All Questions
26
questions
1
vote
0
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36
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Dihedral Groups as Coxeter Groups
I have just completed Exercise 1.2 in the book "Combinatorics of Coxeter Groups" stated below:
Show that there exist Coxeter systems $(W,S)$ and $(W',S')$ with $|S|\neq|S'|$ such that $W\...
- 63
2
votes
0
answers
41
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Relationship between the Strong and Weak Exchange Property of Coxeter Groups
I am a beginner at studying Coxeter group theory, and I am confused with the strong and weak exchange property when reading the book "Combinatorics of Coxeter Groups". Let me first state the ...
- 63
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
$$A\star_{\{(\varphi_1 , C,\varphi_2 )\}}=\langle ...
- 63
2
votes
1
answer
64
views
Does the Coxeter group $C(D_n)$ have any "proper reflection quotients" except $C(A_{n-1})$?
Here, a reflection quotient is a surjective homomorphism between Coxeter groups mapping reflections to reflections. A reflection quotient $C(\Delta) \to C(\Gamma)$ is proper if it is not injective and ...
- 4,786
0
votes
0
answers
70
views
When is the subgroup product of two parabolic subgroups of a Coxeter Group, the Coxeter Group itself?
Let $W$ be a Coxeter Group generated by simple reflections $S$. If $I,J\subseteq S$ and $W_I = \langle s | s \in I \rangle$ when is it true that $W_IW_J = W$?
I am secretly hoping that the answer ...
- 317
2
votes
1
answer
258
views
Crystallographic root system Coxeter Groups
In Humphreys book in "Reflection Groups and Coxeter Groups" he defines a root system $\Phi$ to be crystallographic if it satisfies $\frac{2(\alpha, \beta)}{(\beta, \beta)} \in \mathbb{Z}$ $(\star)$ ...
- 121
2
votes
1
answer
242
views
Let $w_0$ be the element of longest length in a Coxeter group. Show that $l(w_0w)=l(ww_0)=l(w_0)-l(w)$? Find $w_0$ explicitly in $S_n$.
Let $w_0$ be the unique longest element in $W=S_n$. Let us show that $$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
1
vote
0
answers
100
views
Coxeter exchange condition in symmetric group
I would like to prove (for purposes of illustration mainly) that the symmetric group $S_n$ with the set $S$ of adjacent transpositions $(i, i+1)$ is a Coxeter group by proving that it satisfies the ...
- 103
2
votes
0
answers
54
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Embeddings of Coxeter Groups of Rank $3$ into $\text{SO}(3)$
Let $W = \langle x_1, x_2, x_3 \;|\; (x_ix_j)^{m_{ij}} \rangle$ be an irreducible Coxeter group, i.e., the graph with vertices $v_1, v_2, v_3$ and edges between all pairs $(v_i, v_j)$ with $m_{ij} \ne ...
- 4,786
1
vote
1
answer
93
views
Can every positive root of a Coxeter group be written as a simple root and a positive root?
Can every positive root of a Coxeter group be written as a simple root and a positive root? I think that this is possible. For example, in type $B_2$, the set of positive roots are $\alpha_1, \alpha_2,...
- 14.6k
1
vote
0
answers
555
views
About parabolic subgroup of a Weyl group
Let $W$ be a Weyl group/Coxeter group. Let $\Phi$ be the associated root system, fix a positive root system $\Phi^+$ and let
$\Delta$ be the set of simple roots.
Let $W_I$ be the parabolic subgroup ...
- 255
2
votes
0
answers
117
views
Reflection length of the longest word in a Coxeter group.
Let $(W,S)$ be a Coxeter system. The set of reflections of $W$ is $S=\{wsw^{-1}:s \in S, w \in W\}$. The reflection length of $w \in W$ is defined as the minimal number $m$ such that $w = r_1 \cdots ...
- 14.6k
0
votes
1
answer
131
views
In what different terms can Coxeter systems be described?
My starting point is this question: https://mathoverflow.net/questions/214569
As I understand it they say, that the Coxeter matrix is not sufficient to describe the group.
I thought that up to ...
- 103
4
votes
1
answer
121
views
Cardinality of a coxeter group
Let ${G}$ be a Coxeter group with the next presentation
\begin{equation}
G = \left\langle s_1,s_2,\cdots,s_{n-1} : (s_is_{i+1})^3=1 , \ (s_is_j)^2=1 \ ,\ |i-j| > 1 \right\rangle
\end{equation}
...
3
votes
0
answers
169
views
Longest element of a subgroup
Say I have a finite Weyl group, $W$, and a set of generators $S:= \{s_1,...,s_k\}$ (making $W,S$ a coxeter system) and an automorphism $\theta: W\rightarrow W$ which permutes $S$. I know that the ...
- 5,993