All Questions
Tagged with group-theory coxeter-groups
11
questions
5
votes
0
answers
317
views
Mapping $\Delta(2,2,2)\mapsto \Delta(4,4,2)$... [closed]
Looking at the images below, you recognize that the adajency matrix of the graph $A_G$ splits up into three different color submatrices, with $A_G=A_r+A_b+A_d$ (where $d$ is dark, damn...).
It's ...
- 18.6k
13
votes
1
answer
1k
views
Is every finite group of isometries a subgroup of a finite reflection group?
Is every finite group of isometries in $d$-dimensional Euclidean space a subgroup of a finite group generated by reflections?
By "reflection" I mean reflection in a hyperplane: the isometry fixing a ...
- 9,076
4
votes
2
answers
296
views
Signed Permutations and Coxeter Groups
Context: (most of which is pulled from comments and answers to https://mathoverflow.net/questions/431964/signed-permutations-and-operatornameson)
The diagonal subgroup $ C_2^n $ of $ O_n(\mathbb{Z}) $ ...
- 8,542
0
votes
1
answer
70
views
Are complex reflection groups never perfect?
This is a follow-up to
Conceptual reason why Coxeter groups are never simple
A complex reflection group is a finite subgroup of $ U_n $ that is generated by pseudo reflections. A pseudo reflection is ...
- 8,542
10
votes
3
answers
905
views
What sort of groups are generated by a single conjugacy class?
To clarify, I am not looking for a classification but rather for well-researched examples of families of (finitely generated) groups generated by a single conjugacy class.
A collection of examples, ...
- 4,786
7
votes
2
answers
1k
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 $\mathbb{R}^n$, $x(e_k)=e_{x(k)}$...
- 93
7
votes
1
answer
399
views
On groups with presentations $ \langle a,b,c\mid a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=(abc)^s=1\rangle $...
$$
\langle a,b,c\mid a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=1\rangle =\Delta(p,q,r)
$$
This is a presentation of a triangle group $\Delta(p,q,r)$, a special kind of Coxeter group.
EDIT In fact, these are ...
- 18.6k
4
votes
2
answers
175
views
Conceptual reason why Coxeter groups are never simple
Is there a conceptual reason why (non-abelian) Coxeter groups are never simple?
For example is there some obvious normal subgroup that can be defined? Or perhaps it is for some reason clear that the ...
- 8,542
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
1
vote
2
answers
341
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.
- 14.6k
1
vote
1
answer
290
views
Reflection Group of Type $C_n$
In Humphreys' book Reflection Groups and Coexeter groups, he defines a group of type $B_n$ (for $n \ge 2$) in the following way:
Let $V = \Bbb{R}^n$, and define $\Phi$ to be the set of all vectors of ...
- 8,040