All Questions
9
questions
3
votes
1
answer
116
views
Coxeter notation for the symmetries of the maximally symmetric unit-distance embedding of $K_{3,3}$ in $\mathbb R^4$
My Shibuya repository now contains unit-distance embeddings in the plane of all cubic symmetric graphs to $120$ vertices, except the first two ($K_4$ and $K_{3,3}$) which do not have this property, as ...
- 104k
3
votes
1
answer
57
views
Partial Order on a Vector space with a Crystallographic Coxeter Group
Let $V$ be a vector space, and $W$ an irreducible, crystallographic Coxeter group on $V$ with simple root system $\Pi = \{a_1 \cdots a_n \}$.
We define a partial ordering on $V$ by $u \geq v$ iff $u - ...
- 127
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
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
359
views
Conditions for a neat subgroup to act fixed-point free
Given a hyperbolic reflection group $G$ acting on hyperbolic space $\mathbb{H}_n$ by, well, reflections in hyperplanes.
Does a neat subgroup of $G$ act fixed-point free on $\mathbb{H}_n$? If not, ...
user149890
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
1
vote
1
answer
104
views
Orbits in Coxeter Group
I'm interested in computing the orbits in a finite coxeter group, i.e. $D_4$ (see here
).
The orbits of the roots $\mathcal{O}(r_i)$ can be obtained by applying all the reflections
$\displaystyle ...
- 760
1
vote
0
answers
64
views
Fundamental Region $F$ for Coxeter Group $G\subseteq\mathcal{O}(\mathbb{R}^3)$
Let $G\subseteq \mathcal{O}(\mathbb{R}^3)$ (orthogonal transformations). For a reflection $S\in G$ through a hyperplane $\mathcal{P}\subset\mathbb{R}^3$ we call the two unit vectors $\pm r$ that are ...
- 760
4
votes
2
answers
792
views
In a reflection group, the longest word $w_0$ contains all simple reflections
This is Exercise 2 of section 1.8 in Humphreys' "Reflection groups and Coxeter groups", p.16.
The longest word $w_0$ in a finite reflection group $W$ acting on a Euclidean space (with a specified ...
- 415