Semidirect product decomposition of Coxeter groups
Cédric Bonnafé, Matthew J. Dyer
备注：
28 pages, one table. We have added some comments on parabolic subgroups, double cosets representatives, finite and affine Weyl groups, invariant theory, Solomon descent algebra
Let $(W,S)$ be a Coxeter system, let $S=I \dot{\cup} J$ be a partition of $S$ such that no element of $I$ is conjugate to an element of $J$, let $\widetilde{J}$ be the set of $W_I$-conjugates of elements of $J$ and let $\widetilde{W}$ be the subgroup of $W$ generated by $\widetilde{J}$. We show that $W=\widetilde{W} \rtimes W_I$ and that $(\widetilde{W},\widetilde{J})$ is a Coxeter system.
Nonuniqueness of semidirect decompositions of groups is an insufficiently studied question in contrast to direct decompositions. We obtain some results about semidirect decompositions for semidirect products with factors which are nontrivial direct products. We deal with a special case of semidirect product when the twisting homomorphism acts diagonally on a direct product, as well as for the case when the extending group is a direct product. We give applications of these results in the case of generalized dihedral groups and classic dihedral groups $D_{2n}$. For $D_{2n}$ we give a complete de…
We show that a linear decomposition attack based on the decomposition method introduced by the author works by finding the exchanged secret keys in all main protocols using semidirect products of (semi)grops proposed by Kahrobaei, Shpilrain, Habeeb, Koupparis and Lam.
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.
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.
三维流形的“JSJ-分解”的想法始于瓦尔德豪森的工作，后来通过雅科、沙伦和约翰森的工作得到发展。证明了存在一个有限的2边不可压缩环面的集合，它将给定的闭不可约3-流形分成具有强拓扑结构的块。Sela引入了组的JSF分解的思想，这个思想已经在各个方向蓬勃发展。一般的想法是考虑一个特定的群类X和X中的群被另一个类Y中的群分裂。例如，Rips和Sela认为分裂群的无限循环群。对于X中的任意群G，目标是产生G在Y中具有棱道群的唯一群分解图T，使得T揭示G在Y中具有棱道群的所有群分解图.更具体地说，如果V是T的一个顶点群，那么要么没有Y-群可以分裂G和V，要么V有一个特殊的“类曲面群”结构。第二种类型的顶点群通常称为“orbifold群”。对于一个非线性生成的Coxeter系统（W，S），我们给出了W在虚交换子群上的分裂的一个约化的JJ-分解T.我们证明了T是唯一的，每个顶点和棱道群由S的子集生成（因此T是“可视的”）。T的构造是算法的。如果V是S的子集，生成T的一个Orbifold顶点群，则V是K和M的不交并，其中< M >V< K >是虚交换的，V是虚闭曲面群或V是虚自由的，< V >V是和的直积< M >< K >。
Centralisers of semi-simple elements are semidirect products
Université de Picardie-Jules Verne（亚眠皮卡第朱尔维尔纳大学）；LAMFA, CNRS UMR 7352（法国国家科学研究中心联合研究实验室 7352）；Université Paris Cité（巴黎西岱大学）；Sorbonne Université（索邦大学）；IMJ-PRG（数学与理论物理研究所）
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.
Poincare series of Klein groups, Coxeter polynomials, the Burau representation and Milnor invariants
对于二元多面体群的Poincare级数和Coxeter多项式，得到了与Euclid算法和正交多项式理论相近的命题：广义Ebeling公式，分解成分歧的连分数，Christoffel-Darboux恒等式，组合公式.关于亚历山大-康威多项式的因式分解的已知结果允许将庞加莱级数和考克斯特多项式与弦链的Burau表示和Milnor不变量连接起来。一个使用A 'Campo链路和Coxeter链路的重构。利用S. M. Gusein-Zade，F. Delgado和A.坎皮罗
Long fully commutative elements in affine Coxeter groups
Coxeter群$W$的一个元素称为全交换的，如果它的任何两个约化分解可以通过相邻交换生成元的一系列转置来联系。在预印本“Fully commutative elements in finite and affine Coxeter groups”（arXiv：1402.2166）中，R. Biagioli和作者证明了，对于每个不可约的仿射Coxeter群，关于长度的序列计数完全交换元素最终是周期性的。在目前的工作中，我们研究这个序列的周期部分，这些群体中的每一个，特别是我们确定的最小周期。我们还观察到，在类型$A$仿射，我们得到一个循环筛分现象的例子。
An element of a Coxeter group W is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. An element of a Coxeter group W is cyclically fully commutative if any of its cyclic shifts remains fully commutative. These elements were studied in Boothby et al.. In particular the authors enumerated cyclically fully commutative elements in all Coxeter groups having a finite number of them. In this work we characterize and enumerate cyclically fully commutative elements according to their Coxeter length in all finite or aff…
This paper introduces a new class of right-angled Coxeter groups with totally disconnected Morse boundaries. We construct this class recursively by examining how the Morse boundary of a right-angled Coxeter group changes if we glue a graph to its defining graph. More generally, we present a method to construct amalgamated free products of CAT(0) groups with totally disconnected Morse boundaries that act geometrically on CAT(0) spaces that have a treelike block decomposition.
We obtain new presentations for the imprimitive complex reflection groups of type $(de,e,r)$ and their braid groups $B(de,e,r)$ for $d,r \ge 2$. Diagrams for these presentations are proposed. The presentations have much in common with Coxeter presentations of real reflection groups. They are positive and homogeneous, and give rise to quasi-Garside structures. Diagram automorphisms correspond to group automorphisms. The new presentation shows how the braid group $B(de,e,r)$ is a semidirect product of the braid group of affine type $\widetilde A_{r-1}$ and an infinite cyclic group. Elements of $…
For every finite Coxeter group $Γ$, each positive braids in the corresponding braid group admits a unique decomposition as a finite sequence of elements of $Γ$, the so-called Garside-normal form.The study of the associated adjacency matrix $Adj(Γ)$ allows to count the number of Garside-normal form of a given length.In this paper we prove that the characteristic polynomial of $Adj(B_n)$ divides the one of $Adj(B_{n+1})$. The key point is the use of a Hopf algebra based on signed permutations. A similar result was already known for the type $A$. We observe that this does not hold for type $D$. T…
We extend the usual notion of fully commutative elements from the Coxeter groups to the complex reflection groups. Then we decompose the sets of fully commutative elements into natural subsets according to their combinatorial properties, and investigate the structure of these decompositions. As a consequence, we enumerate and describe the form of these elements in the complex reflection groups.
We classify the groups quasi-isometric to a group generated by finite-order elements within the class of one-ended hyperbolic groups which are not Fuchsian and whose JSJ decomposition over two-ended subgroups does not contain rigid vertex groups. To do this, we characterize which JSJ trees of a group in this class admit a cocompact group action with quotient a tree. The conditions are stated in terms of two graphs we associate to the degree refinement of a group in this class. We prove there is a group in this class which is quasi-isometric to a Coxeter group but is not abstractly commensurabl…
