All Questions
6
questions
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
1
vote
0
answers
141
views
Which Coxeter Elements have powers that are the longest element of the (Finite, Irreducible) Coxeter Group?
Let $(W,S)$ be a finite, irreducible Coxeter Group. I thought it was true (from Humphrey's book Ex 2 on page 82) that if the Coxeter Number of $W$, $h$, is even then $$c^{h/2} = \omega_0 \quad \dagger$...
- 317
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
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
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
754
views
The number of reduced expressions for the longest element of $B_n$?
Let $W=W_{\Phi}$ be a reflection group, with root system $\Phi$, and $\Delta=\{\alpha_1, ...,\alpha_n\}\subseteq \Phi$ a simple system. So $W$ is generated by the $s_{\alpha_i}=s_i$ for $i=1,2,...n. $ ...
- 984