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

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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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1answer
26 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
86 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
50 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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17 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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48 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
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3answers
61 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
57 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 ...
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19 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, $W=\langle s_i,i=1,2,\cdots,n | s_i^2 =1\rangle$. ...
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1answer
28 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
91 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
50 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
64 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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13 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
36 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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27 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
37 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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171 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
60 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
65 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 ...
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0answers
40 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 ...
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63 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 ...
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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
89 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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65 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
101 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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1answer
138 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 ...
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54 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 ...
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0answers
180 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
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1answer
208 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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47 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 ...
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1answer
155 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 ...
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2answers
202 views

Reflection groups and symmetric group

Define the action of $S_n$ on $\mathbb{R}^n$: take any $x\in S_n$, consider the mapping $x: \mathbb{R}^n\to\mathbb{R}^n$, $e_1, e_2 ...e_n$ are the standard basis of $\mathbb{R}^n$, ...
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0answers
97 views

lattices in semisimple Lie groups

I would like to learn more on lattices in semisimple Lie groups, especially their relations with Coxeter groups. Does anyone have suggestions of books that could be useful? Thanks!
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1answer
90 views

Why is this map diagonalisable?

In some lecture notes about Reflection Groups, the writer constructs a vector space based on a Coxeter-system. For every $s \in S$ where $(W|S)$ is a Coxeter-system, he calls the related basis vector ...
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178 views

Spherical building of an exceptional group of Lie type

I've read that one of Tits' original motivations for studying buildings was that he wanted to give a unified description of algebraic groups that would allow the definition of exceptional groups such ...