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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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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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1answer
24 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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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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If P(x) and Q(x) are two polynomials such that P(x) | P(Q(x)), what could be the conditions for Q such that P (x) = 0 => Q(x) = x?

If P(x) and Q(x) are two polynomials such that P(x) | P(Q(x)), what are the restrictions for Q such that the statement P (x) = 0 => Q(x) = x to be true (I was thinking the restriction has to be Q to ...
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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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Multiplicities in restricted root systems of split real rank one groups

In Knapps book, Lie groups beyond an introduction, p.372-373, the restricted roots of $SU(n,1)$ and $SO_e(n,1)$ and their multiplicities are computed. Does anyone know a source where this is done for $...
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221 views

Where does the half sum of positive roots live?

Definitions: Let $G$ be a compact, connected, semi-simple (defined below) Lie group with maximal torus $T$, Weyl group $W:=N(T)/T$ and lie algebra $\mathfrak{g}$. Let $\Lambda$ be the dual of $T$, and ...
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Real roots of U(2)

On page 350 of Hall's book real roots of $U(2)$ are listed as $(1, 1)$ and $(-1, -1)$ after identifying the maximal torus algebra $\frak{t}$ of diagonal matrices with $\mathbb{R}^2$. However, my ...
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1answer
51 views

Determinant of adjoint representation

Let $G$ be a semisimple Lie group with Iwasawa decomposition $G=KAN$ and consider the determinant of the adjoint representation $\operatorname{Ad}$ of $AN$. I want to determine what the derived ...
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Decomposing $k$th exterior powers $\Lambda^kV(\omega_1)$

Let $\Phi$ be a $G_2$ root system, $\omega_1$ the fundamental weight corresponding to the shorter root and consider the unique irreducible highest weight module $V = V(\omega_1)$ of highest weight $\...
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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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1answer
54 views

Highest Weights of Defining and Adjoint Representations of $\mathfrak{so}_5$

I am asked to describe the defining representation of $\mathfrak{sp}_4$ in terms of highest weights, and then I am asked to repeat this process for the defining and adjoint representations of $\...
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1answer
20 views

Calculating the Formal character on the irreducible $(n+1)$ dimensional representation of $\mathfrak{sl}_2$

Let $V(n)$ be the unique, irreducible representation of $\mathfrak{sl}_2$ of $(n+1)$-dimensions. Let $\rho$ be the sum of all fundamental weights. I want to calculate the formal character $ch(V(n)) ...
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1answer
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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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1answer
74 views

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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52 views

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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33 views

The definition of a coroot for non-split reductive groups?

Let $G$ be a connected, reductive group over a field $k$, and $A_0$ a maximal split torus of $G$. Let $\Phi = \Phi(G,A_0)$ be the set of roots of $A_0$ in $G$. Then the $\mathbb R$-linear span $\...
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1answer
34 views

If the off-diagonal entries of a positive-definite symmetric matrix $A$ are $\leq 0$, then $A^{-1}$ has positive entries

Let $A \in \operatorname{GL}_n(\mathbb R)$ be a symmetric matrix which is positive definite, i.e. $A = Q^tQ$ for some invertible matrix $Q$. Suppose that the off diagonal entries of $A$ are $\leq 0$. ...
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1answer
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Simple coroots in a non-reduced root system do not form a base?

Let $(V,R)$ be a not necessarily reduced root system, and $(V^{\vee},R^{\vee})$ the dual root system. A chamber of $R$ is a connected component of the complement in $V$ of the hyperplanes $H_{\alpha} ...
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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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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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Questions about parabolic algebra

Let $\mathfrak{g}$ be a complex semisimple Lie algebra and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. We denote by $\Phi$ the system of roots of $\mathfrak{g}$ with respect to $\...
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1answer
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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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Iwasawa integration formula

Let $G=KAN$ be the Iwasawa decomposition of $G$ and $k(g):=k, a(g):=a$ be the corresponding projections onto $K$ resp. $A,g=kan$. Then I want to proof for any continuous $f:K\to\Bbb C$ that $$\int_K f(...
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1answer
99 views

Recover the root system from a root lattice

I am new to Lie algebra and have a maybe naive question about root system. If we have a root system $\Phi$ we can associate it with a lattice $\Lambda(\Phi)$. I want to know how to recover the $\Phi$...
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Reflections along root vectors in a basis generate the Weyl group?

Let $\Phi$ be a root system of type ADE, $\Lambda$ be the lattice in the Euclidean space spanned by $\Phi$. If $\Lambda$ is spanned by $\{v_i\}\subset \Phi$ (as a $\mathbb Z$-module), and let $\...
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How to understand this picture of E8?

This is the first picture you see when you go to the wikipedia page on Lie groups. It is supposedly a diagram of the root system E8. I understand that the 240 vertices of the diagram are supposed to ...
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1answer
39 views

Simple reflections have determinant $-1$

I have read that, given any root system $\Phi$ of a real Euclidean vector space $V$, every (simple) reflection $s_{\alpha} \colon V \to V$ has $$\det s_{\alpha} = -1.$$ I understand why $\det s_{\...
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Pairing half the sum of the roots with a simple coroot

I was calculating something with the root system $A_n$ and I think there might be a more general principle at work. Here is the example: let $G = \operatorname{GL}_5$, with maximal torus $T$ and ...
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1answer
115 views

What is weight lattice modulo coroot lattice?

In Lectures on Tensor Categories and Modular Functors by Bakalov and Kirillov, the $S$ matrix (expression 3.3.7) is expressed in the form $\vert P/kQ^\vee \vert^{-1/2}\times(\cdots)$, where $k\in \...
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1answer
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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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1answer
44 views

Simple root systems are contained in a half-space

Let $V$ be a Euclidean space, that is, a finite dimensional real linear space with a symmetric positive definite inner product $\langle \cdot, \cdot\rangle$. Definition: An (abstract) root system in $...
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Arithmetic Progressions in Abstract Root Systems

I'm interested in proving part (2) of Theorem 10 here: Let $\Delta$ be an abstract root system in [a finite dimensional Euclidean space] $V$. (1) If $\alpha$ and $\beta$ are in $\Delta$, and $...
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1answer
148 views

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 ...
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1answer
57 views

Introduction to Root Systems [closed]

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 ...
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Equivalence of Cartan matrices

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 $R$ ...
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How to Calculate the Size of the Exceptional Irreducible Root Systems

There are three exceptional irreducible root systems $E_6$, $E_7$, and $E_8$ which correspond to the Dynkin diagrams $$ E_6\; \begin{aligned} &\>\bullet \\[-1ex] &\,\,\mid \\[-1ex] \...
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27 views

A subset of root system disjoint from its negative

Let $\Phi$ be a root system in a finite dimensional Euclidean space. Let $S\subset \Phi$ a non-empty subset such that if $\alpha$ is in $S$ then $-\alpha$ is not in $S$. Is it true that there is ...
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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
146 views

Why is the opposite Borel subgroup used?

When reading some papers on flag varieties, I sometimes find some remarks mentioning opposite Borel subgroup. It seems that people do so when they consider algebraic group. To my understanding, it is ...
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1answer
60 views

root system of semi-simple Lie algebra and passing into euclidean space

Let $L$ be a semisimple Lie algebra over $\mathbb{C}$; $H$ maximal abelian subalgebra. So $L$ has decomposition $$L=H\oplus (\oplus_{\alpha\in\Phi}L_{\alpha}).$$ The set $\Phi$ is root system of $L$ ...
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Automorphisms of root system and its action on quotient of weight lattice

Let $\Phi$ be a root system in $E=\mathbb{R}^l$, $\Delta=\{\alpha_1,\cdots,\alpha_l\}$ be a base of $\Phi$. Let $\langle\alpha,\beta\rangle:=\frac{2(\alpha,\beta)}{(\beta,\beta)}$ for $\alpha,\beta\...
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minimal dominant weight and cosets of root lattice

This is a question about abstract root system, root lattice and weight lattice. Most definitions are standard as in Wikipedia or Humphreys' Lie algebra, section 13. If $\Phi$ is an irreducible ...
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Root systems and distinguished bases

I'm working on Root Systems (https://en.wikipedia.org/wiki/Root_system). I learnt the basic definitions of a basis, of the Weyl group, and I read the proof of the existence of such bases for a given ...