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.

263 questions
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Dominant Weyl chamber and co-adjoint orbit for Riemann symmetric pair

Let $K$ be a compact Lie group with Lie algebra $\mathfrak{k}_0$. Suppose that $\sigma$ is an involutive automorphism of $K$ which defines a symmetric pair $(\mathfrak{k}_0,\mathfrak{k}_0^\sigma)$. ...
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Computing all the roots from the Cartan matrix

I have the following Cartan matrix and I have to compute all the roots. I know there are 18. \begin{equation} \begin{bmatrix} 2 & -1 & 0\\ -1 & 2 & -2\\ 0 & -1 & 2 \end{bmatrix}...
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How to compute the $\alpha_i$ string through $\alpha_j$?

If I have a set of simple roots $\{\alpha_1,\ldots, \alpha_n\}$ for a root system $\Phi$. What would be the way to compute the $\alpha_i$ string through $\alpha_j$? Okay, I read this answer proposed ...
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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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How can one determine the dimension of a Lie algebra from the attached root system?

Suppose we know the root system/Cartan matrix of a semisimple Lie algebra. Is there a formula that determines the dimension of the Lie algebra? Thanks in advance.
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Unnecessary condition in the definition of an isomorphism of root systems

Let $R$ be a root system in a vector space $V$ (of characteristic $0$) with coroots $R^{\vee} \subseteq V^*$. Recall that for every $\alpha \in R$ the corresponding coroot $\alpha^{\vee} \in R^{\vee}$ ...
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Notions of Fundamental Groups for semisimple algebraic groups

Let $G$ be a connected, semisimple linear algebraic group over a field $k$. In Springer's Linear Algebraic group, the Fundamental Group of $G$ is defined by a certain quotient of groups $P/Q$, which ...
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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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What is known about such dominant integral weights of compact semisimple Lie groups?

I am interested in special dominant integral weights $\lambda \in \mathfrak{h}^*$, where $\mathfrak{h}$ is a Cartan subalgebra of the Lie algebra $\mathfrak{g}$ of a compact semisimple Lie group $G$. ...
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Understanding how the Lie algebra G2 arises in nature

I'm trying to understand these notes on G2. But I don't understand the very beginning. I have no background in physics at all. What is meant by a "configuration space?" I looked on wikipedia but ...
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Unique faithful $7$-dimensional representation of semisimple Lie Algebra with $G_2$ root system

I am asked to show that if $\mathfrak{g}$ is a semisimple Lie Algebra with root system of type $G_2$, then it has a unique, $7$-dimensional faithful representation. To start, let $\omega_1, \omega_2$...
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Subgraphs of Dynkin Diagrams

Am I right in thinking that if we have two semisimple Lie Algebras $\mathfrak{g}$ and $\mathfrak{h}$ with respective Dynkin Diagrams $A$ and $B$, we may find an injective homomorphism of Lie Algebras ...
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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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The intersection of a maximal toral subalgebra with a simple ideal of a Lie algebra is a maximal toral subalgebra of the simple ideal.

I'm reading Humphreys' Introduction to Lie Algebras and Representation Theory and I have a question about Corollary 14.1, which reads: Humphreys Corollary 14.1. Let $L$ be a semisimple Lie algebra,...
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The lattice generated by $\{w(\rho) - \rho\,\vert\,w\in W\}$

Consider an irreducible root system associated to a complex simple Lie algebra $\mathfrak{g}$. Let $\rho$ be the half sum of positive roots and let $W$ be the Weyl group. Then what is the lattice $L$ ...
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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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If $\Phi$ is irreducible then $\Delta$ is irreducible.

Let $\Phi$ a root system with basis $\Delta$. Show that if $\Phi$ is irreducible then $\Delta$ is irreducible. Comments: Suppose that $\Delta = \Delta_1 \cup \Delta_2$ is a partition of $\Delta$ into ...
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Why is $n_i \frac{\langle \alpha_i , \alpha_i \rangle}{\langle \alpha , \alpha \rangle}$ an integer?

I'm stuck in this question and I would like some help. Question: In a reduced root system let $\alpha= n_1 \alpha_1 + ...+ n_k \alpha_k$ be a root such that each $\alpha_i$ is a simple root. Show ...
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Macdonald's “Symmetric Functions and Hall Polynomials” Section 1.5 Example 9

I'm trying to follow Example 9 in Section 1.5 of Macdonald's book "Symmetric Functions and Hall Polynomials". I have trouble with understanding some points. Before stating my question, I will first ...
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Is the root cone is contained in the weight cone?

Let $G$ be a semisimple algebraic group over a field $k$. Let $A_0$ be a maximal split torus of $G$, and $\mathfrak a_0 = \operatorname{Hom}(X(A_0), \mathbb R)$ the real Lie algebra of $A_0$ with ...
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Definition of a root system generated by roots $a$ and $b$.

In a paper I am looking at, part of a Lemma starts as follows; Let $R$ be an irreducible root system. Suppose that $a, b \in R$ are two roots with $| b | \geq |a |$, choose $k \in \{ 1, 2, 3 \}$ to ...
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How to obtain uniqueness in correspondence between simple systems and positive systems?

In reading the appendix of Lectures on Chevalley Groups by Steinberg, I'm having trouble understanding the uniqueness aspect of Proposition 9 (in both parts). Here is the setup. Let $V$ be an inner ...
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What property of the root system means a Lie algebra has complex structure?

Given just the root system of a Lie algebra. How can we tell if the Lie algebra will or will not admit a complex representation? (e.g. a representation in terms of complex $N\times N$ matrices which ...
In order to understand Lie Algebras and the Weyl Group, I am learning about root systems. Looking for an intuitive explanation of some parts. From here: A subset $R$ of a vector space $V$ is called ...
Suppose that $\mathfrak{g}$ is an $n$-dimensional complex semisimple Lie algebra, with a root system $R$, and suppose that $S = \{\alpha_1,...,\alpha_m\}$ is a base for $R$. The Cartan matrix for $R$ ...