Brent Everitt, John Fountain
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.
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We show that every finitely generated Artin-Tits group admits a finite Garside family, by introducing the notion of a low element in a Coxeter group and proving that the family of all low elements in a Coxeter system (W, S) with S finite includes S and is finite and closed under suffix and join with respect to the right weak order.
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.
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 $G(d, d, n)$.
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 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 Todd-Coxeter Algorithm for Semigroups and Monoids
该研究介绍了计算半群和幺半群同余的Todd-Coxeter算法，并创新性提出了半群场景下对应群算法中Felsch策略的类似策略。
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
本文证明有限Coxeter群中抛物型拟Coxeter元素的反射分解的Hurwitz轨道由生成子群和共轭类多重集完全区分，并分类了满足极小生成集为最小大小的有限Coxeter群。
We introduce a relative Garside element, the quotient of the corresponding Garside elements G1 and G2, for a pair of Artin monoids associated to Coxeter graphs Gamma1 and Gamma2, the second graph containing a new vertex. These relative elements give a recurrence relation between Garside elements. As an application, we compute explicitly the Garside elements of Artin monoids corresponding to spherical Coxeter graphs or the longest elements of the associated finite Coxeter groups.
Reflections and quotients in Coxeter groups
提出了一个公式，将 Coxeter 群元素的左下降集与其在极大商投影的左下降集联系起来，这是涉及商与下降的一般性结果的特例。
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.
On generating functions and automata associated to reflections in Coxeter systems
本文研究Coxeter系统反射的生成函数与自动机，证明反射前缀语言的正则性，并推导出仿射Coxeter群反射Poincaré级数的简单公式。
We construct a new monoid structure for Artin groups associated with finite Coxeter systems. This monoid shares with the classical positive braid monoid a crucial algebraic property: it is a Garside monoid. The analogy with the classical construction indicates there is a ``dual'' way of studying Coxeter systems, where the pair (W,S) is replaced by (W,T), with T the set of all reflections. In the type A case, we recover the monoid constructed by Birman-Ko-Lee
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.
Noncrossing combinatorics of quasi-Coxeter elements
该研究探讨有限Coxeter群绝对序中拟Coxeter元下方区间的组合学，证明其EL-可壳性（E₇、E₈型可能例外），计算ζ多项式，证明Williams关于Coxeter元的猜想，还发现相关区间幺半群的可消去性反例，否定了Baumeister等人的问题。
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.
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.
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 show the analogue of Muhlherr's [Coxeter groups in Coxeter groups, Finite Geom. and Combinatorics, Cambridge Univ. Press (1993), 277-287] for Artin-Tits monoids, and for Artin-Tits groups of spherical type. That is, the submonoid (resp. subgroup) of an Artin-Tits monoid (resp. group of spherical type) induced by an admissible partition of the Coxeter graph is an Artin-Tits monoid (resp. group). This generalizes and unifies the situation of the submonoid (resp. subgroup) of fixed elements of an Artin-Tits monoid (resp. group of spherical type) under the action of graph automorphisms, and the…
Powers of Coxeter elements with unbounded reflection length
本文研究 Coxeter 群中 Coxeter 元的幂的反射长度，证明在辫关系足够大时该长度无界，并建立任意 Coxeter 群与同秩泛 Coxeter 群反射长度函数的联系，同时给出特殊 Coxeter 矩阵情形下的上界。
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…
We propose a slight weakening of the definitions of Artin monoids and Coxeter monoids. We study one `infinite series' in detail.
We derive functional relationships between spherical generating functions of graph monoids, right-angled Artin groups and right-angled Coxeter groups. We use these relationships to express the spherical generating function of a right-angled Artin group in terms of the clique polynomial of its defining graph. We also describe algorithms for computing the geodesic generating functions of these structures.
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.
Reflection Representations of Coxeter Groups and Homology of Coxeter Graphs
本文利用与Coxeter图相关图的整同调群特征标，分类有限秩Coxeter群的广义几何表示，并刻画生成元以反射方式作用的表示。
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.
Stable reflection length in Coxeter groups
本文在Coxeter群中引入稳定反射长度以研究反射长度的渐近行为，建立其与稳定换位子长度和稳定挠长度的联系，并完整刻画了反射长度在幂次上无界的元素。
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.
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 $(W,S)$ 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 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 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 2010, Everitt and Fountain introduced the concept of reflection monoids. The Boolean reflection monoids form a family of reflection monoids (symmetric inverse semigroups are Boolean reflection monoids of type $A$). In this paper, we give a family of presentations of Boolean reflection monoids and show how these presentations are compatible with quiver mutations of orientations of Dynkin diagrams with frozen vertices. Our results recover the presentations of Boolean reflection monoids given by Everitt and Fountain and the presentations of symmetric inverse semigroups given by Popova respecti…
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.
Computing normal forms in groups (or monoids) is in general harder than solving the word problem (equality testing). However, normal form computation has a much wider range of applications. It is therefore interesting to investigate the complexity of computing normal forms for important classes of groups. For Coxeter groups we show that the following algorithmic tasks can be solved by a deterministic Turing machine using logarithmic work space, only: 1. Compute the length of any geodesic normal form. 2. Compute the set of letters occurring in any geodesic normal form. 3. Compute the Parikh-ima…
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.
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 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 $W$.
Prefixes of Coxeter elements are Coxeter elements of the reflection subgroups they generate
本文研究有限秩Coxeter系统中Coxeter元的前缀性质，证明其为对应反射子群的Coxeter元，相关子群由其既约反射分解因子生成，且该分解上的Hurwitz作用可迁，将一般情形归约为Hubery与Krause已处理的特殊情形。
Hurwitz numbers for reflection groups II: Parabolic quasi-Coxeter elements
本文在良生成复反射群中定义并刻画抛物拟 Coxeter 元素，计算其约化反射分解与相对生成集的基数，并将后者与 Frobenius 流形几何不变量关联，为后续统一公式奠定基础。
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…
This is the second in a series of papers that develops the theory of reflection monoids, motivated by the theory of reflection groups. Reflection monoids were first introduced in arXiv:0812.2789. In this paper we study their presentations as abstract monoids. Along the way we also find general presentations for certain join-semilattices (as monoids under join) which we interpret for two special classes of examples: the face lattices of convex polytopes and the geometric lattices, particularly the intersection lattices of hyperplane arrangements. Another spin-off is a general presentation for t…
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…
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. $(G_k,S_k)$
On maximal dihedral reflection subgroups and generalized noncrossing partitions
本文仅基于字的组合学给出了 Dyer 关于任意 Coxeter 群中极大二面体反射子群唯一性结果的新证明，并借此证明了绝对序中长度为3的区间构成格，从而推导出对应的区间群为拟Garside群。
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…
Affine Dual Braid Monoids: Finite Cores, Exceptional Cluster Complexes, and Koszul Resolutions
本研究针对有限秩晶体学仿射Coxeter系统，构造了平凡模的极小线性分次自由分解，证明仿射对偶辫群代数是Koszul的，揭示了例外丛复形的拓扑性质与同调意义。
Affine reflection subgroups of Coxeter groups
本文研究有限秩无限 Coxeter 群中的仿射反射子群，刻画了由它们产生的极限根及群拥有此类子群的条件，并揭示了归一化迷向锥与虚锥的交集由相关极限根组成。
Let $(W, I)$ be a finite Coxeter group. In the case where $W$ is a Weyl group, Berenstein and Kazhdan in \cite{BK} constructed a monoid structure on the set of all subsets of $I$ using unipotent $χ$-linear bicrystals. In this paper, we will generalize this result to all types of finite Coxeter groups (including non-crystallographic types). Our approach is more elementary, based on some combinatorics of Coxeter groups. Moreover, we will calculate this monoid structure explicitly for each type.
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.
For a real reflection group the reflecting hyperplanes cut out on the unit sphere a simplicial complex called the Coxeter complex. Abramenko showed that each reflecting hyperplane meets the Coxeter complex in another Coxeter complex if and only if the Coxeter diagram contains no subdiagram of type $D_4$, $F_4$, or $H_4$. The present paper extends Abramenko's result to a wider class of complex reflection groups. These groups have a Coxeter-like presentation and a Coxeter-like complex called the Milnor fiber complex. Our first main theorem classifies the groups whose reflecting hyperplanes meet …
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.
In this work we study representations of certain Coxeter groups to obtain some properties of the corresponding reflection groups.
There are well known relations between braid groups and symmetric groups, between Artin-Briskorn braid groups and Coxeter groups. Inverse braid monoid the same way is related to the inverse symmetric monoid. In the paper we show that similar relations exist between the inverse braid monoid of type $B$ and the inverse reflection monoid of type $B$. This gives a presentation of the last monoid.
Hurwitz Equivalence of Reflection Factorizations in Types $B$ and $D$
本文证明了 Lewis 关于有限 Coxeter 群中反射因子分解的 Hurwitz 等价性猜想在类型 A、B、D 下成立，并给出了将猜想约化为例外类型有限问题的约化方法。
Birman, Ko and Lee have introduced a new monoid ${\cal B}^{*}_{n}$--with an explicit presentation--whose group of fractions is the $n$-strand braid group ${\cal B}_{n}$. Building on a new approach by Digne, Michel and himself, Bessis has defined a {\it dual} braid monoid for every finite Coxeter type Artin-Tits group extending the type A case. Here, we give an explicit presentation for this dual braid monoid in the case of types B and D, and we study the combinatorics of the underlying Garside structures.
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