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Automorphisms of coxeter groups

Takato Uehara. In this paper, we consider automorphism groups of rational surfaces which admit cuspidal anticanonical curves and have certain nontrivial automorphisms. By applying Coxeter theory, we show that the automorphism groups of the surfaces are isomorphic to the infinite cyclic group. Automorphism groups of rational surfaces. N2 - In this paper, we consider automorphism groups of rational surfaces which admit cuspidal anticanonical curves and have certain nontrivial automorphisms.

AB - In this paper, we consider automorphism groups of rational surfaces which admit cuspidal anticanonical curves and have certain nontrivial automorphisms. Overview Fingerprint. Abstract In this paper, we consider automorphism groups of rational surfaces which admit cuspidal anticanonical curves and have certain nontrivial automorphisms.

automorphisms of coxeter groups

Keywords Automorphism group Cuspidal anticanonical curve Rational surface. Access to Document Link to publication in Scopus. Link to citation list in Scopus. Rational Surface Mathematics. Journal of Pure and Applied Algebra1 Uehara T. Journal of Pure and Applied Algebra. Uehara, Takato. In: Journal of Pure and Applied Algebra.If is a right-angled Coxeter system, then is a semidirect product of the group of symmetric automorphisms by the automorphism group of a certain groupoid.

We show that, under mild conditions, is a semidirect product of by the quotient. We also give sufficient conditions for the compatibility of the two semidirect products. When this occurs there is an induced splitting of the sequence and consequently, all group extensions are trivial. A Coxeter group is determined by its diagram It is known that in certain cases, determines as well see, e. This is the case for right-angled Coxeter groups [ 34 ], where the only relations are for all generators and for some pairs of generators and For right-angled Coxeter groups, it is convenient to consider the Coxeter diagram rather than the classical Coxeter graph : the presence of an edge with endpoints and means that and commute in.

In the present article, we focus on the set whose members are the vertex sets of maximal complete subgraphs. We say that has condition C if there exist and a collection such that C1 for C2 for each the cardinality of the set is odd. Motivated by the results of Tits regarding sequence 1. Clearly 1. For example, is a right-angled Coxeter group and is a nontrivial extension.

On the other hand, if has no center, then finding nontrivial extensions of is surprisingly difficult. Whether 1. The group has been studied extensively in [ 57 ] and is called the group of symmetric automorphisms of We approach the problem of whether 1.

Automorphisms of Right-Angled Coxeter Groups

A positive answer to both a and b implies that 1. We show in Theorem 5. To obtain a splitting of 1.

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Theorem 1. If has condition C and each leaves all vertices of invariant, then 1. It was recently shown [ 8 ] that 1.

However, this result does not lead to a generalization of Theorem 1. Coxeter groups are typically defined by presentations, and there are various conventions for representing such presentations diagramatically.

In this section, we review some definitions and important properties, focusing exclusively on the right-angled case. See [ 9 ] or [ 10 ] for a comprehensive treatment. If is any set, let denote the set of subsets of with cardinality 2. Definition 2. Given a finite set and let denote the undirected graph with vertex set and edge set note that does not have loops or parallel edges.

automorphisms of coxeter groups

As such graphs are often used to represent right-angled Coxeter groups, is called a Coxeter diagram. Given set The graph is the subgraph of spanned by A complete subgraph is maximal if it is not properly contained in any complete subgraph of. The presentation is the Coxeter presentation defined by Definition 2. Remark 2. The Coxeter diagram is not the same as the traditional Dynkin diagram. Indeed, as graphs, the Coxeter and Dynkin diagrams are complementary. Clearly, each Coxeter diagram defines a unique right-angled Coxeter group.

On the other hand, to recover the diagram from a group one must first choose a particular Coxeter presentation. It is natural to wonder whether nonisomorphic diagrams might define isomorphic groups.

The relationship between right-angled Coxeter groups and their diagrams is clarified by the following result. Theorem 2. If and are right-angled Coxeter systems for then there is an automorphism such that.Years ago I stumbled upon the convex regular 4-polytopes - four-dimensional analogues of the Platonic solids. I did not understand the mathematics behind these structures back then - but recently I decided to figure out how this works.

This page is not mobile friendly. While some content might be viewable, the WebGL stuff will likely not work. The page is best viewed with Chrome, and with a discrete GPU enabled for the ray traced stuff. My first encounter with four-dimensional polytopes was Jenn 3D by Fritz Obermeyer.

Jenn describes itself as follows:.

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Jenn is a toy for playing with various quotients of Cayley graphs of finite Coxeter groups on four generators. Jenn builds the graphs using the Todd-Coxeter algorithm, embeds them into the 3-sphere, and stereographically projects them onto Euclidean 3-space. This description made absolutely no sense to me when I first read it - none of these terms were familiar to me. My purpose with this page is to go through this description step-by-step and explain the various terms.

We will also go one step further, and discuss how these structures can be ray traced in realtime. This can be done using a clever inverse construction technique of folding space into a fundamental domain. The cube has several automorphisms - transformations that will map the cube onto itself.

Including the identity transformation, and taking into account that we could also mirror each one of these transformations, we arrive at 48 automorphisms for the cube. Besides the rotation symmetries above, we can also depict the reflection symmetries. Shown to the right are the 9 different such reflection operations, that will map the cube onto itself although in a mirrored version. These 48 transformations form a group : a set of elements, together with a rule for combining any two elements such that the result is again an element in the set.

There must an identity element in the group, and every element must have an inverse element: i. In our case, combining any number of the 48 transformations will result in a transformation that is already present in our set of transformations.

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Likewise, for all rotations and reflections, there exists an inverse transformation for degrees rotations, and for reflections, the transformations are their own inverses.It is known that two Coxeter groups in this family are isomorphic if and only if they admit Coxeter systems having the same rank and the same multiset of finite exponents.

In particular, each group in this family is isomorphic to a group that admits a Coxeter system whose associated labeled graph is a star shaped tree. We give the complete description of the automorphism group of this group, and derive a sufficient condition for the splitting of the automorphism group as a semi-direct product of the inner and the outer automorphism groups.

This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Bahls, P. Imperial College Press, London Bardakov, V. Bell, R. Algebra 1— Springer, New York Bourbaki, N.

Chapters 4—6. Translated from the French Original by Andrew Pressley. Springer, Berlin Hermann, Paris, Brady, N. Dedicata, vol. Brink, B. Brown, K.

On torsion images of Coxeter groups and question of Wiegold

Dyer, J.MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up. But I might as well ask the question more generally. Suppose we have a Coxeter diagram.

This gives a Coxeter group. What are the outer automorphisms of this group? It seems we get one from any symmetry of the diagram; are these all of them?

So, I'm hoping this analogous result is true. But maybe it's too bold a generalization; I'll settle for Coxeter diagrams that come from Dynkin diagrams. No and no. I found this result in this thesis by William Franszen which has much more information about this problemwhich was the second search result I got for "outer automorphisms of coxeter groups.

It's useful to look at a short paper by Tits Sur le groupe des automorphismes de certains groupes de CoxeterJ. Algebrano. While he doesn't treat arbitrary Coxeter groups he does include the ones of interest to you. Probably there is more recent literature coming from the study of the isomorphism problem for Coxeter groups. The article by Tits is partly based on an earlier one that year in the same journal by L.

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James here. Sign up to join this community. The best answers are voted up and rise to the top. What are the outer automorphisms of a Coxeter group? Ask Question.

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Asked 6 years, 1 month ago. Active 6 years, 1 month ago. Viewed times. Improve this question. John Baez. John Baez John Baez Active Oldest Votes. Improve this answer. Qiaochu Yuan Qiaochu Yuan k 32 32 gold badges silver badges bronze badges. Sound right? In any case, the main point is to look at "commuting" subsets of the simple generators.

Jim Humphreys Jim Humphreys Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown.We show that every Coxeter group that is not virtually abelian and for which all labels in the corresponding Coxeter graph are powers of 2 or infinity can be mapped onto uncountably many infinite 2-groups which, in addition, may be chosen to be just-infinite, branch groups of intermediate growth.

Also we answer affirmatively a question raised by Wiegold in Kourovka Notebook. One of the most outstanding problems in Algebra known as the Burnside Problem on periodic groups was formulated by Burnside in and was later split into three branches: the General Burnside Problem, the Bounded Burnside Problem, and the Restricted Burnside Problem. The General Burnside Problem was asking if there exists an infinite finitely generated torsion group. The Bounded Burnside Problem was solved by S.

Novikov and S. The Restricted Burnside Problem was solved by E. The problem of Burnside inspired a lot of activity and new directions of research. Among various problems around the Burnside problem is the problem on minimal values of periods of elements. In the case of the Bounded Burnside Problem the main remaining open question is: what is the minimal n such that the free Burnside group. Is it 5,7,8 or a larger number? By the celebrated result of E. Therefore making the order of any element of G smaller will make the group finite.

The case of p -groups is of special interest because of many reasons. As we will see, the orders 2 and 4 and 8 for the product x y are possible values, while the triple 244 is not possible because the corresponding group is crystallographic. As Coxeter groups are generated by involutions it is natural to investigate their 2-torsion quotients.

A Coxeter group can be described by a Coxeter graph Z. If a Coxeter graph is not connected, then the group C is a direct product of Coxeter subgroups corresponding to the connected components.

Therefore we may focus on the case of connected Coxeter graphs. If we are interested in 2-torsion quotients of Cthen one has to assume that m i j are powers of 2 or infinity. In order for C to have infinite torsion quotients it has to be infinite and not virtually abelian. The list of finite and virtually abelian Coxeter groups with connected Coxeter graphs is well known.

A comprehensive treatment of Coxeter groups can be found in M. Let C be a non virtually abelian Coxeter group defined by a connected Coxeter graph Z with all edge labels m i j being powers of 2 or infinity. Moreover these quotients can be chosen to be residually finite, just-infinite, branch 2-groups of intermediate growth and the main property that distinguishes them is the growth type of the group. The definition of a branch group is a bit involved and we direct the reader to [ 16154 ] for more information on branch groups.It turns out the lights are triggered by the opening of the back doors so if the doors are opened between having been driven the warning lights come on upon start-up.

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automorphisms of coxeter groups

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