# Questions tagged [weyl-group]

This tag is for questions regarding to "Weyl Group", a group associated with a compact Lie group that can either be abstractly defined in terms of a root system or in terms of a maximal torus. More generally, there are Weyl groups associated with symmetric spaces.

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### Isomorphisms of irreducible root systems

Let $E,E'$ be two euclidean vector spaces and $\Phi,\Phi'$ two root systems of $E$ and $E'$, respectively. Let $\varphi:E\to E'$ be an isomorphism of root systems. Applying the definition of root ...
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### Condition for a sum of images of fundamental dominant weights to lie on a wall

Let $\Delta$ be a base of a root system with Weyl group $W$. For $\alpha\in\Delta$, let $\varpi_\alpha$ be the corresponding fundamental dominant weight. Let $w\neq r$ be elements of $W$. I would like ...
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### Irreducible Dual Representation

For a semisimple Lie Algebra $\mathfrak{g}$ with Cartan Subalgebra $\mathfrak{t}$, let $V(\lambda)$ be the unique irreducible highest weight module with highest weight $\lambda$. I am asked to show ...
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### Which roots are fixed by simple reflections of the Weyl Group?

Let $\Phi$ be a root system of a semisimple Lie Algebra, and $W$ it's Weyl group. Let $\Delta = \{ \alpha_1, \dots, \alpha_l \}$ be a root basis, and let $w_i \in W$ be the simple reflection ...
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 ...
### Longest element of Weyl Group for $G_2$
Let $\mathfrak{g}$ be a semisimple Lie Algebra, $\mathfrak{t}$ a Cartan Subalgebra, $\Phi$ the corresponding set of roots, $\Delta \subset \Phi$ a root basis and $W$ the Weyl Group with respect to \$\...