For questions involving abstract root systems, their associated Weyl groups and Dynkin diagrams, as well as their applications to Lie theory, graph theory, or other related fields.
2
votes
0answers
38 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 ...
4
votes
1answer
66 views
Same Dynkin diagrams $\Longrightarrow$ Isomorphic root systems.
I am studying the book Introduction to Lie algebras. In page 122 there is something I don't understand and I am looking for some help.
In the beginning of that page the authors give the definition of ...
2
votes
0answers
37 views
How to find root subgroups
$\newcommand{\GL}{\text{GL}}\newcommand{\diag}{\text{diag}}$For $G = \GL_n(k)$ let $B$ be the upper triangular matrices and $T$ be the diagonal matrices in $G$. In this case I understand that the ...
0
votes
0answers
27 views
Classifying all rank 2 and 3 root systems
I am working with the representation theory of complex simple Lie algebras, and have a question:
It is intuitively clear that the root systems $A_1\times A_1$, $A_2$, $B_2$, and $G_2$ comprise all ...
1
vote
0answers
18 views
Why is the dual space of Cartan subalgebra an irreducible representation of Weyl group
it is proposition 14.31 in Fulton-Harris book. The proof goes like this. Let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$, and assume $\mathfrak{z}\subseteq\mathfrak{h}^*$ were preserved ...
1
vote
1answer
36 views
The Weyl group preserves inner product
Show that the Weyl group $W$ preserves the inner product: $(w(\lambda)\,,\, w(\mu)) = (\lambda, \mu)$ for all $w\in W$ and $\lambda, \mu\in E$.
I know it suffices to check this on reflections ...
1
vote
0answers
35 views
Root system of a Lie Algebra
Could anybody help me to solve this problem with roots system?
Be $\Phi$ an irreducible root system. $\Phi^{+}$ a choice of positives roots in $\Phi$. Prove that if $(\alpha,\beta)\ge0$ $\forall ...
0
votes
1answer
61 views
Dimension of Lie algebra according to root system
I was wondering how is it possible to find the dimension of a semi-simple lie algebra $L$ if its corresponding root system is (lets make it simple) of type $B_2$. We can find the number of roots and ...
1
vote
1answer
62 views
Length of root strings is at most 4
Let $\Phi$ be a root system. In his Introduction to Lie algebras and Representation Theory, J. Humphreys proves that if
$$\beta-p\alpha,\dots,\beta,\dots,\beta+q\alpha$$
is the $\alpha$-root string ...
3
votes
0answers
57 views
$\Delta \subset \Phi$ is a base in a root system imples $\Delta^\vee \subset \Phi^\vee$ is a base in a root system
(the notation here is compatible with J.E. Humphrey's "Introduction to Lie Algebras and Representation Theory")
Let $\Phi \subset E$ be a root system.
Let $\Delta \subset \Phi$ be a base.
I already ...
2
votes
2answers
84 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 ...
1
vote
1answer
67 views
Weyl group of this root system is $S_n$?
Let $E=\{(x_1,\dots,x_n)\in \mathbb{R}^n:\sum x_i=0\}$ is of dimension $n-1$ take root system $R=\{e_1-e_j:1\ge i,j\le n\}$, I know that it is of rank $n-1$ root system, but why its weyl group is ...
0
votes
1answer
30 views
weyl group act by conjugation?
let $W$ be a weyl group and $\alpha\in R$ we have $s_{w(\alpha)}=ws_{\alpha}w^{-1}$, to prove this the author says $ws_{\alpha}w^{-1}$ acts as identity on $wL_{\alpha}=L_{w(\alpha)}$ , and ...
1
vote
1answer
37 views
not all automorphisms of a root system are elements of weyl group
could any one tell me why not all automorphisms of a root system are elements of weyl group? For example in $A_n, n>2$ the automorphism $\alpha\mapsto -\alpha$ is not in the weyl group. I do not ...
