All Questions
Tagged with group-theory coxeter-groups
51
questions with no upvoted or accepted answers
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
4
votes
0
answers
65
views
Is there an algorithm to check that a subgroup of a CAT$(0)$ group is *not* quasiconvex?
Let $G$ be a finitely generated CAT$(0)$ group and $H$ a subgroup. If $H$ is quasiconvex then it is finitely generated, so we can immediately conclude that any non-finitely generated subgroup of $G$ ...
- 2,382
4
votes
0
answers
233
views
How to find generators of translation subgroup of an abstract reflection (coxeter) group
I have an infinite reflection group https://en.wikipedia.org/wiki/Coxeter_group
Take for example the affine groups $[4,4],[4,3,4],[4,3,3,4]$...
I'd like to get an explicit expression for generators ...
- 1,010
4
votes
1
answer
216
views
Question about proof of positive roots under reflection
Let $(W, S)$ be a finite Coxeter system. Furthermore, let $V$ be a real vector space with a (finite) basis $\{ \alpha_s | s \in S \}$.
For every $s \in S$ one can define the reflection $\sigma_s : V ...
- 3,149
4
votes
0
answers
113
views
Reference for reading Dynkin diagrams in Lie theory?
I have learned that given a Dynkin diagram corresponding to a Kac-Moody algebra, I should be able to use the diagram to read off the generators and relations of the Weyl group of that algebra. Each ...
- 304
4
votes
0
answers
64
views
Computing orders of some irreducible finite Coxeter groups
There is a particular method in Reflection groups and Coxeter groups by Humphreys to compute the orders of various irreducible finite Coxeter groups in Chapter 2.11. The method involves using group ...
- 4,443
4
votes
0
answers
208
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 $$r_k=e_{k+1}-e_k~\...
- 4,485
3
votes
0
answers
55
views
A question regarding affine Coxeter groups
Let $\Gamma_n$ denote the isometry group of the regular tessellation of $\mathbb{R}^n$ by $n$-cubes, i.e. $\Gamma_n= \left( \bigoplus\limits_{i=1}^n \mathbb{D}_\infty \right) \rtimes S_n$. Now, let $...
- 33.3k
3
votes
0
answers
112
views
Question about a certain involution on a Coxeter group $W$.
This is a small question that arose while reading the paper On Okuyama's Theorems About Alvis-Curtis Duality by M. Cabanes. It can be read here.
Let $(W,S)$ be a finite Coxeter system, $l$ the length ...
- 12.4k
3
votes
0
answers
79
views
How would one classify point groups?
By point group I mean a finit subgroup of $\mathrm O(\Bbb R^n)$.
Lists of point groups for some small dimensions are found on Wikipedia, but I am not certain about their completeness. As there seem ...
- 30.1k
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
3
votes
0
answers
208
views
Conceptual question about extended affine Weyl group $\hat{W}_a$
Denote an affine Weyl group by $W_a$, and let $\mathcal{H}$ be the collection of hyperplanes $H_{\alpha, k}, \text{ } \alpha \in \Phi,k \in \mathbb{Z}$. I know for a fact that $W_a$ and the extended ...
- 4,443
3
votes
0
answers
193
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 $...
- 31
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
3
votes
1
answer
225
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 ...
- 355