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…
Oshima's Lemma describes the orbits of parabolic subgroups of irreducible finite Weyl groups on crystallographic root systems. This note generalises that result to all root systems of finite Coxeter groups, and provides a self contained proof, independent of the representation theory of semisimple complex Lie algebras.
Let $(W, S)$ be a Coxeter system. We give necessary and sufficient conditions on the Coxeter diagram of $(W, S)$ for $W$ to be relatively hyperbolic with respect to a collection of finitely generated subgroups. The peripheral subgroups are necessarily parabolic subgroups (in the sense of Coxeter group theory). As an application, we present a criterion for the maximal flats of the Davis complex of $(W,S)$ to be isolated. If this is the case, then the maximal affine sub-buildings of any building of type $(W,S)$ are isolated.
Let $(W,S)$ be a Coxeter system of finite rank and let $J,K\subset S$. We study the rationality of the Poincaré series of the set of representatives of minimal length of $(W_J,W_K)$-double cosets of $W$: we conclude that it depends mostly on the rationality of the Poincaré series of the normalizers of finite parabolic subgroups of $W$. For affine Weyl groups, we prove that all these series are rational and we give some explicit examples.
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.
In a recent paper we claimed that both the group algebra of a finite Coxeter group $W$ as well as the Orlik-Solomon algebra of $W$ can be decomposed into a sum of induced one-dimensional representations of centralizers, one for each conjugacy class of elements of $W$, and gave a uniform proof of this claim for symmetric groups. In this note we outline an inductive approach to our conjecture. As an application of this method, we prove the inductive version of the conjecture for finite Coxeter groups of rank up to 2.
For a finite Coxeter group $W$ and $w$ an element of $W$ the `excess' of $w$ is defined to be $e(w) = \min\{\ell(x) + \ell(y) - \ell(w) \; | \; w=xy, \; x^2 = y^2 = 1\}$ where $\ell$ is the length function on $W$. Here we investigate the behaviour of $e(w)$, and a related concept reflection excess, when restricted to standard parabolic subgroups of $W$. Also the set of involutions inverting $w$ is studied.
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…
Let G be an 1-connected simple Lie group, and let x\inG be a group element. We determine the isomorphism type of the centralizer C_{x} in term of a minimal geodesic joinning the group unit e\inG to x. This result is applied to classify the isomorphism types of maximal subgroups of maximal rank of G, and the isomorphism types of parabolic subgroups of G.
We develop combinatorics of parabolic double cosets in finite Coxeter groups as a follow-up of recent articles by Billey-Konvalinka-Petersen-Slofstra-Tenner and Petersen. (1) We construct a double coset system as a generalization of a two-sided analogue of a Coxeter complex and present its order structure with its local dimension function on certain connected components. As applications of double cosets to Bruhat graphs, we also prove: (2) every parabolic double coset is regular, (3) invariance of degree on Bruhat graph on lower intervals as an analogy of the one for Kazhdan-Lusztig polynomial…
The main result of this paper describes the normalizer of a finite parabolic subgroup of a (possibly infinite) Coxeter group. We use this to compute the automorphism groups of some Lorentzian lattices and K3 surfaces.
A model for a finite group is a set of linear characters of subgroups that can be induced to obtain every irreducible character exactly once. A perfect model for a finite Coxeter group is a model in which the relevant subgroups are the quasiparabolic centralizers of perfect involutions. In prior work, we showed that perfect models give rise to interesting examples of $W$-graphs. Here, we classify which finite Coxeter groups have perfect models. Specifically, we prove that the irreducible finite Coxeter groups with perfect models are those of types $\mathsf{A}_{n}$, $\mathsf{B}_n$, $\mathsf{D}_…
It is well known that the normaliser of a parabolic subgroup of a finite Coxeter group is the semidirect product of the parabolic subgroup by the stabiliser of a set of simple roots. We show that a similar result holds for all finite unitary reflection groups: namely, the normaliser of a parabolic subgroup is a semidirect product and in many, but not all, cases the complement can be obtained as the stabiliser of a set of roots. In addition we record when the complement acts as a reflection group on the space of fixed points of the parabolic subgroup.
We refine a conjecture by Lehrer and Solomon on the structure of the Orlik-Solomon algebra of a finite Coxeter group $W$ and relate it to the descent algebra of $W$. As a result, we claim that both the group algebra of $W$, as well as the Orlik-Solomon algebra of $W$ can be decomposed into a sum of induced one-dimensional representations of element centralizers, one for each conjugacy class of elements of $W$. We give a uniform proof of the claim for symmetric groups. In addition, we prove that a relative version of the conjecture holds for every pair $(W, W_L)$, where $W$ is arbitrary and $W_…
In this paper, we prove that all finitely generated malnormal subgroups of one-ended right-angled Coxeter groups are strongly quasiconvex and they are in particular quasiconvex when the ambient groups are hyperbolic. The key idea is to prove all infinite proper malnormal subgroups of one-ended right-angled Coxeter groups are join-free and then prove the strong quasiconvexity and the virtual freeness of these subgroups. We also study the subgroup divergence of join-free subgroups in right-angled Coxeter groups and compare them with the analogous subgroups in right-angled Artin groups. We charac…
This paper provides some evidence for conjectural relations between extensions of (right) weak order on Coxeter groups, closure operators on root systems, and Bruhat order. The conjecture focused upon here refines an earlier question as to whether the set of initial sections of reflection orders, ordered by inclusion, forms a complete lattice. Meet and join in weak order are described in terms of a suitable closure operator. Galois connections are defined from the power set of W to itself, under which maximal subgroups of certain groupoids correspond to certain complete meet subsemilattices of…
In recent papers we have refined a conjecture of Lehrer and Solomon expressing the character of a finite Coxeter group $W$ acting on the $p$th graded component of its Orlik-Solomon algebra as a sum of characters induced from linear characters of centralizers of elements of $W$. Our refined conjecture relates the character above to a component of a decomposition of the regular character of $W$ related to Solomon's descent algebra of $W$. The refined conjecture has been proved for symmetric and dihedral groups, as well as finite Coxeter groups of rank three and four. In this paper, the second in…
We study the buildings in which parallelism of residues is an equivalence relation. If the building admits a group action, we describe how parallel residues are related to residues with equal stabilizers. This permits to retrieve the fact that in a Coxeter group or in a graph product, intersections of parabolic subgroups are parabolic.
