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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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 ...
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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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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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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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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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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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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&...
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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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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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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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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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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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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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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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Intuition behind calculating weight systems using SU(2) strings of roots?

I am in a weird sitution in my Groups course where I can 'blindly follow the rules' in, for instance, finding a weight system from its highest weight by representing the weights in their Dynkin basis, ...
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How to derive the mentioned relations (in part a-b-c of example 9) from the Weyl denominator formula?

I want to understand Example 9 (a-b-c) of section 1.5 of the book "Symmetric functions and Hall polynomials (https://math.berkeley.edu/~corteel/MATH249/macdonald.pdf)" (pages 78-79): In ...
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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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Change of basis in root systems

It is for sure s really easy question, but I need to ask. Let $\Phi \subset \mathbb R^n$ be an abstract root system. We denote by $\{e_1,\ldots,e_n\}$ the canonical basis of $\mathbb R^n$ and by $\{\...
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Simple Lie algebras have irreducible root systems

Lemma 2.1. Let $g$ be a simple Lie algebra with Cartan subalgebra $h$. Then the root system $\Delta$ corresponding to $h$ is irreducible. Proof. Suppose $\Delta$ decomposes as $\Delta_1\cup\Delta_2$ ...
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Isomorphisms of irreducible root systems

Let $E,E'$ be two euclidean vector spaces and $\Phi,\Phi'$ two root systems of $E$ and $E'$, respectively. Let $\varphi:E\to E'$ be an isomorphism of root systems. Applying the definition of root ...
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Uniqueness of roots system of a semisimple Lie algebra

Take $L$ a $\mathbb{C}$-Lie algebra with finite dimension and semisimple. Now take $H \subset L$ a maximal toral subalgebra we can define the roots system associated to $(L,H)$ as the set of maps $H\...
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An Exercise on Abstract Weight Theory (Humphrey Exercise 13.8)

The exercise goes like this: Let $\Phi$ be an irreducible root system. Prove that each $\lambda_i$ is of the form $\sum_j q_{ij}\alpha_j$, where all $q_{ij}$ are positive rational numbers. Here, as ...
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What does it mean for a root to be an element of the Lie algebra?

My understanding of roots in Lie algebra theory is as follows: given a Cartan subalgebra $\mathfrak{h}\subset\mathfrak{g}$ with basis $\{H_i\}_{i=1}^{r}$, and root vectors $\{X_\alpha\}_\alpha$, the ...
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Condition for a sum of images of fundamental dominant weights to lie on a wall

Let $\Delta$ be a base of a root system with Weyl group $W$. For $\alpha\in\Delta$, let $\varpi_\alpha$ be the corresponding fundamental dominant weight. Let $w\neq r$ be elements of $W$. I would like ...
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Cartan integers are preserved by isomorphism

I have this proposition and the demonstration, but there is some parts I don't understand, $E$ is a euclidean space and $\Phi$ is a root system with base $\Delta$. Proposition: Let $\Phi' \subset E'$ ...
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I don´t understand root systems

I don´t understand root systems. The Wikipedia (and my university lectures) say it is some configuration of vectors with certan properties. The root vectors should span the whole space, which I ...
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A question about roots and root spaces of semisimple Lie algebras

In the book "Lie Groups: Beyond and Introduction" by Anthony Knapp, the author describes roots of a semisimple (A Lie algebra without any non-zero solvable ideal) Lie algebra as elements of ...
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A description of the $A_n$-root system in terms of $\mathbb{R}^{n}$

The root system $(V,\Delta)$ of type $A$ is usually defined as a subset of $\mathbb{R}^{n+1}$: $$ \{ e_i - e_j ;\, 1 \leq i,j \leq n+1,\, i \neq j\}. $$ However these vectors span an $n$-dimensional ...
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Dominance of $w\mu$ for dominant cocharacter $\mu$

NOTE: The question has now been posted on MathOverflow: Dominance of $w\mu$ for dominant cocharacter Let $G$ be a reductive group over a field $k$ of characteristic zero with maximal split torus $T$ ...
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Dominance order on cocharacter group $X_*(T)$

Let $G$ be a reductive group over a field $k$ of characteristic zero with maximal split torus $T$ of rank $n$ and a Borel $B \supset T$ defining a set of simple roots $\Delta$. By $X_*(T)$ we denote ...
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Understanding how to find roots of Lie algebra $\mathfrak{su}(N)$

I am having difficulties understanding why the roots of $\mathfrak{su}(N)$ algebra are $\alpha_{ij}=e_i-e_j$. This is (roughly) what my professor said: We define the ladder operators in $\mathfrak{su}...
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Help with a proof of a lemma from Humphreys about reflections [closed]

I want to understand this proof, I don't get why he says that the minimal polinomial of $\tau$ divides $(T-1)^l$. And at the end how can I explain why gcd$(T^k-1,(T-1)^l)=T-1$?
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Sum of tensor products of positive roots of $\mathfrak{su}(n)$

Let $\Delta^+$ denote a choice of positive roots for the Lie algebra $\mathfrak{su}(n)$. The Weyl vector, $\rho$, is half the sum of the positive roots \begin{equation} 2\rho = \sum_{\beta \in \...
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Sum of cubes of positive roots, $\alpha(H)^3$, of arbitrary Cartan element

Let $\mathfrak{h}$ the Cartan subalgebra of a simply-laced (ADE type) Lie algebra of rank $r$, $\mathfrak{g}$, and $\Phi$ the associated root system. Furthermore, let $\Phi^+$, $\Pi$, denote the ...
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How do I prove a set is a root system?

I will start by setting up the problem then I will ask my question regarding it. Problem: Let $E=\{x=\sum_{i=1}^{n+1} x_{i}\epsilon_{i} \in \mathbb{R}^{n+1}|\sum_{i=1}^{n+1}x_{i}=0 \}$, where $\{\...
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Constructing a surjective intertwining map onto Verma module

I want to prove the following result: Theorem: Let $\mathfrak{g}$ be a semi-simple Lie algebra, $\mathfrak{h}$ be a Cartan subalgebra, $\mu \in \mathfrak{h}$, and $W_{\mu} = \mathfrak{U}_{\mathfrak{g}...
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1 answer
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Exponentiating Lie algebra element does not result in algebraic group element

This question is about a particular case of the special orthogonal groups considered in section 23.4 of Borel's Linear Algebraic Groups. Let $k$ be a field of characteristic $\neq 2$. Let $F$ be a ...
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Decomposition of finite-dimensional representation of semisimple Lie-Algebra into irreducible subrepresentations.

This is my first post on MathSE, since I could not find a helpful answer to my question on here yet. Let $(L,[\cdot,\cdot])$ be a finite-dimensional semisimple complex Lie-Algebra, $H\subseteq L$ a ...
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Weyl groups sends Weyl Chamber onto another:

I came across two statements while studying Weyl group of root Systems. First one: The Weyl group say $W$ sends one Weyl Chamber onto another. If $\gamma$ is regular in a Euclidean Space $E$, we have $...
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Relation between product of reflections and angle

I am currently reading Chapter 3 Root Systems of John Humphreys book on Lie Algebras. It's known that Weyl Group $W$ is generated by set of reflections. If I consider an arbitrary element of Weyl ...
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Automorphism groups of which lattices act irreducibly on the ambient Euclidean space

Let $V$ be a finite-dimensional real inner product space and let $L \subset V$ be a lattice of full rank. Consider the finite group $$ \mathrm{Aut}(L) = \{ f \in \mathrm{O}(V) \mid f(L) = L \}, $$ ...
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Recovering roots from Cartan integers

I am currently reading the chapter 3- Root Systems of John Humphreys book on Lie Algebra. I am trying to recover my set of roots from knowledge of Cartan integers. (I am considering the base $\...
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Question regarding root systems of a complex Lie algebra

Let $\mathfrak g$ be a semi-simple Lie algebra with Cartan subalgebra $\mathfrak h$ and root system $\Phi$. I have seen it stated that for any $\alpha \in \Phi$, $-\alpha \in \Phi$, and that this is a ...
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