All Questions
13
questions
1
vote
0
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65
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Angles of the Fundamental Alcove (Chamber?)
I am trying to calculate the angles of the fundamental alcove (chamber?) for the root systems of type $B_2$, $C_2$, and $G_2$; the fundamental alcove (chamber?) forms a triangle in these cases, so I ...
- 8,040
1
vote
1
answer
175
views
Reflection Group of Type $D_n$
Here is the description of the reflection group of type $D_n$ in Humphreys' book Reflection Groups and Coxeter Groups:
($D_n$, $n \ge 4)$ Let $V = \Bbb{R}^n$, and define $\Phi$ to be the set of ...
- 8,040
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
1
vote
1
answer
168
views
Simple Reflections on Simple Roots
I have two related questions concerning simple reflections and simple roots. Let $\Phi$ be a root system for a reflection group $W$, let $\Pi \subset \Phi$ be a positive system, and let $\Delta$ be a ...
- 8,040
2
votes
0
answers
22
views
System of representatives in reflection groups and subgroups
I'm working on a paper from Steinberg (1974), "On a theorem of Pittie". The paper is mainly about roots and reflection groups. I'm having trouble understanding the proof of lemma 2.5(a) ...
- 21
0
votes
1
answer
40
views
There is an element of every possible length in $[W_{\theta} \backslash W]$
Let $(W,S)$ be the Weyl group of a root system with base $\Delta$, and let $\theta \subset \Delta$. Let $W_{\theta}$ be the group generated by $\theta$. It is a general result that in every right ...
- 34.3k
1
vote
0
answers
555
views
About parabolic subgroup of a Weyl group
Let $W$ be a Weyl group/Coxeter group. Let $\Phi$ be the associated root system, fix a positive root system $\Phi^+$ and let
$\Delta$ be the set of simple roots.
Let $W_I$ be the parabolic subgroup ...
- 255
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
0
answers
255
views
If $G$ is an algebraic group of adjoint type, is $\alpha_s^\vee(-1)=1$?
Suppose $G$ is a semisimple algebraic group of adjoint type. If $(W,S)$ is the Coxeter system and $s\in S$, and $\alpha_s$ is a simple root with corresponding coroot $\alpha_s^\vee$, is it true that $\...
- 660
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
1
vote
1
answer
60
views
In type $B_n$, is $s_\beta(\alpha)=\alpha+\sqrt{2}\beta$?
In a Coxeter group acting on a vector space $V$, if $\alpha$ is a simple root, there is a simple reflection $s_\alpha$ defined on $V$ by
$$
s_\alpha(\lambda)=\lambda-2B(\alpha_s,\lambda)\alpha_s
$$
...
- 903
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
1
answer
482
views
the orbit of a root under operations of irreducible crystallographic group?
Suppose we have an irreducible crystallographic coxeter group G acting in a vector space V, how can we show that the orbit of an ...
- 842