Tagged Questions

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.

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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 ...
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$\Delta \subset \Phi$ 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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 ...