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Questions tagged [weyl-group]

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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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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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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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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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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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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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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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Dominant weights in root system

Suppose a root system is given. Let $\gamma$ be an integral, dominant weight and $\rho$ be the half sum of positive roots. I have been told that one can choose an integral, dominant weight $\mu$ such ...
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The ratio of Raw Maxima of Mahonian numbers in terms of groups

There is a clear explanation that the polynomial $$(1+x)(1+x+x^2)...(1+x+x^2+...+x^n)$$ with Mahonian numbers gives the Poincare series for the symmetric group $S_{n+1}$ considered as a Coxeter group ...
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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 $\...
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Restriction of Bruhat order to stabilizer of a vector

Let $W$ be a Weyl group (maybe better a Coxeter group, i.e. a group with action on vector space $V$ generated by reflections with some conditions). Consider $v \in V$ be a vector. Let $\text{Stab}_v \...
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Deduction concerning Weyl chambers

In Humphreys' book on linear algebraic groups, he includes the following argument. Here $G$ is a connected algebraic group and $\mathfrak{B}$ denotes the collection of Borel subgroups of $G$. For ...
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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 $S_k \times GL(V)$ ...
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238 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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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\...
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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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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}....
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1answer
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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 $w_{0}\in W$ ...
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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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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 "...
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1answer
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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 $...
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$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 $W$). ...
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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. ...
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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 ...
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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 ...
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1answer
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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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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 ...
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1answer
138 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{...
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1answer
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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} \...
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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. ...
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50 views

How to justify that any set of coxeter generators for a Weyl group are simple

Let $G$ be a group with a $\mathrm{BN}$-pair, and let $W:=N/(B\cap N)$ be the Weyl group of $G$, where $W$ is generated by a set $S$ of simple roots (as in the definition of a $\mathrm{BN}$-pair) and $...
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Inverse Galois Problem: the two-fold central extension of $Sp_6(\mathbb F_2) \cong [W(E_7),W(E_7)]$

I came across the following problem when I was trying to construct a certain type of homomorphisms from $\Gamma_{\mathbb Q}$ to $E^{sc}_7(\mathbb F_p)$ for any prime $p$: Is the double cover of $Sp_6(...
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Conjugacy classes of Weyl group of type B

I'm really stuck in proving the fact that the conjugacy classes of B_n are deterimined by a positive and negative cycle stractures. Can you give me any hint. I think if I can use the fact that $g(i_1,...
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Reduced decomposition of an element in the Weyl group smaller length than it should have

I am making some mistake computing an example with root systems, but I have so far been unable to find it. Let $J$ be the $6$ by $6$ matrix with $1$s on the antidiagonal, and take $G = \textrm{SO}_6(\...