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 weights of a representation of $U(N)$

Let $R$ be a finite-dimensional irreducible representation of $U(N)$, with the set of weights $W_R$. Each element of $W_R$ is a vector of length $N$ with integer entries. Firstly, I would like to know ...
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How to calculate volume of root lattices $A_n$

Is there anywhere in the bibliography an explanation as to why the volume of the root lattice $A_n$ is $\sqrt{n+1}$? $$A_n = \biggl\{(x_0,x_1,\dots,x_n) \in \mathbb{Z}^{n+1} : \sum_{i=0}^{n} x_i = 0\...
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Generators and raising/lower operators from simple roots

I have three simple roots for a real Lie algebra: $$\alpha_1 = (1,0,0), \quad \alpha_2 = (-\frac{1}{2},-\frac{1}{\sqrt{2}},-\frac{1}{2}), \quad \alpha_3=(0,0,1).$$ The question says “Assume that $E_{i\...
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How do roots of a root system correspond to symmetric points of the associated Weyl group

So I've read that the Weyl group of the $A_3$ root system is the full symmetry group of a tetrahedron $T_h$, $C_3$ is $O_h$, and $H_3$ is $I_h$. I've read some complicated proofs of this but I don't ...
2 votes
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66 views

Can this linear algebra / geometry lemma in Humphreys be proven by induction?

Many questions have been asked on this site about the proof of Lemma 9.1 on page 42 of Humphreys's Lie algebra book. The full proof is posted here. I understand the proof, but I don't like it and ...
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5 votes
1 answer
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Connection between exponents of a root system and solutions to linear systems over finite fields

Let $h_1, \ldots, h_r$ be linear forms in variables $x_1, \ldots, x_n$ with integer coefficients. Let $\mathbb F_q$ denote the finite field with $q = p^e$ elements. I am asked to prove that except in ...
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Every root is both preprojective and preinjective in the root systems of Dynkin diagrams

I have a definition for preprojective and preinjective roots in a root system as: If $\mathsf{R}$ is a root system and $C$ is Coxeter element adapted to an orientation of a quiver $Q$ without oriented ...
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Computation of roots and Borel subgroups

I have a question about the computation of positive roots/coroots and Borel subgroups of the indefinite group $G = \operatorname{GSO}(n, 2)$ for some $n \in \mathbb{Z}^{+}$, where the form defining ...
3 votes
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$w(\Delta) = \Delta \implies w = \mathrm{id}$ in Weyl group

Consider $\Phi$ a root system (definition of Erdmann) with basis $\Delta \subseteq \Phi$ (subset which is a basis of $E$ and each element of $\Phi$ is a linear combination with only non negative ...
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Why isn't the Weyl group of a root system defined as the isometry group of that system?

I know that the Weyl group of a root system is a subgroup of its isometry group, but (as in the case of $A_2$) it isn't always the whole isometry group. Why isn't the Weyl group defined as the ...
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Standard reference for a fundamental theorem on classification of root lattices

In Schuett-Shioda's Mordell-Weil Lattices, the authors refer to a fundamental theorem on root lattices: Theorem 2.25 Any positive-definite even integral root lattice is isometric to an orthogonal sum ...
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2 votes
1 answer
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Inner product of highest root.

In Macdonalds book on affine Hecke algebras and orthogonal polynomials, it is stated, without proof, that $\langle \varphi^\vee,\alpha\rangle \in \{0,1\}$ for all positive roots $\alpha$ not equal to $...
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Is there a simple proof of every Borel subalgebra containing a Cartan subalgebra is standard?

If a Borel subalgebra $\mathfrak b$ of a semisimple Lie algebra $\mathfrak g$ contains a Cartan subalgebra $\mathfrak h$, then we have a root space decomposition $$ \mathfrak b = \mathfrak h \oplus \...
1 vote
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Exercise 13.5 from Humphreys

I am trying to solve the following exercise from Humphreys: Let $\Phi$ be a root system, let $\Lambda_r$ and $\Lambda$ be its root lattice and weight lattice respectively. Prove that any subgroup $\...
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Two different notions of maximal roots of a root system

There is a partial order on a root system given by $\alpha \prec \beta$ if and only if $\beta-\alpha \in R^{+}$, and Humphreys (10.4. Lemma A) proves that there is a unique maximal root with respect ...
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Is normalizer of a torus subgroup the Weyl group of a root system?

Let $G$ be a connected compact Lie group and $S$ a torus subgroup contained in the maximal torus $T$. Denote by $R_+$ the set of positive roots on $\mathfrak{t}$. Let $N_G(S)$ and $Z_G(S)$ be the ...
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how to find reduced words for each element of reflection (Coxeter) group in GAP

I have a finite reflection (or Coxeter) group defined abstractly through the standard presentation $$(s_i s_j)^{c_{ij}}=1$$ For each of its elements I want to find the number of reduced words equal to ...
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Question about definition of Verma modules

I am studying Verma modules (reading Dixmier´s Enveloping Algebras) and have a question regarding the definition as a quotieng of the enveloping algebra. Let $g$ be a Lie algebra and $h$ its Cartan ...
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Cartan matrices are independent of choice of simple systems

Suppose we have a Euclidean space $V$ with a bilinear product $(-,-)$ and suppose that $\Phi$ is an abstract crystallographic root system in $V$ with a simple system $\Delta=\{\alpha_i, \dots, \...
2 votes
2 answers
107 views

Only Weyl group of rank two root system can be dihedral

Let $\Phi$ be a (reduced, crystallographic) root system, and $W$ its Weyl group. Is it possible to prove that if we know $W$ is dihedral, then the rank of $\Phi$ is two, Without using the ...
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Root decomposition of lie algebra from the decomposition of the complexification

Currently I'm reading Representations of Compact Lie Groups (Brocker) and in V-2.1 Brocker writes a decomposition of $\mathfrak{g}$ and $\mathfrak{g_\mathbb{C}}$ by the weight spaces. The book marks: $...
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Reference for the bijection of parabolic subgroups $P$ containing $B$ and subsets of the set of simple roots $\Delta(B)$

Let $G$ be a split reductive group over a perfect field $k$ (not necessarily algebraically closed) with split maximal torus $T$ and Borel $B \supset T$. Then there is(/should be) an inclusion-...
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Intersection of standard parabolic subgroups

Let $G$ be a connected split reductive group over a field k of characteristic zero with split maximal torus $T$ and Borel $B \supset T$. Additionally let $\Phi$ be the root system corresponding to $T$ ...
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Finding a dominant root $\alpha$ for a semisimple, irreducible Lie-algebra $\mathfrak{g}$.

I´m working myself right now through this article and have trouble understanding a part of the proof of Proposition 6.4, case II) on page 345-346. I try to give a short outline of the situation: ...
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1 answer
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Possible lenghts in an irreducible root system

If a root system $R$ is irreducible (not a product of two root systems) then $R$ does not contain three vectors of pairwise different lengths. To show this do we need just to compute all the angles ...
2 votes
1 answer
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Claims on Root systems

For a root system R prove or disprove: a. Assume that the angle θ between the roots α and β is obtuse (θ > π/2) Then α+β ∈R. b. The angle θ between α and β is π/2 . Then α+β is not a root. c. If ...
2 votes
1 answer
198 views

Which Dynkin diagram is being spoken about here? Why is there a double line?

I'm confused about the following comment in Knapp's Lie Groups 2ed, page 397. Here, $\Delta$ is a root system associated to a complex semisimple Lie algebra, $\alpha, \beta$ are orthogonal roots and ...
1 vote
1 answer
119 views

What is a simple component of a root system?

The above is from Knapp's Lie Groups; Beyond an introduction', 2ed, page 397. Question 1: What is a simple component of a reduced root system? Or, more specifically, given a root system $\Delta$ and ...
1 vote
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On axiomatic definition of affine root systems

In Macdonald's book Affine Hecke Algebras and Orthogonal Polynomials, chapter 1 introduces affine root systems. I will recall the definition here: Let $E$ be a non-zero real Euclidean space (finite ...
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Set of roots such that sum of them with given positive root is not a root

Let $\Phi$ be a indecomposable root system of rank greater than 1 and fix a positive root $\alpha$. How can one describe the set $ B = \{ \beta \in \Phi \;| \; \alpha + \beta \notin \Phi \}$? It's ...
1 vote
1 answer
148 views

The effect of a Cayley transform on a Cartan subalgebra

I'm trying to understand equation 6.65b from Knapp's 'Lie groups,' 2ed. Setup: Let $\mathfrak{g}_0$ be a real semisimple Lie algebra with an involution $\theta$. Let $B$ be a bilinear, symmetric, non-...
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1 answer
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Root system for Lie algebras: If $\alpha, \beta, \alpha - \beta \in \Phi$ are all roots, is $(\alpha, \beta) > 0$?

Exercise 8.11 of Humphreys' Introduction to Lie Algebras and Representation Theory asks to prove that if $\alpha, \beta \in \Phi$ and $(\alpha,\beta) > 0$, then $\alpha - \beta \in \Phi$ (...
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1 vote
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Angles of the Fundamental Alcove (Chamber?)

I am trying to calculate the angles of the fundamental alcove (chamber?) for the root systems of type $B_2$, $C_2$, and $G_2$; the fundamental alcove (chamber?) forms a triangle in these cases, so I ...
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1 vote
1 answer
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Reflection Group of Type $D_n$

Here is the description of the reflection group of type $D_n$ in Humphreys' book Reflection Groups and Coxeter Groups: ($D_n$, $n \ge 4)$ Let $V = \Bbb{R}^n$, and define $\Phi$ to be the set of ...
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1 vote
1 answer
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Reflection Group of Type $C_n$

In Humphreys' book Reflection Groups and Coexeter groups, he defines a group of type $B_n$ (for $n \ge 2$) in the following way: Let $V = \Bbb{R}^n$, and define $\Phi$ to be the set of all vectors of ...
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6 votes
2 answers
240 views

Explicit matrices $h_\alpha$ that correspond to the long roots $\alpha$ in a classical compact simple Lie algebra over the reals

While it is easy to find many sources that give expressions for the (co)roots of an abstract root system, it is less easy to find a reference that gives explicit matrices that are the "coroots&...
1 vote
1 answer
258 views

How are weights and roots connected? (In the context of the semisimple Lie algebras)

I am studying representations of complex semisimple Lie algebras, their root system, weight spaces etc. I am a very beginner so this question is about relating key definitions to one another. Question ...
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Reduced adapted expression for the longest element given an orientation for a Dynkin graph

In Kirillov's book "Quiver representation and quiver varieties" in page 45 there's a Theorem 3.33 that he say is due to Lusztig that says: Given an orientation $\Omega$ of a Dynkin graph $Q$,...
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1 answer
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About imaginary root of a symmetrizable Kac Moody algebra

The question is from Infinite dimensional Lie algebras by Victor G Kac (ex 5.17, Page 74) Let $\mathfrak{g}=\mathfrak{g}(A)$ be a symmetrizable Kac Moody algebra and let $\Delta$ be its set of roots. ...
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If $\Delta$ is decomposable, why does $\mathfrak{g}$ have a corresponding decomposition into direct sum of ideals.

Assume that $\mathfrak{g}$ is a finite-dimensional semisimple Lie algebra over an algebraically closed field $F$ of characteristic $0, \mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$ and $\...
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1 vote
1 answer
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Simple Reflections on Simple Roots

I have two related questions concerning simple reflections and simple roots. Let $\Phi$ be a root system for a reflection group $W$, let $\Pi \subset \Phi$ be a positive system, and let $\Delta$ be a ...
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1 vote
2 answers
134 views

Property of order 2 automorphism of simple Lie algebra

If $\sigma$ is an automorphism of simple Lie algebra, of order 2, such that we fixed : $\sigma (x_\beta)=-y_\beta$ for each simple root of a given base $\Delta$, is it also necessarily true that $\...
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Difference of two simple roots is zero (?)

I am studying simple Lie algebras and, on page 129 of the book "Group Theory A physicist's Survey", by Pierre Ramond, it is claimed at the bottom of the page that << Hence the ...
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1 answer
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Determinie the multiplicity of a root in $\mathfrak{g}(A)$ where $2\times 2$-matrix $A$ given (or arbitrary)

I'm trying to solve the exercises on Infinite-dimensional Lie algebras by Victor G.Kac. Exercise 1.6.: Let $A = \begin{pmatrix}2&-3\\-3&2\end{pmatrix}$. Show that $\operatorname{mult} (2\...
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Counting and finding root subsystems

Let $\Phi$ be an irreducible root system. A root subsystem of $\Phi$ is a subset $\Psi \subseteq \Phi$ which is a root system. One can find the possible types of root subsystems of $\Phi$ by deleting ...
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1 vote
1 answer
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Type of a root (sub)-system

Let $E:=\{x\in \mathbb{R}^{l+1}:x_1+x_2+\cdots + x_{l+1}=0\}$ and let $\Phi\subseteq E$ denote its root system of type $A_l$ given the basis $\Delta=\{e_i-e_{i+1}, 1\leq i \leq l\}$ and with $\{e_i\}$ ...
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1 answer
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When is the Lie bracket of two root spaces nonzero?

Let $L$ be a semisimple Lie algebra with root space decomposition $L = L_0 \oplus \bigoplus_{\alpha \in \Phi} L_\alpha$. For roots $\alpha, \beta \in \Phi$, we always have $[L_\alpha, L_\beta] \...
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Determining the Weyl group from a given root system

1. Definitions For $V$ a vector space over $\mathbb C$ we call a subset $R \subset V$ an abstract root system if: (1) The set $R$ is finite, spans $V$ and $0 \notin R$. (2) For every $\alpha$ in $R$ ...
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1 answer
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A simple question on root systems

I read the Wikipedia page about root systems. It is very easy to verify that the 2 following images are showing some root systems. https://en.wikipedia.org/wiki/Root_system These 2 pictures are from ...
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System of representatives in reflection groups and subgroups

I'm working on a paper from Steinberg (1974), "On a theorem of Pittie". The paper is mainly about roots and reflection groups. I'm having trouble understanding the proof of lemma 2.5(a) ...
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