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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1answer
48 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
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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0answers
39 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 $G$.
...
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37 views
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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0answers
45 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
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1answer
29 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
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1answer
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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0answers
30 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 ...
1
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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 ...
3
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0answers
25 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
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1answer
29 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
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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 $\...
1
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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)) ...
2
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1answer
32 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 ...
2
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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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1answer
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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0answers
53 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
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0answers
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 $\...
1
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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 $\leq 0$. ...
0
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1answer
17 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
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0answers
29 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
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2answers
52 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 ...
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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
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0answers
12 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
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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
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0answers
101 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
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0answers
26 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
$$\int_K f(...
2
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1answer
99 views
Recover the root system from a root lattice
I am new to Lie algebra and have a maybe naive question about root system.
If we have a root system $\Phi$ we can associate it with a lattice $\Lambda(\Phi)$. I want to know how to recover the $\Phi$...
6
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1answer
196 views
Reflections along root vectors in a basis generate the Weyl group?
Let $\Phi$ be a root system of type ADE, $\Lambda$ be the lattice in the Euclidean space spanned by $\Phi$. If $\Lambda$ is spanned by $\{v_i\}\subset \Phi$ (as a $\mathbb Z$-module), and let $\...
4
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0answers
57 views
How to understand this picture of E8?
This is the first picture you see when you go to the wikipedia page on Lie groups. It is supposedly a diagram of the root system E8. I understand that the 240 vertices of the diagram are supposed to ...
1
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1answer
39 views
Simple reflections have determinant $-1$
I have read that, given any root system $\Phi$ of a real Euclidean vector space $V$, every (simple) reflection $s_{\alpha} \colon V \to V$ has $$\det s_{\alpha} = -1.$$
I understand why $\det s_{\...
0
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0answers
33 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
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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 \...
0
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1answer
40 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 ...
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0answers
34 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 ...
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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
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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 $...
2
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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
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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 ...
2
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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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0answers
28 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$ ...
3
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1answer
55 views
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]
\...
0
votes
1answer
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 ...
2
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0answers
71 views
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 $L=H\oplus \amalg_{\alpha\in\Phi} L_{\alpha}.$
...
3
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2answers
90 views
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
$$L=H\oplus \amalg_{\alpha\in\Phi} L_{\alpha}....
6
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1answer
146 views
Why is the opposite Borel subgroup used?
When reading some papers on flag varieties, I sometimes find some remarks mentioning opposite Borel subgroup. It seems that people do so when they consider algebraic group. To my understanding, it is ...
2
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1answer
60 views
root system of semi-simple Lie algebra and passing into euclidean space
Let $L$ be a semisimple Lie algebra over $\mathbb{C}$; $H$ maximal abelian subalgebra. So $L$ has decomposition
$$L=H\oplus (\oplus_{\alpha\in\Phi}L_{\alpha}).$$
The set $\Phi$ is root system of $L$ ...
5
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0answers
115 views
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\...
1
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0answers
75 views
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 ...
1
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0answers
28 views
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 ...
