0
votes
1answer
24 views

coxeter graph and root system

I want to show that a coxeter graph $\Gamma$ is connected if and only if its root system $\Phi$ is irreducible. So let $\Delta$ be a simple system of $\Phi$, and $\Delta$ is also our simple system. ...
1
vote
1answer
87 views

angles between simple roots are obtuse, problem with proof

Let $\Phi$ be a root system in the following sense: (1) $\Phi \subset \mathbb{R}^n$ consists of a finite number of nonzero vectors, (2) for each $\alpha \in \Phi$, $\Phi \cap \mathbb{R} \alpha = ...
1
vote
1answer
44 views

Angle between roots in a root system

Let $\Phi$ be a root system in the following sense: (1) $\Phi \subset \mathbb{R}^n$ consists of a finite number of nonzero vectors, (2) for each $\alpha \in \Phi$, $\Phi \cap \mathbb{R} \alpha = ...
3
votes
2answers
44 views

Conjugacy of simple system in a root system

I'll set up the problem, then ask the question. Let $V$ be a finite dimension vector space over $\mathbb{R}$ and let $\Phi$ be a root system in $V$, i.e. (1) $\Phi \cap \mathbb{R} \alpha = ...
2
votes
0answers
53 views

Quaternion, Dihedral groups and A-D-E classification

$\bullet$ What is the role of Quaternion group $H$ and dihedral groups $D_n$ in A-D-E classification? $\bullet$ Is Quaternion group $H$ in $A$ (special linear Lie algebra of traceless operators) or ...
2
votes
0answers
65 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 ...
2
votes
2answers
144 views

Greatest elements in crystallographic root systems

I have a question regarding a remark in the book "Reflection Groups And Coxeter Groups" by James E. Humphreys (unfortunately the book is not to be found as a whole on google books or such). In ...