All Questions
Tagged with group-theory coxeter-groups
127
questions
15
votes
2
answers
1k
views
Finite/Infinite Coxeter Groups
In the same contest as this we got the following problem:
We are given a language with only three letters letters $A,B,C$. Two words are equivalent if they can be transformed from one another using ...
- 23.5k
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
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
10
votes
1
answer
249
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 ...
- 55.9k
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
2k
views
What is a Coxeter Group?
I've recently started investigating abstract algebra and have now stumbled upon "Coxeter Groups", which are a mystery to me.
I've read that Coxeter Groups
have something to do with reflections (in ...
- 559
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
7
votes
1
answer
431
views
Relation between reflection group and coxeter group
Reflection group is defined see https://en.wikipedia.org/wiki/Reflection_group.
An abstract Coxter group is defined to have generators $s_1$, $s_2$, ..., $s_n$ and relations $s^2_i=e$, $(s_is_j)^{m_{...
- 1,160
7
votes
1
answer
128
views
Condition implying $N(H)/H$ a Coxeter group?
I'm interested in which finite groups can arise as
$$
N(H)/H
$$
for $ H $ a connected subgroup of a compact connected simple Lie group $ G $.
One obvious family of examples is take $ H $ to be the ...
- 8,542
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
6
votes
1
answer
604
views
Degrees of Fundamental Invariants of Coxeter Groups $A_n$
I think I misunderstood something simple but not sure what.
According to https://en.wikipedia.org/wiki/Coxeter_element,
The invariants of the Coxeter group acting on polynomials form a polynomial ...
- 2,672
6
votes
1
answer
376
views
Centralizers of reflections in parabolic subgroups of Coxeter groups
Let us consider a (not necessarily finite) Coxeter group $W$ generated by a finite set of involutions $S=\{s_1,...,s_n\}$ subject (as usual) to the relations $(s_is_j)^{m_{i,j}}$ with $m_{i,j}=m_{j,i}$...
- 2,697
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
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
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