# Questions tagged [root-systems]

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.

313 questions
Filter by
Sorted by
Tagged with
80 views

### If $\lambda \prec \eta$ are positive roots, then there exists another root $\zeta$ such that $\lambda \prec \zeta \prec \eta$.

Let $\Phi$ be an irreducible root system with a base $\Delta$ and $\lambda, \eta$ be positive roots such that $\lambda \prec \eta$ and $ht(\eta)-ht(\lambda)\geq 2$. Question: Does there exist another ...
20 views

49 views

62 views

20 views

38 views

### Exercise on the highest root

Problem : Let $\phi$ be an irreducible root system and $\phi^{+}$ a choice of positive roots. Prove that if $(\alpha, \beta) \geq 0$, $\forall \beta \in \phi^{+}$, then $\alpha$ is the highest root ...
30 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 ...
32 views

### Reference for application of the theory of weights and weight-spaces to $\mathfrak{su}(2)$ and $\mathfrak{su}(3)$

In the last couple of weeks I've been reading J.E. Humphreys' "Introdcution to Lie Algebras and Representation Theory" and after finishing Chapter 3 I had a chat with my TA about the topics*. He told ...
115 views

### Extending base vector field via tensorproduct from $\mathbb{Q}$ to $\mathbb{R}$

Let $L$ be a finite dimensional vector space over some field $\mathbb{F}$ ($\mathrm{char}(\mathbb{F})=0$), $H\subset L$ a subspace of $L$ with $\mathrm{dim}_{\mathbb{F}}(H)=h$ and let $H^*$ be its ...
108 views

40 views

### Is the sum of coroots the coroot of the sum?

While studying the book Introduction to Lie Algebras and Representation Theory, by Humphreys, I've came across a problem that seems simple, but I just cannot figure out: If $\alpha, \beta \in \Phi$,...