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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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Weyl group of $D_n$

We have that the root system of the Dynkin diagram of type $D_n$ can be realized by considering the standard inner product in $\mathbb{R}^n$ and the vectors $\{e_1 - e_2, \ldots, e_{n-1} - e_n, e_{n-1}...
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Lifting of elements in Weyl groups of $B_2$ and $C_2$ type

It is well known that given a maximal torus $T$ in a linear Lie group $G$, the corresponding Weyl group can be defined as $N_G(T)/T$. As the title suggests, I am now working on the Weyl groups of $B_2$...
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Spinorial representation of Weyl group of SO(8)

First of all, physicist here, doing my best to post this question as rigorously as possible. My question is: how can I construct a spinorial representation of the Weyl group of $SO(8)$? Attempt at a ...
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Is it true that if we have $R\subset E$ an irreducible root system then $E$ is an irreducible representation of the Weyl group?

Recently in my class of lie algebra the profesor said the following statement If we have $R\subset E$ (with $E$ The Euclidean space) an irreducible root system then $E$ is an irreducible ...
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About elements in affine extended Weyl group

I'm not sure how to prove a statement about extended Weyl groups. Let $V$ be a finite vector space over $\mathbb{R}$, with a positive definite symmetric bilinear form (·,·), R ⊂ V be a reduced ...
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Correspondence Lie group/Torus and Lie algebra/CSA

I'm having trouble understanding the relation between Lie groups and Lie algebras. I'm studying on this article of Kostant, Section 2.1. So $K$ is compact connected simply connected simple Lie group, $...
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The Weyl group of type $D_2$, as a subset of $S_4$

My question is related to this post. I am trying to compare the Weyl groups associated to root systems of type $D_2$ and $A_3$ respectively. I know that the simple roots of $D_2$ are $e_1-e_2$ and $...
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Length function on Weyl group $S_n$ gives part of the Pascal's triangle

$$ \begin{matrix} \ell&\mathbf 0&\mathbf 1&\mathbf 2&\mathbf 3&\mathbf 4&\mathbf 5&\mathbf 6&\mathbf 7&\mathbf 8&\cdots\\ \hline \color{gray}{A_0|}&\color{...
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One question about reductive group

I'm reading Rapoport's paper "On the classification and specialization of Fisocrystals with additional structure", and I'm confused for one detail. Here G is a reductive group over $F$, $F$ ...
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Weyl chambers under group inclusion

Let $T$ be a fixed maximal torus of a compact Lie group $G$, and let the Weyl group $W=W(G,T)$ act on $\mathfrak t$, the Lie algebra of $T$, by fixing an identification of $\mathfrak t$ and its dual. ...
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Sum of weights of a representation of $U(N)$

Let $R$ be a finite-dimensional irreducible representation of $U(N)$, with the set of weights $W_R$. Each element of $W_R$ is a vector of length $N$ with integer entries. Firstly, I would like to know ...
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How do roots of a root system correspond to symmetric points of the associated Weyl group

So I've read that the Weyl group of the $A_3$ root system is the full symmetry group of a tetrahedron $T_h$, $C_3$ is $O_h$, and $H_3$ is $I_h$. I've read some complicated proofs of this but I don't ...
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Are Weyl algebra $A_1$ and it's opposite algebra isomorphic?

Let $A$ be a noncommutative ring and $ab=c$ in $A$. $A'$ is it's opposite ring if $ba=c$ in $A'$. If $A$ is a Weyl algebra $A_1$, are $A$ and $A'$ isomorphic? I have an idea, but I think it's wrong. ...
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Connection between exponents of a root system and solutions to linear systems over finite fields

Let $h_1, \ldots, h_r$ be linear forms in variables $x_1, \ldots, x_n$ with integer coefficients. Let $\mathbb F_q$ denote the finite field with $q = p^e$ elements. I am asked to prove that except in ...
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$w(\Delta) = \Delta \implies w = \mathrm{id}$ in Weyl group

Consider $\Phi$ a root system (definition of Erdmann) with basis $\Delta \subseteq \Phi$ (subset which is a basis of $E$ and each element of $\Phi$ is a linear combination with only non negative ...
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Why isn't the Weyl group of a root system defined as the isometry group of that system?

I know that the Weyl group of a root system is a subgroup of its isometry group, but (as in the case of $A_2$) it isn't always the whole isometry group. Why isn't the Weyl group defined as the ...
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Inner product of highest root.

In Macdonalds book on affine Hecke algebras and orthogonal polynomials, it is stated, without proof, that $\langle \varphi^\vee,\alpha\rangle \in \{0,1\}$ for all positive roots $\alpha$ not equal to $...
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Two different notions of maximal roots of a root system

There is a partial order on a root system given by $\alpha \prec \beta$ if and only if $\beta-\alpha \in R^{+}$, and Humphreys (10.4. Lemma A) proves that there is a unique maximal root with respect ...
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Is normalizer of a torus subgroup the Weyl group of a root system?

Let $G$ be a connected compact Lie group and $S$ a torus subgroup contained in the maximal torus $T$. Denote by $R_+$ the set of positive roots on $\mathfrak{t}$. Let $N_G(S)$ and $Z_G(S)$ be the ...
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What is the longest Weyl element of $SO(2n+1)$?

Let $w_{2n+2}$ is the matrix with ones on the non-principal diagonal and zeros elsewhere. Let $V$ be the $2n+2$-dimensional quadratic space with the symmetric bilinear form $\left<,\right>$ ...
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Particular Weyl group longest word

I was wondering if there is an algorithm for computing the Weyl group longest word starting with a particular choice. For example, let $w_1$ and $w_2$ be defined as follows: $$w_1= s_{i-1} s_{i-2} … ...
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What is the name of the "duality" Dynkin diagram automorphism

In a Dynkin diagram there is a special involution often constructed as follows: Let $C$ be the Weyl chamber corresponding to our diagram. There is a unique element of the Weyl group $w$ for which $wC =...
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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 does the weyl group act on weights\roots

Let the Weyl group be: $$W=N(T)/T$$ where $T$ is the maximal torus of some lie group $G$ and $N(T)$ is the normalizer of $T$. I saw that in this question that the Weyl group acts on weights by: $$(w.\...
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On axiomatic definition of affine root systems

In Macdonald's book Affine Hecke Algebras and Orthogonal Polynomials, chapter 1 introduces affine root systems. I will recall the definition here: Let $E$ be a non-zero real Euclidean space (finite ...
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Conditions for finiteness of a reflection group

Given a subgroup $G$ of $O(V)$ that is generated by reflections ($V$ a Euclidean space), under what conditions can we determine that $G$ is finite? I realize that if the angles between any two roots ...
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Counting and finding root subsystems

Let $\Phi$ be an irreducible root system. A root subsystem of $\Phi$ is a subset $\Psi \subseteq \Phi$ which is a root system. One can find the possible types of root subsystems of $\Phi$ by deleting ...
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How to find the longest element in a double coset of a Weyl group using SageMath?

I asked a question about computing longest element in a double coset in AskSage It has not been answered for a long time. So I asked it here. Let $W$ be a finite Coxeter group. Denote by $W_I$ the ...
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Determining the Weyl group from a given root system

1. Definitions For $V$ a vector space over $\mathbb C$ we call a subset $R \subset V$ an abstract root system if: (1) The set $R$ is finite, spans $V$ and $0 \notin R$. (2) For every $\alpha$ in $R$ ...
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Defining the Weyl group of type $D_n$ as a subgroup of Weyl group of type $B_n$ in GAP

I am looking to define the Weyl group of type $D_n$ as a subgroup of Weyl group of type $B_n$ in the software GAP. In general, one can define these groups separately. For example, let's say ...
Riju's user avatar
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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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Normalizer of a torus

Let $ G $ be a connected compact Lie group of rank $ r $ . Let $ T $ be an $ m $ dimensional (proper) closed connected subgroup of $ G $. Let $ N(T) $ be the normalizer in $ G $ of $ T $ $$ N(T):=\{g \...
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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 $S_n$...
Riju's user avatar
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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 ...
Tereza Tizkova's user avatar
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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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Longest element of $D_n$ and the set of positive roots [duplicate]

Personally I am not very familiar with group theory and need some clarifications. Let's look at $D_n$ and its longest elemements. According to OEIS A162206 the triangle begins: $1$; $1;2;1$; $1;3;5;6;...
Mikhail Gaichenkov's user avatar
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Weyl groups sends Weyl Chamber onto another:

I came across two statements while studying Weyl group of root Systems. First one: The Weyl group say $W$ sends one Weyl Chamber onto another. If $\gamma$ is regular in a Euclidean Space $E$, we have $...
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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 ...
Joshua Ruiter's user avatar
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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 ...
UtilityMaximiser's user avatar
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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 ...
cip's user avatar
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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 ...
Sabino Di Trani's user avatar
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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 ...
Ted Jh's user avatar
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How to find rational elements of order 3 in the Lie group $A_2$

I'm new to Lie theory and I'm trying to find all the rational elements of order 3 in the Lie group $A_2$. I found all the elements of order 3 contained in the fundamental region, which is defined by $...
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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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