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.

3
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0answers
40 views

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 $(\mathfrak{k}_0,\mathfrak{k}_0^\sigma)$. ...
4
votes
1answer
38 views

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}...
1
vote
0answers
37 views

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 ...
0
votes
1answer
34 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 ...
0
votes
1answer
30 views

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.
2
votes
0answers
27 views

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 $\alpha^{\vee} \in R^{\vee}$ ...
0
votes
0answers
53 views

Carter's construction of the Chevalley group (Case $A_1$)

I am currently working through Carter's book "Simple Groups of Lie Type", trying to explicitly construct the generators of the Chevalley group of the root system $A_1$. In the context of $A_1$ let $\...
1
vote
0answers
28 views

Root system of Lie algebra: right angle => not simple? [duplicate]

Why is it that if the roots of a Lie algebra form angles of 90 degrees, the Lie algebra is not simple? Is it because the said roots commute with each other and so the Lie algebra can be broken down to ...
7
votes
0answers
109 views

Which root lattices have a theta series with this property?

Suppose $\Lambda$ is a root lattice (the integral lattice generated by a crystallographic root system). Consider its theta series $$\theta_{\Lambda}(q) = \sum_{a\in \Lambda} q^{(a,a)/2},$$ where $(\...
1
vote
1answer
65 views

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 ...
1
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1answer
32 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
votes
0answers
46 views

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 $G$. ...
1
vote
0answers
51 views

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 ...
3
votes
1answer
35 views

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 $...
2
votes
1answer
250 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 ...
1
vote
0answers
33 views

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 ...
2
votes
1answer
75 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 ...
3
votes
0answers
26 views

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 $\...
2
votes
1answer
39 views

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 $\omega_1, \omega_2$...
3
votes
1answer
77 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 $\...
1
vote
1answer
26 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)) ...
2
votes
1answer
37 views

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 ...
1
vote
1answer
111 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 ...
2
votes
1answer
88 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,...
0
votes
1answer
53 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 $L$ ...
1
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0answers
63 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 ...
0
votes
0answers
41 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 $\...
1
vote
1answer
42 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 $\leq 0$. ...
0
votes
1answer
23 views

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} ...
0
votes
0answers
39 views

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 ...
1
vote
2answers
58 views

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 ...
0
votes
0answers
22 views

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 \...
0
votes
0answers
16 views

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 $\...
4
votes
1answer
82 views

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 ...
4
votes
0answers
116 views

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 ...
1
vote
0answers
27 views

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(...
2
votes
1answer
112 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 $\Phi$...
6
votes
1answer
200 views

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 $\...
4
votes
0answers
61 views

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 ...
1
vote
1answer
47 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_{\...
0
votes
0answers
36 views

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 ...
3
votes
1answer
133 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 \...
0
votes
1answer
43 views

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 ...
1
vote
0answers
35 views

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 ...
1
vote
0answers
29 views

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 ...
2
votes
1answer
51 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 $...
2
votes
0answers
17 views

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 $...
3
votes
1answer
161 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 ...
2
votes
1answer
60 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 ...
1
vote
0answers
30 views

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 $R$ ...