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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71 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. ...
6
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1answer
129 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 ...
2
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1answer
56 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 $H$. $G$ ...
1
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1answer
41 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 ...
2
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1answer
32 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 $W(G_1)\times W(G_2)$ ...
1
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1answer
54 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+\...
3
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0answers
63 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 ...
0
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1answer
36 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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28 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 $a_{ij} = 0 \iff a_{ji} = 0$. ...
1
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1answer
96 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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0answers
24 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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0answers
33 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 $T$ ...
2
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1answer
98 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,...
1
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1answer
120 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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14 views

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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0answers
15 views

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 ...
3
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1answer
135 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 ...
1
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1answer
40 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 $w\rho-\rho$ ...
3
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0answers
52 views

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 ...
1
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1answer
53 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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0answers
36 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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0answers
27 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 (...
0
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1answer
92 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 ...
3
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0answers
81 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 ...
1
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1answer
62 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 $W(E_6)\subset G$....
0
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1answer
91 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?
0
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1answer
126 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). ...
0
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0answers
104 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\...
4
votes
1answer
415 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 ...
1
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1answer
316 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 ...
1
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0answers
156 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 ...
2
votes
2answers
162 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 $\...
2
votes
1answer
181 views

Schur-Weyl Duality (Example for k = 3)

I am an undergraduate student interested in representation theory. I know that you can decompose the iterated tensor product $(C^n)^{\otimes k}$ into the direct sum of irreducible $S_k \times GL(V)$ ...
1
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2answers
583 views

What is the order of the Weyl group?

Let $W$ be the Weyl group of any of the classical Lie algebras $A_n,B_n,C_n,D_n$. What is $|W|$? A naïve calculation suggests that $$ \begin{aligned} A_n&\colon\ |W|=(n+1)!\\ B_n&\colon\ |W|=...
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0answers
55 views

Conjugating the longest element of a parabolic subgroup by longest element in larger parabolic?

This is a minor detail in a paper I'm reading. Let $W$ be a Weyl group with simple roots $\Pi$, and suppose $J\subseteq\Pi$. Let $w_I$ denote the longest element of the parabolic subgroup $W_I$ for $I\...
3
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1answer
130 views

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 $L=H\oplus \amalg_{\alpha\in\Phi} L_{\alpha}.$ ...
3
votes
2answers
190 views

Automorphism in Weyl group of root system and its extension to 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}....
2
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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 $w_{0}\in W$ ...
1
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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 ...
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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 "...
1
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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 $...
0
votes
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 $W$). ...
1
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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. ...
4
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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 ...
2
votes
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 ...
3
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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.
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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 ...
1
vote
1answer
264 views

Weyl Operator and Field Operator

Given the creation$a^*$ 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{...
2
votes
1answer
46 views

Exercise on saturated Tits system

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} \...
4
votes
2answers
76 views

Weyl group of a quotient by a central subtorus.

This is probably an easy question, hanging on some minor detail, but i cannot find a proof for it. I try working through T.A. Springer's book "Linear algebraic groups" and i got stuck at remark 7.1.4. ...