Questions tagged [root-systems]

For questions involving abstract root systems, their associated Weyl groups and Dynkin diagrams, as well as their applications to Lie theory, graph theory, or other related fields.

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Sum of cubes of positive roots, $\alpha(H)^3$, of arbitrary Cartan element

Let $\mathfrak{h}$ the Cartan subalgebra of a simply-laced (ADE type) Lie algebra of rank $r$, $\mathfrak{g}$, and $\Phi$ the associated root system. Furthermore, let $\Phi^+$, $\Pi$, denote the ...
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41 views

How do I prove a set is a root system?

I will start by setting up the problem then I will ask my question regarding it. Problem: Let $E=\{x=\sum_{i=1}^{n+1} x_{i}\epsilon_{i} \in \mathbb{R}^{n+1}|\sum_{i=1}^{n+1}x_{i}=0 \}$, where $\{\...
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Constructing a surjective intertwining map onto Verma module

I want to prove the following result: Theorem: Let $\mathfrak{g}$ be a semi-simple Lie algebra, $\mathfrak{h}$ be a Cartan subalgebra, $\mu \in \mathfrak{h}$, and $W_{\mu} = \mathfrak{U}_{\mathfrak{g}...
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43 views

Exponentiating Lie algebra element does not result in algebraic group element

This question is about a particular case of the special orthogonal groups considered in section 23.4 of Borel's Linear Algebraic Groups. Let $k$ be a field of characteristic $\neq 2$. Let $F$ be a ...
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38 views

Decomposition of finite-dimensional representation of semisimple Lie-Algebra into irreducible subrepresentations.

This is my first post on MathSE, since I could not find a helpful answer to my question on here yet. Let $(L,[\cdot,\cdot])$ be a finite-dimensional semisimple complex Lie-Algebra, $H\subseteq L$ a ...
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14 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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16 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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Automorphism groups of which lattices act irreducibly on the ambient Euclidean space

Let $V$ be a finite-dimensional real inner product space and let $L \subset V$ be a lattice of full rank. Consider the finite group $$ \mathrm{Aut}(L) = \{ f \in \mathrm{O}(V) \mid f(L) = L \}, $$ ...
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36 views

Recovering roots from Cartan integers

I am currently reading the chapter 3- Root Systems of John Humphreys book on Lie Algebra. I am trying to recover my set of roots from knowledge of Cartan integers. (I am considering the base $\...
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33 views

Question regarding root systems of a complex Lie algebra

Let $\mathfrak g$ be a semi-simple Lie algebra with Cartan subalgebra $\mathfrak h$ and root system $\Phi$. I have seen it stated that for any $\alpha \in \Phi$, $-\alpha \in \Phi$, and that this is a ...
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23 views

Root system of group of symplectic similitudes $GSp_{2n}$

Let $G = \operatorname{GSp}_{2n}$ be the group of symplectic similitudes. I am trying to work out certain cocharacters and have a question on the root system of this group: A maximal torus $T$ in $G$ ...
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33 views

Centre of the algebra $\mathbb{Z} [\hat{W}]$ of the affine Weyl group

For $W$ the Weyl group of some root datum $(P, R, P^\vee, R^\vee)$ (here $P$ is the weight lattice, $R$ the root lattice, and I will write the group structure additively) associated to an algebraic ...
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55 views

The proof of simple roots generating the root systems in T.A. Springer's Linear Algebraic Groups

In Springer's book Linear Algebraic Groups, the author presented Theorem 8.2.8. about simple roots, root system and Weyl group, as below: Here $R$ is the root system, $D$ is the set of simple roots, $...
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real infinitesimal weights in Representations Of Compact Lie Groups.

I'm currently reading Representations Of Compact Lie Groups by T. Bröcker and T. Tom Dieck. In the section to Representations and Lie Algebras (p.112) they introduce the notion of (infinitesimal) ...
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Root system : Constructing the Dynkin diagram

this is the image of the root system of $A_2$ i found online can someone explain to me how we construct the Dynkin diagram from this and why it's different from $D_2$ Also, why the root systems are ...
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Finding the closest point in a root lattice

Let $L_n$ be a crystallographic root lattice, embedded inside $\mathbb{R}^n$. This means that $L_n$ is the $\mathbb{Z}$-span of the simple roots $\alpha_1, \ldots, \alpha_n \in \mathbb{R}^n$, which ...
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Schur-Positivity proof using Kashiwara crystals.

It is well known that the polynomial $$(x_1+\cdots+x_n)^d$$ is Schur positive with coefficients $c_\lambda$ in the Schur expansion equal to the dimension of the irreducible representation $\lambda$ of ...
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25 views

Weyl group $W$ is a normal subgroup of $Aut(R)$

Let $R$ be a root system and $W$ be the Weyl group of $R$. Then we have that the group $W$ is a normal subgroup of the group $Aut(R)$ of automorphisms of $V$ ($V$ a finite dimensional vector space) ...
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29 views

Two roots are proportional in this case

In the book of Serre's "Complex Semisimple Lie Algebra", we found the following on the page 29: Can we rescale the image on math.stackexchange? I don't understand there the last part. How ...
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32 views

Dimension of maximal toral subalgebra and number of its roots

Let $L$ be a semisimple Lie algebra, $H$ be a maximal toral subalgebra of $L$ and $\Phi$ be the set of roots relative to $H$. Then $$L=H\bigoplus(\bigoplus\limits_{\alpha\in\Phi}L_{\alpha}).$$ We also ...
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47 views

Help with the invariant quadratic forms and the Weyl Group

I am reading the 5the chapter of J-P. Serre's Complex Semisimple Lie Algebras book and I don't understand the following proof given: Why can we easy conclude that with the fact that $W$ is finite? ...
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36 views

Symmetry in the complex semisimple lie algebra - help to understand definition

I got stucked in the definition of "symmetry" in the chapter of Lie Algebras to understand later the root systems. Well in the script they used the following definition: Let $\alpha \in V\...
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25 views

Order of the fundamental Group of a root system and the determinant of the Cartan Matrix

Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbb{C}$. Let for some fixed cartan subalgebra $\mathfrak{h}$, $\Phi$ be the root system of $\mathfrak{g}$. Let $\Delta=\{\...
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159 views

Structure of the Weil restriction

I am trying to understand how Weil restriction affects the structure theory of a reductive group over the two fields involved. As a toy example, I looked at the following: Consider $SL_2$ as an ...
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29 views

Worked Examples of Root System Calculation

I am looking for some worked examples of questions like "Compute the root system of the special orthognal Lie algebra $\mathfrak{so}_{2n}$". I understand the theory but find the computations ...
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47 views

$\Phi$-extreme weights and the Weyl group orbit of the highest weight

$\newcommand{\g}{\mathfrak{g}}$ Let $P(\g)$ be the weight lattice of $\g$ a semisimple Lie algebra over $\mathbb{C}$, and $P_{++}(\g)$ the set of dominant integral weights. A subset $\Psi \subset P(\g)...
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Levi subgroups and subsystems of root systems

Let $G$ be a connected reductive algebraic group over a local field $F$ with fixed maximal torus $T$, and denote by $R = R(G,T)$ the set of roots of $T$ in $G$, namely, the set of all nontrivial ...
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41 views

Integrality property of the root system of a semisimple Lie algebra

For $\mathfrak{g}$ a semisimple Lie algebra, $\mathfrak{h}$ a choice of Cartan subalgebra, $\Phi$ the set of roots, there is the integrality property that if $\alpha$ is a root then $c \alpha \in \Phi$...
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76 views

Maximally compact Cartan subalgebra of $SO(p,q)$ and its adjoint representation

This is a somewhat technical question so please bear with me. However, apparently my understanding leads to a contradiction, so I must be missing something basic, and I would be grateful to gain ...
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24 views

How to prove that the root systems of Coxeter groups are discrete?

First I want to know the definition of "discrete" here. I guess it can be inferred from the bilinear form on the Coxeter datum. But it is still hard to organize the prove by finding the ...
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How does one find subalgebras using the Dynkin diagram of a Lie algebra

The following is taken from these lecture notes on page 8.16: [...] $\mathfrak{so}(p+q)$ has as subalgebras $\mathfrak{so}(p)$ and $\mathfrak{so}(q)$ as well as their direct sum $\mathfrak{so}(p)\...
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62 views

A problem on weights of an irreducible representation

Let $\mathfrak{g}$ a semisimple Lie algebra with a Cartan subalgebra $\mathfrak{h}$. Let $\phi$ a root system with base $\Delta$ And $V$ an irreducible representation of $\mathfrak{g}$. Let $\Gamma \...
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40 views

Positive Cartan integers in $A_2$

It is well known that if $\alpha$ and $\beta$ are a set of simple roots for the root system of type $A_2$ and if the angle between them is $\frac{\pi}{3}$, then the Cartan integers $\langle \alpha, \...
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Abelian ideals of Borel subalgebra of $\mathfrak{sl}(n)$

Let $g$ a simple Lie algebra with root space decomposition $g = \mathfrak{h} \oplus \bigoplus_{\alpha \in \phi} g_{\alpha}$, where $\mathfrak{h}$ is a Cartan subalgebra and $\phi$ is the root system. ...
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140 views

Cartan subalgebras of classical Lie algebras are diagonal matrices

I want to prove that Cartan subalgebras of $A_l, B_l, C_l, D_l$ consist of the respective diagonal matrices. As for $A_l$: let $\mathfrak{h}$ the set of diagonal matrices of $\mathfrak{sl}(l+1)$. We ...
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104 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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65 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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29 views

bad primes and torsion primes

Let $Φ$ be a root system of type $G_2$, with base ${α, β}$ where $α$ is short. Then $Ψ_1 :=$ ±{$α, 3α + 2β$} is a closed subsystem of type $A_1A_1$, and clearly $|\mathbb{Z}Φ/\mathbb{Z}Ψ_1| = 2$; ...
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48 views

Inclusion of Dynkin diagrams implies inclusion of respective root systems

I want to prove that every inclusion of Dynkin diagrams determines an inclusion of their corresponding root systems. My idea is to use the fact that if two root systems have the same Cartan matrix, ...
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42 views

Computational procedure to find the basis of a Lie algebra generated by a finite collection of operators

Is there a procedure for finding the basis of a Lie algebra generated by a (known) finite set $A_{1}, \ldots, A_{K}$ of skew-hermitian operators on an Hermitian space $\mathbb{C}^{K}$? The Lie ...
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51 views

Positive bases with respect to another fixed base

Let $\Delta$ be a base of a root system, i.e. a subset of a root system $ E =span \phi$ with $E$ Euclidean space, such that $\Delta$ is a basis of $E$ and every root can be written as a linear ...
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If $\lambda = \sum k_i \alpha_i$ and $P_\lambda \subseteq \cup_{\alpha \in \Phi} P_\alpha \implies \lambda = c\alpha$ for some $\alpha \in \Phi$.

This is an exercise 10.10 in Humphreys book on Lie algebras. Let $\Phi$ be a root system lying in the euclidean space $E$ and let $\Delta = \{\alpha_1,\cdots,\alpha_\ell\}$ be a basis for $\Phi$. ...
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161 views

How to understand the Galois *-action on a Dynkin diagram

Let $L/k$ be a (Galois) quadratic field extension, and let $\sigma \in \operatorname{Gal}(L/k)$ be the nontrivial automorphism. Let $h$ be a Hermitian form on $L^{4}$, and let $G = \operatorname{SU}_{...
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27 views

Showing that $R = \{ \pm(e_i-e_j) : 1 \leq i < j \leq l+1\}$ is a root system

Let $e_i$ be the standard unit basis vector in $\mathbb{R}^{l+1}$ I'm trying to show that the $R = \{ \pm(e_i-e_j) : 1 \leq i < j \leq l+1\}$ is a root system for $E = spanR = \{ \Sigma \alpha_i ...
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90 views

Cartan Matrix from commutation relations

Let a set of elements, $T^i_j$, with $i,j=1,\cdots,n$ satisfying the $\mathcal{su}(n)$ algebra $$ [T^i_j, T^k_l] = \delta^k_j T^i_l - \delta^i_l T^k_j\,,\qquad (T^i_j)^\dagger = T^j_i. $$ There are $n^...
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1answer
166 views

Which root systems admit a proper root subsystem with full span?

Let $\Phi$ be an irreducible root system of $\mathrm{rank}(\Phi) = n$. I am allowing the case where $\Phi$ is not reduced. Say that a subset $\Psi \subset \Phi$ is a root subsystem if it is a root ...
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51 views

Higher-dimensional reflections

I would like to know if it is possible to find a hierarchy on reflections in the following sense: Let $V$ be a finite-dimensional euclidean vector-space with standard inner product $\langle\, \_ \,,\, ...
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68 views

A root system of vectors in R^n admits a simple system; but does every finite set of vectors

I'm reading about root systems in the context of finite reflection groups. As I understand it, every root system (a set $\Phi$ of vectors in $R^n$ with some nice properties) admits a simple system, i....
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39 views

Show that if $B$ is a base for a root system, then the angle between any two distinct elements of $B$ is obtuse.

Show that if $B$ is a base for a root system, then the angle between any two distinct elements of $B$ is obtuse. A subset $B$ of a root system $R$ is a base for the root system $R$ if: $B$ is a ...
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79 views

Weights of a faithful representation and roots of $\mathfrak{g}$

I fail to understand an elementary statement about roots of a semisimple Lie algebra $\mathfrak{g}$ and weights of a faithful representation. There are two sources for the classification of simple ...

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