For questions about Coxeter groups, an abstract group that admits a formal description in terms of reflections.

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On groups with presentations $ \langle a,b,c|a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=(abc)^s=1\rangle $…

$$ \langle a,b,c|a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=1\rangle =\Delta(p,q,r) $$ This is a presentation of a triangle group $\Delta(p,q,r)$, a special kind of Coxeter group. What about the following ...
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Mapping $\Delta(2,2,2)\mapsto \Delta(4,4,2)$…

Looking at the images below, you recognize that the adajency matrix of the graph $A_G$ splits up into three different color submatrices, with $A_G=A_d+A_b+A_d$ (where $d$ is dark, damn...). It's ...
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15 views

Intersection of halfspaces and hyperplanes

If $H_1$, $H_2$ and $H_3$ are hyperplanes in an $n$ dimensional vector space $V$ then I want to prove that the linear span of $ H_1\cap H_2^+\cap H_3$ is $H_1\cap H_3$. Clearly the span is contained ...
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38 views

Coxeter presentation of Hyperoctahedral group

I know that the hyperoctahedral group $(\mathbb{Z}/2\mathbb{Z})^n \rtimes S_n$ has the presentation $\langle s_{\text{1}},\ldots,s_n:s_{\text{1}}^{\text{2}}=s_i^2=1, ...
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Generating set of Coxeter group from a $(B,N)$-pair?

Suppose we have a $(B,N)$-pair for a group $G$. The group $W:=N/(B\cap N)$ is known to be generated by a set $S$ of involutions. Apparently $$ S=\{w\in W:B\cup BwB\text{ is a group}\}. $$ The reason ...
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1answer
69 views

A subset of roots whose mutual angles agree with those of a simple system

I would appreciate help/hints solving the following exercise from Humphreys book "Reflection Groups and Coxeter Groups", page 11, exercise 1. Let $\Phi$ be a root system of rank $n$ of unit ...
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82 views

Every positive system of roots contains a unique simple system.

The following question is in effort to understand a proof to a theorem appearing in "Reflection Groups and Coxeter Groups" by Humphreys on page 8. Let $\Phi$ be a root system in the euclidean space ...
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1answer
31 views

Lie algebra-like structure corresponding to noncrystallographic root systems

In the classification of Coxeter groups, or equivalently root systems: $$A_n, B_n/C_n, D_n, E_6, E_7, E_8, F_4, G_2, H_2, H_3, H_4, I_2(p)$$ with $p \geq 7$, the last four fail to generate any ...
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47 views

A question about Coxeter groups.

Let $G$ be a group, and $x,y,z \in G - \{1\}$, where $1$ is the identity element of $G$. Assume that $x^2=y^2=z^2=1$, $xy =yz$ and $xz$ is of infinite order. Can $G$ be a Coxeter group? Can $G$ be ...
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When is a right-angled Coxeter group one-ended?

Let $\Gamma$ be a simplicial graph (ie. without multiple edes nor loops). We define the associated right-angled Artin group $A(\Gamma)$ by the presentation $$\langle v \in V(\Gamma) \mid [u,v]=1 \ ...
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Reduce expresion

How can I reduce this the following expresion $$\sum_{k = 1}^m \frac{x_1 \sin \left( \frac{2k\pi}{m} \right) - x_2 \cos \left( \frac{2k\pi}{m} \right)}{x_1 \cos \left( \frac{2k\pi}{m} \right) + x_2 ...
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1answer
77 views

Centralizers of reflections in parabolic subgroups of Coxeter groups

Let us consider a (not necessarily finite) Coxeter group $W$ generated by a finite set of involutions $S=\{s_1,...,s_n\}$ subject (as usual) to the relations $(s_is_j)^{m_{i,j}}$ with ...
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20 views

When is a stabilizer group a reflection group?

Let $G$ be a compact, connected Lie group and $K$ a closed, connected subgroup. If $K = T$ is a maximal torus, it is well known that $W := N_G(T)/Z_G(T) = N_G(T)/T$ is a finite reflection group, the ...
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What decides if a Coxeter Group is “crystalline” or “non-crystalline”?

I am currently writing a research essay concerning Crystallographic theories applied to Virology. Do keep in mind that I am not a mathematician and I know very little about Coxeter groups in general. ...
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The product of Weyl groups is the Weyl group of the product

Let $G$ and $H$ be compact, connected Lie groups. Write $W(-)$ for the Weyl group. Then it is not hard to see that $$W(G \times H) \cong W(G) \times W(H).$$ Where can I find a published statement of ...
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26 views

An injection of Weyl groups

I've shown, quite accidentally, that Weyl group of $F_4$ injects into the Weyl group of $E_6$ as the subgroup of elements normalizing a maximal torus $T^4$ of $F_4$. One might a priori expect other ...
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32 views

Affine Hecke algebras and Lusztig relations

I study the book "Affine Hecke algebras and orthogonal polynomials" by I.G. Macdonald. He propose a formula in section $4.2$, especially formula $(4.2.9)$. This formula is the following: ...
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1answer
89 views

Weyl groups: correspondence of reflections and roots?

If W is the Weyl group of some ADE-type Lie algebra, and w is an element corresponding to a reflection (not just an involution), does it necessarily correspond to a root?
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Isomorphic Coxeter Groups

Is it true that two Coxeter groups having the same Coxeter matrix (equivalently, the same Coxeter graph) isomorphic? Because otherwise, the definition of a Coxeter group from its Coxeter matrix does ...
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1answer
41 views

Use GAP3 program for computing the h-polynomials of Lusztig

I am using GAP to compute things in Kazhdan-Lusztig theory, especially using the package "Chevie". According to the GAP3 Manual, the base change between the "C"-basis and "T"-basis can be computed, ...
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1answer
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Texts on Coxeter groups

I'm looking for an introductory text on Coxeter groups. It can assume undegraduate knowledge of Algebra (Groups up to and including the Sylow theorems in Fraleigh, elementary knowledge of rings, ...
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Lusztig's $h$-function of a dihedral group

Following the notations in Hecke algebras with unequal parameters, let $(W,S,L)$ be a weighted Coxeter system, and $H$ be the corresponding Hecke algebra with $\{c_w |w \in W\}$ the Kazhdan-Lusztig ...
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Create Platonic solids from the coxeter group (vertexes & edges & faces)

How can one define vertexes, edges and faces from the Coxeter group? For example, for all platonic solids? I would like to create a general function that takes the Coxeter diagram as input, and gives ...
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34 views

Simple systems are the “smallest”

Let $\Phi$ be a root system of a finite reflection group $W$. Let $\Delta$ be a simple system in $\Phi$. I want to prove that $\Delta$ is "the smallest" set which generates $W$, more precisely: There ...
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1answer
249 views

Is every finite group of isometries a subgroup of a finite reflection group?

Is every finite group of isometries in $d$-dimensional Euclidean space a subgroup of a finite group generated by reflections? By "reflection" I mean reflection in a hyperplane: the isometry fixing a ...
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1answer
87 views

Is every finitely generated Kleinian group commensurable to a Coxeter group?

Or, to a finitely generated reflection group? Here, I do not insist that the Coxeter group is represented as a hyperbolic reflection group. If not, what is an counterexample? And what is a ...
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22 views

Embed one Coxeter System into another

What is a good reference that explains all the braid relations and diagrams for Coxeter systems concisely? In particular, how do I embed $H_3$ inside $D_6$, or $H_4$ inside $E_8$? Any hints?
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62 views

No cycles in finite coxeter graphs

Is there an elementary (no consideration of root systems involved) proof of the fact that the graph of an finite coxeter system doesn't entail any cycle? I got as far as this: If there were any cycle ...
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3answers
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Right-angled Artin groups are residually finite

I know that residual finitness of RAAGs (Right-Angled Artin Groups) follows from linearity, but does there exist a more direct proof, maybe simpler? EDIT: I added a proof based on cube complexes ...
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Representation of cell location in hyperbolic plane

I want to represent an order-5 square tiling (image from Wikipedia; more text below image): Obviously for a simple grid I can uniquely refer to a given square by its ...
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A question on universal Coxeter group

In the set up of Lusztig's Hecke algebra with unequal parameters, let $W$ be a universal Coxeter group with finite many simple reflections, that is, $W=\langle s_i,i=1,2,\cdots,n | s_i^2 =1\rangle$. ...
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coxeter graph and root system

I want to show that a coxeter graph $\Gamma$ is connected if and only if its root system $\Phi$ is irreducible. So let $\Delta$ be a simple system of $\Phi$, and $\Delta$ is also our simple system. ...
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angles between simple roots are obtuse, problem with proof

Let $\Phi$ be a root system in the following sense: (1) $\Phi \subset \mathbb{R}^n$ consists of a finite number of nonzero vectors, (2) for each $\alpha \in \Phi$, $\Phi \cap \mathbb{R} \alpha = ...
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1answer
110 views

Angle between roots in a root system

Let $\Phi$ be a root system in the following sense: (1) $\Phi \subset \mathbb{R}^n$ consists of a finite number of nonzero vectors, (2) for each $\alpha \in \Phi$, $\Phi \cap \mathbb{R} \alpha = ...
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1answer
102 views

Isn't the picture on Wikipedia about Weyl Chambers wrong?

Wikipedia's article on Weyl groups shows an example of a root system and the corresponding fundamental chambers (in my understanding, also known as fundamental regions or fundamental domains). In this ...
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Subspace invariant under irreducible Coxeter group

I'm trying to show that if $G$ is an irreducible Coxeter group, then it acts irreducibly on vector space $V$. That is, $V$ has no nontrivial $G$-invariant subspaces. I started by assuming that $V$ has ...
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1answer
56 views

Proof about coxeter groups

Prove: If a coxeter system $(W,S)$ is reducible, then it is the product of parabolic subgroups. Reducible system means the coxeter diagram is disconnected. Parabolic subgroup: let $S$ be the set ...
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What is the Coxeter diagram for?

I understand that Coxeter diagrams are supposed to communicate something about the structure of symmetry groups of polyhedra, but I am baffled about what that something is, or why the Coxeter ...
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reflection representation of an arbitrary Coxeter group

A reflection group has a reflection representation in the natural sense. But what is the reflection representation for a Coxeter group if its simple roots cannot be regarded as reflections in the ...
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1answer
52 views

Weyl group and weight lattice chambers.

Consider two simple Lie groups $G_1$ and $G_2$. Let $G_1$ have $W_1$ as a Weyl group and $G_2$ have $W_2$ as a Weyl group. Is it true that the Weyl group of $G_1 \times G_2$ is $W_1 \times W_2$? ...
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283 views

The Tits Cone - Geometric Understanding

I know that the definition of the Tits cone is $Y=\bigcup_{w\in W}{wC}$ with W the Coxeter Groups and C the fundamental chamber. One theorem says that Y is the whole space if W is finite. But how can ...
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Why are all symmetry groups of regular polytopes are finite Coxeter groups.

Why are all symmetry groups of regular polytopes are finite Coxeter groups?
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A question on Coxeter groups

Let $W$ be a Coxeter group, and $S$ be its set of simple reflections. For any $w \in W$, define $\mathcal L(w)$ $\mathcal L(w) = \{s \in S| sw<w \}.$ Is it true: If $w \in W$, $s_1,s_2 ...
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Questions about Lusztig's $\mathbf a$-function

In chapter 13 of Lusztig's Hecke Algebra with Unequal Parameters, the function $\mathbf a$ is defined to be $$\mathbf a(z) = \max_{x,y} \deg h_{x,y,z},$$ for any $z$ in the Coxter group, where the ...
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Weyl group of a non-symmetrizable Generalized Cartan Matrix

Let $A$ be a generalized Cartan matrix on the index set $I$. Define the Weyl group of $A$ as the Coxeter group on the basis $I$ with $m(i,j)=2,3,4,6,\infty$ according to whether $A_{ij} A_{ji}$ is ...
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What is the structure of the Coxeter groups of type $\text{D}_n$

I am curious on the structure of the Coxeter group $G$ of type $\text{D}_n$. Here I let $\{e_1,\cdots,e_n\}$ be the standard basis of the vector space $\mathbb{R}^n$. Then I choose ...
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1answer
114 views

reflection groups and hyperplane arrangement

We know that for the braid arrangement $A_\ell$ in $\mathbb{C}^\ell$: $$\Pi_{1 \leq i < j \leq \ell} (x_i - x_j)=0,$$ $\pi_1(\mathbb{C}^\ell - A_\ell) \cong PB_\ell$, where $PB_\ell$ is the pure ...
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1answer
105 views

the orbit of a root under operations of irreducible crystallographic group?

Suppose we have an irreducible crystallographic coxeter group G acting in a vector space V, how can we show that the orbit of an ...
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122 views

List of all elements of the Weyl group of type $C_3$.

What is the list of all elements of the Weyl group of type $C_3$ in terms of simple refletions $s_1, s_2, s_3$? There are 48 elements in the group. Thank you very much.
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259 views

Irreducible Representations of Finite Coxeter Groups

The Coxeter group is defined as $$S = \langle s_i : s_i^2 = (s_i s_j)^{m_{ij}} = 1 \rangle $$ Does it have an irreducible representation of dimension >2 for $S$ finite? Is there a reference on ...