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

learn more… | top users | synonyms

2
votes
3answers
54 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 ...
2
votes
1answer
49 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
votes
0answers
15 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
votes
1answer
20 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. ...
1
vote
1answer
84 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 = ...
1
vote
1answer
39 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
58 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 ...
0
votes
0answers
12 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 ...
1
vote
1answer
32 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 ...
1
vote
0answers
22 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 ...
1
vote
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 $W_1 \times W_2$? ...
1
vote
0answers
128 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 ...
1
vote
1answer
52 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?
1
vote
1answer
58 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
votes
0answers
39 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 ...
1
vote
0answers
55 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
votes
0answers
49 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
85 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
0answers
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 ...
1
vote
2answers
94 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
124 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
votes
0answers
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 ...
2
votes
0answers
169 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
189 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 ...
1
vote
0answers
44 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 ...
7
votes
1answer
149 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 ...
5
votes
2answers
188 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 $\mathbb{R}^n$, ...
3
votes
0answers
93 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!
3
votes
1answer
89 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 ...
5
votes
0answers
165 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 ...