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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21 views

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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39 views

Normalizer of an abelian subgroup

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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24 views

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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2answers
56 views

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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2answers
133 views

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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45 views

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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1answer
57 views

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;...
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17 views

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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19 views

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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25 views

Compute the lengths of Weyl group elements for all possible simple systems

So each choice of simple system for a root system determines a set of simple reflections. The length of an element of weyl group is the minimal number of simple reflections required to express it. My ...
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35 views

Computation of Weyl Groups Orbits in Sage.

We are in the context of Simple Lie algebras (over $\mathbb{C}$). Let $W$ be a Weyl Group and consider the parabolic subgroup $W_I$ for some set I of nodes of the Dynkin Diagram associated to W. I ...
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38 views

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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38 views

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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230 views

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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1answer
120 views

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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69 views

Unique element of the Weyl group sending $\phi^{+}$ to $\phi^{-}$

Let $W$ be the Weyl group acting on the root system $\phi$, with base $\Delta$ and let $C = C(\Delta)$ be the fundamental Weyl chamber. I want to prove that there exists a unique element $w_0 \in W$ ...
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68 views

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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1answer
44 views

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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34 views

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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48 views

Dynkin Diagrams and Weyl Groups

I've been studying Lie Groups and Lie Algebras for while and I got introduced to Dynkin diagrams. I wanted to know where Dynkin diagrams are used in physics, how are they used and what do they mean ...
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92 views

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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1answer
137 views

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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1answer
138 views

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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1answer
218 views

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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1answer
41 views

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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1answer
108 views

If the Weyl group $\mathcal W$ is a normal subgroup of $\mathrm{Aut}(\Phi)$, how can the Weyl group of $A_2$ be dihedral of order $6$?

If the Weyl group $\mathcal W$ is a normal subgroup of $\mathrm{Aut}(\Phi)$, how can the Weyl group of $A_2$ be dihedral of order $6$? The roots of $A_2$ are $\{\pm \alpha, \pm \beta, \pm(\alpha+\...
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78 views

Question about a certain involution on a Coxeter group $W$.

This is a small question that arose while reading the paper On Okuyama's Theorems About Alvis-Curtis Duality by M. Cabanes. It can be read here. Let $(W,S)$ be a finite Coxeter system, $l$ the length ...
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1answer
71 views

Literature for the longest Element of the Weylgroup for $GL(n,K)$. [closed]

I am looking for literature where I can find how the longest Element of the Weyl group looks like for $G=GL(n,K)$ over the diagonal matrices in $G$. I don't even need a proof. But I have no idea where ...
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1answer
141 views

Faithfullness of Weyl group action

Let $I$ be a finite indexing set, and $A \in \operatorname{Mat}_I(\mathbb{Z})$ be a generalised Cartan matrix, i.e. $a_{ii} = 2$, $a_{ij} \leq 0$ for $i \neq j$, and $a_{ij} = 0 \iff a_{ji} = 0$. ...
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1answer
499 views

Weyl group of type $A_n$

Let $E$ be the subspace of $\mathbb{R}^{n+1}$ for which the coordinates sum to $0$ and let $\Phi$ be the set of vectors in $E$ of length $\sqrt{2}$ and which are integers vectors. It is known that $\...
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26 views

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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1answer
104 views

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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1answer
131 views

Structure constants $\frac{N_{a,b}}{\langle c,c\rangle}=\frac{N_{b,c}}{\langle a,a\rangle}=\frac{N_{c,a}}{\langle b,b\rangle}$ for roots $a+b+c=0$

Let $\Phi$ be a root system of a finite-dimensional semisimple complex Lie algebra. Let $a,b,c\in\Phi$ st. $a+b+c=0$. I want to show that $\frac{N_{a,b}}{\langle c,c\rangle}=\frac{N_{b,c}}{\langle a,...
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1answer
237 views

Exercise on root system of type $A_n$

Problem Let $n$ be a positive integer and let $\phi$ be a root system of type $A_n$. Let $\Delta = \{ \alpha_1, .. , \alpha_n \}$ be a base, such that the Dynkin diagram is a string enumerated from $...
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1answer
422 views

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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1answer
75 views

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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1answer
59 views

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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39 views

How to calculate mean and variance in the Weyl group $A_n$?

Let the generation function for $A_n$ is the polynomial $\prod_{k=1}^{n}\frac {(1-x^{k+1})} {(1-x)}$ as defined in OEIS. For eg. for $n=5$ we have numbers: $1, 5, 14, 29, 49, 71, 90, 101, 101, 90, 71,...
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32 views

Is combinatorial info for permutahedra arising from ADE Weyl groups known?

Given an ADE Dynkin diagram, we also have a corresponding Weyl group. For example, the $A_n$ diagrams give the symmetric groups. Applying the Weyl group elements to a generic point in the root space (...
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1answer
150 views

Free presentations of Weyl groups

Can someone please provide me with a reference for the fact that the Weyl group $W$ associated to a root system $\Phi$ can be realised as a Coxeter group? This means that a Weyl group $W$ has a ...
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139 views

Reduced Expression for Reflection in Weyl Group

Let $W$ be a Weyl group of a semisimple Lie algebra $\mathfrak{g}$. Let $\beta$ be a positive root, $\alpha$ a simple root such that $\beta=w(\alpha)$. It is a straightforward fact that we can ...
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1answer
107 views

The automorphism group of the lattice $E_6$

Let $E_6$ be the root lattice, and $G$ be its automorphism group as a lattice (i.e. as $\mathbb Z$-module together with the inner product). Let $W(E_6)$ be the Weyl group. Apparently $W(E_6)\subset G$....
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1answer
260 views

About Weyl group [closed]

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. Let $W$ be its Weyl group. I would like to know whether $W$ is always finite? If so, why?
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1answer
214 views

Finding the Weyl group from a Cartan matrix

I am looking to establish the relationship between Weyl groups and semisimple Lie algebras. So far I have found the root space decomposition (I am using $\mathfrak{sl}(3, \mathbb{C}) $as my example). ...
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150 views

Vectorization identity: Weyl matrices

Take a ring $Z_n = \{0,1,...,n-1\}$. All operations occur modulo $n$. Consider the $n^{th}$ roots of unity $\omega_n = \exp\left(\frac{2\pi i}{n}\right)$. Now define $$X = \sum_a \vert{a+1}\rangle\...
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1answer
787 views

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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1answer
502 views

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 ...
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266 views

About parabolic subgroup of a Weyl group

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 ...
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2answers
341 views

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 $\...