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

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4
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2answers
32 views

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 \ ...
-1
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0answers
26 views

efficient estimate for closeness of 2 points

Hi I am writing a computerprogram and I need a sensable and efficient estimate to determine when 2 points can be called the same. I have some coxeter group. It acts on $\Bbb R^n$ and I want to ...
0
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0answers
19 views

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 ...
4
votes
1answer
57 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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0answers
11 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 ...
4
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2answers
57 views

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. ...
2
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0answers
20 views

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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0answers
22 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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0answers
37 views

Reflection in the highest root

Is there some canonical (reduced) expression (in terms of simple reflections) for the reflection associated to the highest (short) root? Is this the same as the longest element of the Weyl group? ...
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0answers
31 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: ...
0
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1answer
51 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?
2
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1answer
43 views

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 ...
2
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1answer
38 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, ...
3
votes
1answer
30 views

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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0answers
24 views

Length function

Let $W$ be a Coxeter group with simple system $S$, positive system $P$ and root system $R$. Then $S\subset P\subset R$. Let $\lambda:R\rightarrow\{0,1\}$ be the characteristiv function of $P$, in ...
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0answers
26 views

Generic algebra of a Coxeter Group

This question is about Chapter $7$ Exercise $2$ of Humphreys "Reflection Groups and Coxeter Groups". Let $(W,S)$ be a Coxeter System, then we define $E:=\bigoplus_{w\in W}{A\cdot T_w}$ for $A$ a ...
3
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0answers
20 views

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 ...
2
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0answers
31 views

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 ...
0
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1answer
29 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 ...
4
votes
1answer
183 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 ...
1
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1answer
76 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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0answers
21 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?
2
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0answers
57 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 ...
2
votes
3answers
74 views

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 ...
3
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1answer
69 views

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 ...
2
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0answers
28 views

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, $W=\langle s_i,i=1,2,\cdots,n | s_i^2 =1\rangle$. ...
0
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1answer
47 views

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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1answer
109 views

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
78 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 = ...
1
vote
1answer
87 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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0answers
19 views

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
47 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 ...
26
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2answers
724 views

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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0answers
55 views

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
48 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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0answers
247 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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1answer
94 views

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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1answer
80 views

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 ...
2
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0answers
45 views

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 ...
4
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0answers
87 views

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 ...
4
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0answers
61 views

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 ...
3
votes
1answer
104 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 ...
2
votes
1answer
99 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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2answers
115 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.
4
votes
1answer
202 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 ...
3
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0answers
59 views

The universal space of a Coxeter group

Consider a Coxeter group $(W,S)$ and a topological space $X$. We define a mirror structure on $X$ as a locally finite family $(X_{s})_{s\in S}$ of closed subspaces of $X$. Let's consider $W$ with the ...
2
votes
0answers
232 views

Significance of deletion and exchange conditions in reflection groups

I am having trouble warping my head around the exchange and deletion conditions in finite reflection groups (i.e.Coxeter groups). It is mentioned as the "characterising property of coxeter groups ...
2
votes
1answer
289 views

Converting a (signed) permutation to a reduced word

I vaguely know that by looking at the inversions of a permutation, you can write down the reduced word expressing the permutation as a product of adjacent transpositions $s_i = (i,i+1)$. However, I ...
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0answers
53 views

Doubt in the proof of Conjugacy of positive systems under reflection group

I am stuck at a small thing in the proof of conjugacy of positive systems under a finite reflection group. I am using the notation and definitions used in the text by James E. Humphreys. I reproduce ...
8
votes
1answer
163 views

Description of flipping tableau for inversions in reduced decompositions of permutations

Short version: Is there a graphical description of the possible orders in which inversions can appear in a reduced decomposition of a permutation? Something akin to the definition of standard Young ...