# Questions tagged [weyl-group]

A Weyl Group is 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. They can also be viewed as a special type of finite Coxeter group, i.e. a group generated by reflections which, in the case of Weyl groups, acts discretely by isometries on a sphere in some dimension.

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### The number of regular elements in a maximal torus is number of the elements of the Weyl group

I'm reading "Representations of Finite and Compact Groups (by Barry Simon)" and struggling with a sentence(p214). The sentence is In fact, for regular classes (those with one and so only ...
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### Only Weyl group of rank two root system can be dihedral

Let $\Phi$ be a (reduced, crystallographic) root system, and $W$ its Weyl group. Is it possible to prove that if we know $W$ is dihedral, then the rank of $\Phi$ is two, Without using the ...
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### Question concerning positive Weyl chamber

I would like to ask for a hint for exercise 22.5 in Bump's book "Lie groups". The setting is as follows: Let $G$ be a (semisimple, connected, simply connected) compact Lie group, choose a ...
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### Longest element of Weyl group of a simple Lie algebra action on Weyl chambers

Let, $\mathfrak{g}$ be a complex simple Lie algebra with Weyl group $W$,also let $\omega_0$ be the longest element of the Weyl group. We Know that Weyl group acts on the set of Weyl chambers freely ...
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### How to take the imprimitive unitary reflection groups as input in GAP?

I am wondering about how to take the group $G(m,p,n)$ as input in GAP. The groups $G(m,p,n)$ appear in the classification of unitary reflection groups. The group $G(1,1,n)$ is the symmetric group $S_n$...
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### Equivalent definition of Weyl group?

I am new to representation theory and only know an informal definition of Weyl group - it is a group of isometries generated by some transformations (I think reflections) of hyperplanes associated to ...
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### I don´t understand root systems

I don´t understand root systems. The Wikipedia (and my university lectures) say it is some configuration of vectors with certan properties. The root vectors should span the whole space, which I ...
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### Dominance of $w\mu$ for dominant cocharacter $\mu$

NOTE: The question has now been posted on MathOverflow: Dominance of $w\mu$ for dominant cocharacter Let $G$ be a reductive group over a field $k$ of characteristic zero with maximal split torus $T$ ...
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### Relation between product of reflections and angle

I am currently reading Chapter 3 Root Systems of John Humphreys book on Lie Algebras. It's known that Weyl Group $W$ is generated by set of reflections. If I consider an arbitrary element of Weyl ...
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### Qustions on the orbits of weyl group and group actions

I am an undergraduate in physics and know little about math. I know about some basic ideas of Lie groups and Lie algebras like roots, weyl group, weyl chambers but I am ignorant about complexification,...
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### Conjugating a root subgroup by a Weyl group element

Fix a field $k$. This is perhaps unnecessary, but assume $\operatorname{char} k = 0$. Let $G$ be a reductive isotropic quasi-split algebraic $k$-group. Let $S \subset G$ be a maximal split torus (of ...
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### Properties of the Weyl vector $\rho = \frac{1}{2} \sum_{\alpha > 0} \alpha$

Let $G$ be a compact connected Lie group with Lie algebra $\mathfrak{g}$. The group has a maximal torus $T$ with Lie algebra $\mathfrak{t}$. Let $\rho = \frac{1}{2}\sum_{\alpha > 0} \alpha$ be the ...
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### Computing the longest element of the Weyl group

I want to compute the longest element $w_0$ of the Weyl group $W$ for $A_2$, $B_2$ and $G_2$. I saw this has already been asked before here for the case of $G_2$, but the answers are still not very ...
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### Program for computations with Weyl Groups.

Let $W$ a Weyl group and $W_I$ the parabolic subgroup associated to the subset $I$ of simple roots. Currently I am facing with the problem of computing explicitly the set of minimal coset ...
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### The definition of simplicial hyperplane arrangements

I'm struggling to understand a statement in Sec 2.2 of "A simplicial complex of Nichols algebras" by Cuntz and Lentner. Simplicial arrangements are sets of hyperplanes in real vector spaces ...
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