All Questions
7
questions
7
votes
0
answers
108
views
Proving that this relation implies another relation on the Coxeter group [4,3,3,4].
I have a group with five generators $\sigma_i$, and the following relations:
\begin{split}
\sigma_i^2 = \varepsilon \\
|i-j| \neq 1 \implies (\sigma_i\sigma_j)^2 = \varepsilon \\
(\sigma_0\sigma_1)^4 =...
- 2,258
1
vote
1
answer
106
views
Why isn't a coxeter group a HNN-extension?
A doctorate told me to think about why there is no mapping from coxeter groups to $\mathbb{Z}$. This makes sense since HNN-extensions are of the form
$$A\star_{\{(\varphi_1 , C,\varphi_2 )\}}=\langle ...
- 63
1
vote
2
answers
119
views
Number of fixed points of generators of reflections (Coxeter) group
Say I have a group with presentation like
$$\langle s,t,u \mid s^2,t^2,u^2,(st)^2,(su)^3,(ut)^4\rangle,$$
faithful on set $S$ with exactly one orbit ($|S|$ is known). How could I determine $|\text{Fix}...
2
votes
1
answer
242
views
Let $w_0$ be the element of longest length in a Coxeter group. Show that $l(w_0w)=l(ww_0)=l(w_0)-l(w)$? Find $w_0$ explicitly in $S_n$.
Let $w_0$ be the unique longest element in $W=S_n$. Let us show that $$l(ww_0)=l(w_0)-l(w)$$ for any $w \in W$.
We proceed by induction on $l(w)$. First, let $S$ be the generating set for $W$. In ...
- 229
0
votes
0
answers
63
views
Coxeter graph of the group $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \dots \times \mathbb{Z}_2$
I am reading the first chapter of Combinatorics of Coxeter Groups by A.Björner and F.Brenti. In the first example they say that the graph with $n$ isolated vertices (no edges) is the Coxeter graph of ...
- 621
4
votes
1
answer
499
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Coxeter presentation of Hyperoctahedral group $(\mathbb{Z}/2\mathbb{Z})^n \rtimes S_n$.
I know that the hyperoctahedral group $(\mathbb{Z}/2\mathbb{Z})^n \rtimes S_n$ has the presentation
$$\langle s_{\text{1}},\ldots,s_n\mid s_{\text{1}}^{\text{2}}=s_i^2=1, (s_1s_2)^4=(s_is_{i+1})^3=(...
- 1,810
3
votes
0
answers
273
views
About the order of Coxeter groups
I have a question about Coxeter groups with $3$ generators:
Suppose, as a group, $G$ is generated by $a,b$ and $c$, with the relations $a^2 =b^2 =c^2 =1$, $(ab)^m = (bc)^n = (ca)^p =1$ where $2 \...
- 1,155