1
vote
0answers
38 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
55 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 ...
1
vote
1answer
33 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 ...
4
votes
0answers
50 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
88 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
96 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.
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
195 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
152 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
190 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 $\mathbb{R}^n$, ...