# 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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### Describing an action on the Weyl group $W(T)=N_G(T)/T$ for different maximal tori $T$

While trying to understand more the structure of reductive group, I came upon the situation that I describe below. I can't find a mistake in my dissertation, however I arrive to an absurd conclusion. ...
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### Is the Weyl denominator globally well defined on $T$?

The Weyl denominator function on $T$, the maximal torus of a compact connected Lie group $G$ is given by (for $H \in \mathrm{Lie}(T)$) $$\delta(\exp(H)) = \sum_{w \in W} \det(w) e^{\rho(w(H))}$$ where ...
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### Harish Chandra isomorphism:Invariant polynomial functions

I am trying to read the complete proof of Harish Chandra Isomorphism theorem from the book of Humphreys. Notations: $L$ is a finite dimensional semisimple Lie algebra with Cartan subalgebra $H$. $G$ ...
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### Action of Weyl group on $GL_n$

Let $K$ be a field with $\operatorname{char}(K)=0$ and $G=\operatorname{GL}_n$ defined over $K$. The Weyl group $W$ of $G$ is isomorphic to the symmetric group $S_n$. I'm now wondering if $W$ acts ...
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### Is the longest Weyl transformation of a product group the pair of the longests?

More precisely, if $G_1,G_2$ are two Lie groups and $T_1,T_2$ are maximal tori respectively, then $W(G_1\times G_2)\cong W(G_1)\times W(G_2)$. The pair of longest elements in $W(G_1)\times W(G_2)$ ...
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### Values of $b(h_a)-a(h_b)$ with $a,b$ being roots in root system of a Lie algebra

Let $a,b$ be roots in a root system of a finite dimensional complex semisimple Lie algebra. I want to determine the possible values of $b(h_a)-a(h_b)$. The difference equals zero when $a=b$ (clearly) ...
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### Weyl group of a compact Lie group vsWeyl group of a root system

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$ and $T$ be a maximal torus with Lie algebra $\mathfrak{t}$. I read that the Weyl group $W$ of $G$ is "the group of automorphisms of $T$ ...
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### Computing elements of Weyl Groups

I am doing some reading on Weyl groups and I wanted to be able to compute the elements of such groups, expressed in terms of simple reflections and so I decided to write some code for that. I started ...
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### Induced representations of parabolic subgroups in Weyl groups

I am trying to learn about Weyl groups, parabolic subgroups and their (induced) characters. I have come across a puzzling computation on characters of Weyl groups without any indication to what ...
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### How does the Weyl group of a simple Lie algebra act on fundamental weights?

Given a simple lie algebra $\mathfrak{g}$ with root system $R$, the Weyl group $W$ acts on $R$ by definition. The action of any simple reflection $r_i \in W$ on any simple root $\alpha_j \in R$ is ...
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### About expressing $w\rho-\rho$ as a sum of roots

Let $\Phi$ be a root system, $\Phi^+$ be the positive system, $\rho$ be the half sum of positive roots, and $W$ be the Weyl group of $\Phi$. I remember that there is a way to express $w\rho-\rho$ ...
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### Conjugacy classes of rational tori in Symplectic group

Rational conjugacy classes of Frobenius stable tori (in a finite group of Lie type) are in bijection with Frobenius-conjugacy classes of the corresponding Weyl group.When the group is the Symplectic ...
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### Root Systems: existence of a root give two other roots

Let $\Omega$ be a root system and fix a set of positive roots. Let $\gamma \in \Omega$ be a positive root, $\alpha \in \Omega$ a simple positive root, and $s_{\alpha}$ the assosociated reflection in ...
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### Realizing automorphism in Weyl group as automorphism of Lie algebra

$L$ a simple Lie algebra over $\mathbb{C}$ (finite dim.), $H$ a maximal toral, $\Phi$ the root system relative to $H$. Then $L$ has Cartan decomposition $L=H\oplus \amalg_{\alpha\in\Phi} L_{\alpha}.$ ...
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$L$ a simple Lie algebra over $\mathbb{C}$ (finite dim.), $H$ a maximal toral, $\Phi$ the root system relative to $H$. Then $L$ has Cartan decomposition $$L=H\oplus \amalg_{\alpha\in\Phi} L_{\alpha}.... 1answer 81 views ### Find a special element in weyl group For Lie algebra \mathfrak{sl}_{n}, weyl group W of it is generated by S_{\alpha_{1}},S_{\alpha_{2}},... where \alpha_{i}'s are simple roots. How can we find explicit formula for w_{0}\in W ... 1answer 37 views ### Identify some Coxeter group As we all know, the weyl group of lie algebra of B_{2} type is \left\{s_{1},s_{2}|s_{1}^{2}=1, s_{2}^{2}=1, (s_{1}s_{2})^{4}=1\right\}. How can we identify this with Z^{2}_{2}\rtimes S_{2}? If ... 0answers 42 views ### Eigenvalues of elements of a Weyl group I'm currently reading the "Reflection groups and Coxeter groups" textbook by Humphreys, but I'm struggling to understand a paragraph at the beginning of section 3.9. For a Weyl group W, it states "... 1answer 91 views ### Weyl chambers associated to a root system Following Humphreys' Lie algebra, let \Phi be a root system in euclidean space E=\mathbb{R}^n. For every root \alpha\in\Phi let P_{\alpha} be the hyperplane orthogonal to \alpha. Then ... 1answer 64 views ### sn: W\rightarrow \{1,-1\},sn(\sigma)=(-1)^{l(\sigma)} is a homomorphism Define sn: W\rightarrow \{1,-1\} by sn(\sigma)=(-1)^{l(\sigma)}. Prove that sn is a homomorphism (where W is the Weyl group and l is the length function of reduced expressions in W). ... 0answers 47 views ### The relationship between the Weyl group and Isometries of a Maximal Flat Let M be a symmetric space of the noncompact or compact type with G := \text{Isom}(M)^0. It's Weyl group is defined as$$ W := N_G(A) / Z_G(A), $$where A \leq G is a maximal abelian subgroup. ... 2answers 381 views ### How much can you say of a Lie algebra knowing the Weyl Group? The question is exactly the one stated in the title: "How much can you say of a semisimple Lie algebra knowing just the Weyl Group?". Then if you prefer I have few more restrictions: 1) Knowing ... 1answer 111 views ### an element of the Weyl group fixing a vector in the fundamental Weyl chamber I am trying to solve exercise 10.12 from humphreys lie algebra book. I need to prove that if an element \sigma of the Weyl group is such that \sigma v=v for v a vector in the fundamental Weyl ... 1answer 132 views ### Weyl Group Intuition I am just learning about root systems for the first time and I am wondering how people visualize intuitively the notion of a root system and the Weyl group. 0answers 193 views ### The weyl group as a dihedral group I am traying to solve problem 9.4 of humphreys lie algebra book and I need to show that the Weyl groups of A_1 \times A_1 A_2 B_2 and G_2 are dihedral of order 4,6,8,12. My question is, how ... 1answer 264 views ### Weyl Operator and Field Operator Given the creationa^* and annihilation a operators on Fock Space I have the following statement.$$ e^{itN}\Phi(f)e^{-itN}=\Phi(e^{it}f)$$where we have the following definitions$$\Phi(z) = \frac{...
I'm trying to work through the following exercise in Bourbaki, Lie Groups and Lie Algebras. For part (a), we are assuming $b \in B$ normalizes $N$, and we are supposed to show that \$bnb^{-1}n^{-1} \...